Volume 23, Number 1, Spring/Summer 2023, Pages 1–189
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The Petra Šparl Award 2022
The Petra Šparl Award was established in 2017 to recognise in each even-numbered year
the best paper published in the previous five years by a young woman mathematician in
one of the two journals Ars Mathematica Contemporanea (AMC) and The Art of Discrete
and Applied Mathematics (ADAM). It was named after Dr Petra Šparl, a talented woman
mathematician who died mid-career in 2016 after a battle with cancer.
The first award was made in 2018 to Dr Monika Pilśniak (AGH University, Poland), for a
paper she published in AMC 13 (2017) on the distinguishing index of graphs, and then in
2020 two awards were made: one to Dr Simona Bonvicini (Università di Modena e Reggio
Emilia, Italy), for her contributions to a paper in AMC 14 (2018) on solutions of some
Hamilton-Waterloo problems, and one to Dr Klavdija Kutnar (University of Primorska,
Slovenia) for her contributions to a paper in AMC 10 (2016) on odd automorphisms in
vertex-transitive graphs.
Nominations for the 2022 award were invited in 2021, and all cases were considered by a
committee (consisting of the three of us, listed below). There were just five nominations,
and as in previous rounds we considered the nomination statements, comments by co-
authors, reports from referees, and the papers themselves, before making a decision.
This time one nomination stood out from the others, and led to our selection of the winner
of the Petra Šparl Award for 2022 as Dr Jelena Sedlar (of the University of Split, Croatia),
for her single-author paper ‘On Wiener inverse interval problem of trees’, published in Ars
Mathematica Contemporanea 15 (2018) 19–37.
In this paper the candidate resolved two open conjectures posed in the literature regarding
the Wiener index of trees, which are of considerable interest in mathematical chemistry. It
was already known that for connected graphs on n vertices, this index could take at least
n3
6 + O(n
2) consecutive integer values, and the conjectures concerned the values of the
Wiener index for trees on n vertices, essentially stating that the Wiener index could take
almost all values between the minimum and maximum. Dr Sedlar verified and improved
both of these conjectures for the case where n is even, and also proved a corrected version
of them for the case when n is odd, using a clever construction that recursively increases
the value of the Wiener index by 4.
As judges we agreed with the nominator and referee that the candidate’s work in this pa-
per was mathematically both very elegant and very intricate, reflecting her mathematical
prowess.
We heartily congratulate Dr Sedlar, who will be awarded a certificate and invited to give a
lecture in the Mathematics Colloquium at the University of Primorska, and to give lectures
at the University of Maribor and the University of Ljubljana.
Finally, we encourage nominations for the next Petra Šparl Award in 2024, as well as
submissions of high quality new papers that will be worthy of consideration for future
awards.
Marston Conder, Asia Ivić Weiss and Aleksander Malnič
Members of the 2022 Petra Šparl Award Committee
iii

Contents
The Sierpiński product of graphs
Jurij Kovič, Tomaž Pisanski, Sara Sabrina Zemljič, Arjana Žitnik . . . . . . 1
Total graph of a signed graph
Francesco Belardo, Zoran Stanić, Thomas Zaslavsky . . . . . . . . . . . . 27
Domination type parameters of Pell graphs
Arda Buğra Özer, Elif Saygı, Zülfükar Saygı . . . . . . . . . . . . . . . . . 45
Finitizable set of reductions for polyhedral quadrangulations of
closed surfaces
Yusuke Suzuki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
The fullerene graphs with a perfect star packing
Lingjuan Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
On the Aα-spectral radius of connected graphs
Abdollah Alhevaz, Maryam Baghipur, Hilal Ahmad Ganie,
Kinkar Chandra Das . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
A note on the neighbour-distinguishing index of digraphs
Éric Sopena, Mariusz Woźniak . . . . . . . . . . . . . . . . . . . . . . . . 121
Complexity of circulant graphs with non-fixed jumps,
its arithmetic properties and asymptotics
Alexander Mednykh, Ilya Mednykh . . . . . . . . . . . . . . . . . . . . . 129
On generalized Minkowski arrangements
Máté Kadlicskó, Zsolt Lángi . . . . . . . . . . . . . . . . . . . . . . . . . 145
Braid representatives minimizing the number of simple walks
Hans U. Boden, Matthew Shimoda . . . . . . . . . . . . . . . . . . . . . . 163
Volume 23, Number 1, Spring/Summer 2023, Pages 1–189
v

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.01 / 1–25
https://doi.org/10.26493/1855-3974.1970.29e
(Also available at http://amc-journal.eu)
The Sierpiński product of graphs*
Jurij Kovič
Institute of Mathematics, Physics, and Mechanics,
Jadranska 19, 1000 Ljubljana, Slovenia, and
University of Primorska, FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia
Tomaž Pisanski
University of Primorska, FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia, and
Institute of Mathematics, Physics, and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
Sara Sabrina Zemljič
Cambridge Quantum Computing Ltd,
32 St James’s Street, London, SW1A 1HD, United Kingdom
Arjana Žitnik †
University of Ljubljana, Faculty of Mathematics and Physics,
Jadranska 19, 1000 Ljubljana, Slovenia, and
Institute of Mathematics, Physics, and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
Dedicated to Professor Wilfried Imrich on
the occasion of his 80th birthday.
Received 9 October 2019, accepted 30 January 2022, published online 1 September 2022
Abstract
In this paper we introduce a product-like operation that generalizes the construction of
the generalized Sierpiński graphs. Let G,H be graphs and let f : V (G) → V (H) be a
function. Then the Sierpiński product of graphs G and H with respect to f , denoted by
G ⊗f H , is defined as the graph on the vertex set V (G) × V (H), consisting of |V (G)|
*The authors would like to thank Wilfried Imrich and Primož Šparl for fruitful discussion and Susan Deborah
Cook and Luke Morgan for their help in proofreading. They would also like to thank the referees for reading
the manuscript carefully and for their detailed comments which helped to improve the presentation of the paper
significantly. In particular they would like to thank one referee for pointing out that additional assumptions
of connectedness of the graph H in Lemma 2.5(ii) and 2-connectedness of the graph G in Theorem 2.13 and
its corollaries are needed. This work was supported in part by ‘Agencija za raziskovalno dejavnost Republike
Slovenije’ (Slovenian Research Agency) via Grants P1-0294, J1-9187, J1-7051, J1-7110, J1–1691 and N1–0032
and in part by H2020 Teaming InnoRenew CoE.
†Corresponding author.
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 23 (2023) #P1.01 / 1–25
copies of H; for every edge {g, g′} of G there is an edge between copies gH and g′H of
form {(g, f(g′), (g′, f(g))}.
Some basic properties of the Sierpiński product are presented. In particular, we show
that the graph G ⊗f H is connected if and only if both graphs G and H are connected
and we present some conditions that G,H must fulfill for G ⊗f H to be planar. As for
symmetry properties, we show which automorphisms of G and H extend to automorphisms
of G ⊗f H . In several cases we can also describe the whole automorphism group of the
graph G⊗f H .
Finally, we show how to extend the Sierpiński product to multiple factors in a natu-
ral way. By applying this operation n times to the same graph we obtain an alternative
approach to the well-known n-th generalized Sierpiński graph.
Keywords: Sierpiński graphs, graph products, connectivity, planarity, symmetry.
Math. Subj. Class. (2020): 05C76, 05C10, 05C40, 20B25
1 Introduction
The motivation for this paper is the study of Sierpiński graphs and their generalizations.
Although the body of this work is essentially self contained, a few remarks about the role
of Sierpiński graphs seem to be appropriate. The family of Sierpiński graphs Snp was
first introduced by Klavžar and Milutinović in [16] as a variant of the Tower of Hanoi
problem. The Sierpiński graphs can be defined recursively as follows: S1p is isomorphic
to the complete graph Kp and Sn+1p is constructed from p copies of S
n
p by adding exactly
one edge between every pair of copies of Sn+1p in a well-defined manner. The Sierpiński
graphs S13 , S
2
3 , and S
3
3 are depicted in Figure 1. In the “classical” case, when p = 3, the
Sierpiński graphs are isomorphic to the Hanoi graphs. More about Sierpiński graphs and
their connections to the Hanoi graphs can be found in the recent second edition of the book
about the Tower of Hanoi puzzle by Hinz et al. [10].
Sierpiński graphs have been extensively studied in most graph-theoretical aspects as
well as in other areas of mathematics and even psychology. Some notable papers are
[11, 13, 15, 17, 18, 22, 26, 27]. An extensive summary of topics studied on and around
Sierpiński graphs is available in the survey paper by Hinz, Klavžar and Zemljič [12]. In
that paper the authors introduced Sierpiński-type graphs as graphs that are derived from or
lead to the Sierpiński triangle fractal.
These families of graphs have been generalized by Gravier, Kovše and Parreau to a
family called generalized Sierpiński graphs [8]. Instead of the complete graph, an arbi-
trary graph G is used as a base graph to form a self-similar graph in the same way as the
Sierpiński graphs are derived from the complete graph: the generalized Sierpiński graph
S1G is isomorphic to the graph G. To construct the n-th iteration generalized Sierpiński
graph SnG for n > 1, take |V (G)| copies of S
n−1
G and add to the labels of vertices in copy x
of Sn−1G the letter x at the beginning. Then, for any edge {x, y} of G, add an edge between
the vertex xy . . . y and the vertex yx . . . x. See Figure 2 for an example of the second iter-
ation generalized Sierpiński graph, where the base graph is the house graph, i.e. the cycle
on five vertices with a chord.
E-mail addresses: jurij.kovic@siol.net (Jurij Kovič), tomaz.pisanski@upr.si (Tomaž Pisanski),
sara.zemljic@gmail.com (Sara Sabrina Zemljič), arjana.zitnik@fmf.uni-lj.si (Arjana Žitnik)
J. Kovič et al.: The Sierpiński product of graphs 3
Figure 1: The Sierpiński graphs S13 , S
2
3 , and S
3
3 .
4
5
1 2
3
44
45
41 42
43
54
55
51 52
53
14
15
11 12
13
24
25
21 22
23
34
35
31 32
33
Figure 2: The generalized Sierpiński graphs S1G and S
2
G when G is the house graph.
4 Ars Math. Contemp. 23 (2023) #P1.01 / 1–25
The generalized Sierpiński graphs have been extensively studied in the past few years
on topics including chromatic number, Randić index, vertex cover number, clique number,
domination number and metric properties; see, for example, [5, 7, 24, 25].
At this point we would like to mention another approach towards the Sierpiński graphs.
The graphs Sn3 appear naturally as subgraphs of repeated truncations of cubic graphs. More
generally, by applying truncation to a p-valent vertex of a graph n times, i.e. by replacing
each p-valent vertex of the graph by the complete graph Kp repeatedly, the corresponding
part of the obtained graph looks like Snp ; see Pisanski and Tucker [23] and Alspach and
Dobson [1]. The truncation operation on graphs has been generalized in several ways; see
Boben, Jajcay and Pisanski [2] and Exoo and Jajcay [6]. In [4], Eiben, Jajcay and Šparl
study automorphisms of generalized truncations. These constructions show significant sim-
ilarities to our construction, although they are distinct in general. A related construction,
called the clone cover, is considered by Malnič, Pisanski and Žitnik in [20].
In this paper we generalize the generalized Sierpiński graphs even further. Notice that
SnG can be viewed as an operation of S
n−1
G and G. This was a motivation for our introduc-
tion of the Sierpiński product of graphs. If we take any two graphs G and H , the resulting
product locally has the structure of H , but globally it is similar to G. The formal definition
of the Sierpiński product is given in Section 2.
The Sierpiński product shows some features of classical graph products, for instance
the vertex set of the Sierpiński product of two graphs G and H is V (G)×V (H). However,
to define the Sierpiński product of two graphs G and H , one needs some extra information
besides G and H . This information can be encoded as a function f : V (G) → V (H).
Furthermore, the product is defined so that we can extend it to multiple factors. We will
see that by definition the Sierpiński product of two graphs is always a subgraph of their
lexicographic product. Such layer-like structure also plays an important role in studying
symmetries of the Sierpiński product. For extensive information about graph products, see
the monograph by Hammack, Imrich and Klavžar [9].
The paper is organized as follows. In Section 2 we give a formal definition of the
Sierpiński product of two graphs G and H with respect to f : V (G) → V (H); this product
is denoted by G⊗fH . We explore some graph-theoretical properties such as connectedness
and planarity of a Sierpiński product. In particular, we show that G ⊗f H is connected if
and only if both G and H are connected and we present some necessary and sufficient
conditions that G and H must fulfill in order for G ⊗f H to be planar. In Section 3 we
study symmetries of the Sierpiński product of two graphs. We focus on the automorphisms
of G⊗f H that arise from the automorphisms of its factors and study the group generated
by these automorphisms. In several cases we can also describe the whole automorphism
group of G ⊗f H . Finally, in Section 4 we consider the Sierpiński product of more than
two graphs. A special case of n equal factors is considered.
2 Definition of the Sierpiński product and basic properties
Let us first review some necessary notions. All the graphs we consider are undirected,
simple and finite. Let G be a graph and x be a vertex of G. By N(x) we denote the set
of vertices of G that are adjacent to x, i.e. the neighbourhood of x. Vertices in this paper
will usually be tuples, but instead of writing them in vector form (xm, . . . , x1), we will
sometimes write them as words xm . . . x1 (especially in figures). More precisely, vertices
(0, 0, 0) or (0, (0, 0)) will simply be denoted by 000, except in the case when we will want
J. Kovič et al.: The Sierpiński product of graphs 5
to emphasize their origins. The number of vertices of a graph G, i.e. the order of G, will be
denoted by |G|, and the number of edges of G, i.e. the size of G, will be denoted by ||G||.
For other graph theory concepts not defined here we refer the reader to [21].
Definition 2.1. Let G,H be graphs and let f : V (G) → V (H) be a function. Then the
Sierpiński product of the graphs G and H with respect to f , denoted by G⊗f H , is defined
as the graph on the vertex set V (G)× V (H) with two types of edges:
• {(g, h), (g, h′)} is an edge of G ⊗f H for every vertex g ∈ V (G) and every edge
{h, h′} ∈ E(H),
• {(g, f(g′), (g′, f(g))} is an edge of G⊗f H for every edge {g, g′} ∈ E(G).
If V (G) ⊆ V (H) and f is the identity function on its domain, we will skip the index
f and denote the corresponding Sierpiński product of G and H simply by G ⊗ H . Note
that there are no restrictions on the function f : V (G) → V (H). However, sometimes it
is convenient to assume that for every g, g1, g2 ∈ V (G), g1 ̸= g2, the following property
holds: if g1, g2 ∈ N(g), then f(g1) ̸= f(g2). In this case we say that f is locally injective.
a2 a3
a1
a4
b2 b3
b1
b4
c2 c3
c1
c4
1a 1b
1c
2a 2b
2c
3a 3b
3c
4a 4b
4c
Figure 3: The graph K3⊗f1 K4, where V (K3) = {a, b, c}, V (K4) = {1, 2, 3, 4}, f1(a) =
1, f1(b) = 2, f1(c) = 3, and the graph K4 ⊗f2 K3, where f2(1) = a, f2(2) = b, f2(3) =
c, f2(4) = c. The inner edges are solid while the connecting edges are dashed. The solid
subgraphs are aH, bH, cH for H = K4 and 1H, 2H, 3H, 4H for H = K3.
For any vertex g of G, the subgraph of G⊗f H induced by the set of vertices {(g, h) |
h ∈ V (H)} is called the subgraph associated with g and is denoted by gH . We may view
the graph G⊗fH as partitioned into graphs gH , one for every vertex g of G, and connecting
for every edge {g, g′} ∈ E(G) the corresponding vertex f(g) in g′H to the vertex f(g′) in
gH . The edges of G ⊗f H naturally fall into two classes. The edges connecting different
subgraphs gH are called connecting edges, while the edges inside some subgraph gH are
called inner edges. Given gH , we may add its neighbourhood, i.e. all connecting edges
and their endvertices, to obtain a graph denoted by gHN . If we identify all newly added
vertices to a single vertex, denoted by gH , we obtain a graph, denoted by H + g.
Example 2.2. Figure 3 (left) shows the Sierpiński product of K3 and K4 with respect
to the following function f1. The vertices of K3 are labelled with letters a, b, c, the ver-
tices of K4 are labelled with numbers 1, 2, 3, 4 and f1 : V (K3) → V (K4) is given by
6 Ars Math. Contemp. 23 (2023) #P1.01 / 1–25
f1(a) = 1, f1(b) = 2, f1(c) = 3. Figure 3, right, shows the Sierpiński product of K4
and K3 with respect to f2 : V (K4) → V (K3) defined by f2(1) = a, f2(2) = b, f2(3) =
c, f2(4) = c. This shows that the Sierpiński product is not commutative. Figure 4 depicts
examples of the graphs gHN and H + g in K3 ⊗f1 K4 and K4 ⊗f2 K3.
a2
b2 b3
b1
b4
c2 1a 1b
1c
2a
3a
4a
b2 b3
b1
b4
bK4 1a 1b
1c
1K3
Figure 4: Top: the graphs bHN for H = K4 and 1HN for H = K3. Bottom: the graphs
H + b for H = K4 and H + 1 for H = K3.
2.1 Basic properties of the Sierpiński product of graphs
We now state some observations regarding the structure of the Sierpiński product of two
graphs. We omit most of the proofs, since they follow straight from the definition.
Proposition 2.3. Let G,H be graphs and let f : V (G) → V (H) be a function. Then the
following statements hold.
(i) If |G| = 1, then G⊗f H is isomorphic to H .
(ii) If |H| = 1, then G⊗f H is isomorphic to G.
Lemma 2.4. Let G,H be graphs and let f : V (G) → V (H) be a function. Let G′, H ′
be subgraphs of G,H , respectively, and let f ′ be the restriction of f to V (G′) such that
Im(f ′) ⊆ V (H ′). Then the graph G′ ⊗f ′ H ′ is a subgraph of the graph G⊗f H .
The following result explains the role of the graphs G and gH in G⊗f H .
Lemma 2.5. Let G,H be graphs, and let f : V (G) → V (H) be a function. Then the
following statements hold.
J. Kovič et al.: The Sierpiński product of graphs 7
(i) For every vertex g of G, the subgraph gH of G⊗f H is isomorphic to H .
(ii) If, in addition, the graph H is connected, then the graph G is a minor of G⊗f H .
(iii) In particular, if the graph H is connected and the graph G⊗f H is planar, then the
graphs G and H are planar.
Notice that planarity of both factors G and H in (iii) easily follows from the fact that
H is a connected subgraph, and G is a minor of G⊗f H . However, if H is not connected,
the conclusion in (ii) that G is a minor of G ⊗f H does not necessarily follow, since we
cannot simply contract every subgraph gH of G⊗f H to a single vertex. This is shown by
the following example.
Example 2.6. Take G to be K5 with vertices from 1 to 5 and H to be two isolated vertices,
denoted by a and b, and then let f map 1 and 2 from K5 to a of H and the remaining three
vertices of K5 to b of H . The resulting graph G ⊗f H is isomorphic to the disjoint union
of K2,K3 and K2,3 and is therefore planar. Hence, it cannot have K5 as a minor.
Proposition 2.7. Let G,H be graphs and let f : V (G) → V (H) be a function. Then the
graph G⊗f H has |G| · |H| vertices and ||H|| · |G|+ ||G|| edges. In particular, G⊗f H
has ||H|| · |G| inner edges and ||G|| connecting edges.
Lemma 2.8. Let G and H be graphs and let f : V (G) → V (H) be a function. Then the
following holds.
(i) For every g, g′ ∈ V (G) there exists at most one edge connecting graphs gH and
g′H . In addition, graphs gH and g′H are joined by an edge if and only if vertices g
and g′ are adjacent in graph G.
(ii) If a vertex g ∈ V (G) has d neighbours in G, then there are d vertices and d edges
belonging to gHN that are not a part of gH . These d edges are connecting edges.
(iii) The function f is locally injective if and only if the connecting edges form a matching
in G⊗f H or, equivalently, if and only if every vertex of G⊗f H is an endvertex of
at most one connecting edge.
Proof. (i) Only an edge of form {(g, f(g′)), (g′, f(g))} can connect graphs gH and g′H
and this happens if and only if there exists an edge between g and g′. Hence, there exists a
bijective correspondence between the connecting edges of G⊗f H and the edges of G.
(ii) This follows from (i) and the fact that the connecting edges with an endvertex in gH
are in bijective correspondence with the edges connecting g to its neighbours in G.
(iii) For every g ∈ V (G), the connecting edges leading from gH to the rest of the graph
G ⊗f H have distinct endvertices outside gH . If f is locally injective, they have distinct
endvertices within gH . It follows that no two connecting edges share a common endvertex,
so the connecting edges form a matching in G ⊗f H . Conversely, if f fails to be locally
injective, then at least two connecting edges share a common endvertex.
The lexicographic product of two graphs G and H is the graph G ◦H with vertex set
V (G) × V (H) and two vertices (g, h) and (g′, h′) are adjacent in G ◦ H if and only if
either g is adjacent with g′ in G or g = g′ and h is adjacent with h′ in H . For g ∈ V (G),
we denote by gH the subgraph of G ◦H induced by the set {(g, h)|h ∈ V (H)}. Then the
8 Ars Math. Contemp. 23 (2023) #P1.01 / 1–25
graph G ◦H consists of |G| copies of H , and for every edge {g, g′} in G, every vertex of
gH is connected to every vertex in g′H . Therefore, the next result follows straight from
Definition 2.1.
Proposition 2.9. Let G and H be graphs and let f : V (G) → V (H) be a function. Then
the graph G⊗f H is a spanning subgraph of the lexicographic product G ◦H .
Note that for different functions f, f ′, the graphs G ⊗f H and G ⊗f ′ H may be iso-
morphic or nonisomorphic.
Proposition 2.10. Let G,H be graphs and let f : V (G) → V (H) be a function. Let
α ∈ Aut(G), β ∈ Aut(H) and f ′ = β ◦ f ◦ α. Then G⊗f H is isomorphic to G⊗f ′ H .
Proof. Define the function γ : V (G⊗f H) → V (G⊗f ′ H) by γ(g, h) = (α−1(g), β(h))
for g ∈ V (G) and h ∈ V (H). Since α, β are bijections, γ is also a bijection. Since β is an
automorphism, γ maps inner edges to inner edges.
Take a connecting edge in G ⊗f H , say {(g, f(g′)), (g′, f(g))}, where {g, g′} ∈
E(G). Then γ(g, f(g′)) = (α−1(g), β(f(g′)) and γ(g′, f(g)) = (α−1(g′), β(f(g)).
Since f ′(α−1(g)) = β(f(α(α−1(g)))) = β(f(g)) and f ′(α−1(g′)) = β(f(α(α−1(g′)))) =
β(f(g′)), we see that γ also maps a connecting edge to a connecting edge. Therefore γ is
an isomorphism.
Corollary 2.11. Let G be a graph and let f ∈ Aut(G). Then G ⊗ G is isomorphic to
G⊗f G.
In the remainder of this section we consider when the Sierpiński product of two graphs
is connected.
Proposition 2.12. Let G and H be graphs and let f : V (G) → V (H) be a function. Then
G⊗f H is connected if and only if G and H are connected.
Proof. Suppose G and H are connected. Then G⊗f H is connected by Definition 2.1 and
Lemma 2.5.
Conversely, suppose G ⊗f H is connected. Pick two vertices g and g′ from G. Then
a path from gH to g′H in G ⊗f H corresponds to a path in G from g to g′. Therefore,
G is also connected. Suppose now that H is not connected. Denote by H1 a connected
component of H such that V1 = {g ∈ G| f(g) ∈ V (H1)} is nonempty. Take any vertices
g ∈ V1 and h ∈ V (H1). If there exists an edge of form {(g, h), (g′, f(g))}, then h = f(g′),
so g′ ∈ V1. Note that f(g) ∈ V (H1). Therefore, all the neighbours of (g, h) belong either
to gH1 or to g′H1 for some g′ ∈ V1. It follows that there are no edges between the set of
vertices {(g, h) ∈ V (G ⊗f H)| g ∈ V1 and h ∈ V (H1)} and the rest of the vertices of
G⊗fH . So G⊗fH is not connected. This is a contradiction, so H must be connected.
In [19], Klavžar and Zemljič have characterized which generalized Sierpiński graphs
are k-connected and k-edge connected. Unfortunately, their results do not carry over di-
rectly to the Sierpiński product of distinct graphs since the factors may have different con-
nectivity properties. Moreover, the result does not depend only on the connectivity of its
factors, but also on the choice of the function f . For instance, C5 ⊗ C5 is 2-connected.
However, for a constant function f , the product C5 ⊗f C5 is only 1-connected.
J. Kovič et al.: The Sierpiński product of graphs 9
2.2 Planarity
In this section we study planarity of the Sierpiński product G ⊗f H of graphs G and H
with respect to a function f : V (G) → V (H). We have already mentioned in Lemma 2.5
that for a connected graph H , planarity of the graph V (G⊗f H) implies planarity of both
factors. The next theorem characterizes when a Sierpiński product G⊗fH is planar when f
is a locally injective function. Recall that the construction of a graph H+g was introduced
in the beginning of Section 2.
Theorem 2.13. Let G be a 2-connected graph or G = K2, let H be a connected graph and
let f : V (G) → V (H) be a locally injective function. Then the graph G⊗f H is planar if
and only if the following three conditions are fulfilled:
(i) the graphs G and H are planar,
(ii) for every g ∈ V (G) the graph H + g is planar,
(iii) there exists an embedding of the graph G in the plane with a chosen orientation
which has the following property: for every g ∈ V (G), with g1, g2, . . . , gk being the
cyclic order of vertices around the vertex g, there exists an embedding of the graph
H + g in the plane such that the cyclic order of vertices around the vertex gH in
graph H + g is f(gk), f(gk−1), . . . , f(g1).
Proof. The fact that f is a locally injective function simplifies the arguments. Namely, all
the vertices in N(g) are distinct for every g ∈ V (G).
(⇒) First, assume that the graph G ⊗f H is planar. Then, the graphs G,H are planar
by Lemma 2.5 and (i) is established.
Suppose G ⊗f H is embedded in the plane. Note that for every g ∈ V (G), the
embedding of G ⊗f H induces a planar embedding of gH . For every g ∈ V (G), let
N(gH) = {g′H|g′ ∈ N(g)} denote the collection of graphs g′H that are adjacent to gH .
Since the graph G is 2-connected, or G = K2, and the graph H is connected, all the graphs
from N(gH) are inside the same face Fg of the graph gH (otherwise the vertex g would be
a cut vertex in G). For a fixed g0 we may assume that Fg0 is the outer face of g0H . If not,
we take a different stereographic projection of G⊗f H . But then, for every g ∈ V (G), the
corresponding face Fg is the outer face of gH . Hence, we may contract every graph gH to
a single vertex. From now on we assume that the graph G⊗f H is embedded in the plane
as we just explained and choose an orientation of the plane. If we contract every subgraph
gH of G ⊗f H to a single vertex, we obtain a minor of G ⊗f H which is isomorphic to
G. Its planar embedding is determined from the planar embedding of G⊗f H . Hence, for
every vertex of G the cyclic order of its neighbours is defined.
Now again take the embedding of G ⊗f H as described above. Take an arbitrary
g ∈ V (G) and consider the subgraph gHN that inherits the planar embedding and has
all of its pending edges attached to the vertices in the outer face of gH . This establishes
a planar embedding of H + g, which, in turn, is obtained from gHN by a suitable vertex
identification. This proves (ii).
Now it is easy to see that the embedding of G, obtained from G ⊗f H by contracting
every copy of H to a single vertex, fulfills (iii). Namely, the cyclic order g1, g2, . . . , gk of
vertices around a vertex g of G in this embedding corresponds to the ordering of the vertices
(g, f(g1)), (g, f(g2)), . . . , (g, f(gk)) around the outer face of gH in the planar embedding
of G ⊗f H . The cyclic order of vertices around gH in the planar embedding of gH + g
10 Ars Math. Contemp. 23 (2023) #P1.01 / 1–25
described above is then (g, f(gk)), (g, f(gk−1)), . . . , (g, f(g1)). Therefore, an appropriate
embedding of H + g exists for every g ∈ V (g).
(⇐) The converse is proved by construction. By (i), the graphs G and H are planar.
Moreover, by (ii), for every g ∈ V (G), the graph H + g is planar. Using (iii), we embed
the graph G in the plane. Then for each vertex g of G, we replace the vertex g by the planar
embedding of the graph gH , induced by the embedding of H + g from (iii). Again by
(iii), it is possible to connect the copies of H among themselves in such a way that a planar
embedding of the resulting graph, isomorphic to G⊗f H , is obtained.
We believe that Theorem 2.13 holds also if the function f is not locally injective. How-
ever, the arguments in the proof become much more involved in this case.
Remark 2.14. Note that the condition in Theorem 2.13 that the graph G is 2-connected
is essential. For example, take G to be the graph obtained from K4 (whose vertices are
denoted by 1, 2, 3, 4) by adding a new vertex (named 5) to it and joining it to the vertex 4
only. The graph G⊗G is then planar, but G+4 is not and the condition (ii) in Theorem 2.13
does not hold. See Figure 5.
2 3
4
1
5
2 3
4
1
5 4G
Figure 5: The graph G is planar but not 2-connected. The graph G+ 4 contains a subdivi-
sion of K5 and is not planar.
The next result is evident from the proof of Theorem 2.13.
Corollary 2.15. Let G be a 2-connected graph or G = K2, let H be a connected graph
and let f : V (G) → V (H) be a locally injective function. If the graph G ⊗f H is planar,
then for every g ∈ V (G) there exists an embedding of the graph H in the plane such that
the vertices {f(g′); g′ ∈ N(g)} lie on the boundary of the same face.
Using Theorem 2.13 we can now determine when the graph G ⊗ G is planar for a
connected graph G. We also give a sufficient condition for the graph G⊗f H to be planar
when G ̸= H . By a block we mean a maximal connected subgraph of a given graph that
has no cut-vertices. Note that a block with more than two vertices is 2-connected.
Theorem 2.16. Let G be a connected graph. Then the graph G⊗G is planar if and only
if every block of G is outerplanar or equal to K4.
Proof. Note that for every block B of G, the identity function V (B) → V (B) is locally
injective. So we may use Theorem 2.13 and Corollary 2.15 for every block of G.
J. Kovič et al.: The Sierpiński product of graphs 11
First assume that the graph G ⊗ G is planar. Let B be a block of G. Suppose that B
is planar but it is not outerplanar or equal to K4. Then it contains a subdivision of K2,3
or a subdivision of K4 (with at least one additional vertex) as a subgraph, see [3]. Such a
graph B always contains a vertex such that in every planar embedding of B, not all of its
neighbours will be on the boundary of the same face. Therefore B ⊗ B is not planar by
Corollary 2.15. On the other hand, B ⊗ B is a subgraph of G ⊗G by Lemma 2.4, so it is
planar. A contradiction. Therefore, the graph B must be outerplanar or equal to K4.
We prove the converse by induction on the number of blocks of the graph G. If G has
just one block that is outerplanar or equal to K4, then the conditions (i), (ii), (iii) from
Theorem 2.13 are fulfilled, so G⊗G is planar.
Suppose now that G has more than one block and that every block of G is outerplanar
or equal to K4. Let B be a block that corresponds to a leaf in the block-cut tree of G and
let v be the only cut vertex of G contained in B. Let G′ be the graph obtained from G by
deleting all the vertices of B with the exception of v. Then the graph G′ has one block
less than G and, by the induction hypothesis, the graph G′ ⊗ G′ is planar. Take a planar
embedding of G′ ⊗ G′. We obtain a planar embedding of G ⊗ G in the following way.
We take a planar embedding of B in which v appears on the boundary of the outer face;
moreover if B is outerplanar, we take an embedding of B such that every vertex of B lies
on the boundary of the outer face. For every g ∈ V (G′), we insert a copy of B in a face of
gG′ that contains the vertex (g, v) and then identify the vertex (g, v) with the vertex v of
B. We denote the graph obtained so far by K. Observe that K is embedded in the plane.
To the graph K we still need to add a copy of G for every vertex of B except v. These
copies of G are connected only to the copies of G in G ⊗ G corresponding to the vertices
of B. Choose an orientation of the plane. Now we consider two cases.
First, let B = K4, with vertices v, v1, v2, v3. Any three vertices of B are on the bound-
ary of a same face in every embedding of B in the plane. Therefore, in the subgraph vG of
K there exists a face such that the vertices (v, v1), (v, v2), (v, v3) all lie on its boundary;
without loss of generality we may assume that they are arranged in this order (with respect
to the chosen orientation of the plane). We may embed the graph v1G in the plane such that
the vertices (v1, v), (v1, v3), (v1, v2) lie on the outer face in this order, likewise, we may
embed the graph v2G in the plane such that the vertices (v2, v), (v2, v1), (v2, v3) lie on the
outer face in this order, and we may embed the graph v3G in the plane such that the vertices
(v3, v), (v3, v2), (v3, v1) lie on the outer face in this order. Now place viG in the face of the
subgraph vG of K that contains the vertices (v, v1), (v, v2), (v, v3) close to vertex (v, vi)
and connect vertices (v, vi) and (vi, v), for i = 1, 2, 3. The graphs viG are embedded
in the plane such that it is possible to also connect the pairs of vertices (v1, v2), (v2, v1),
(v1, v3), (v3, v1) and (v2, v3), (v3, v2) without crossings. This gives us a planar embedding
of the graph G⊗G.
Next, let B be outerplanar. The subgraph vG of K is embedded in the plane and it
contains a copy of B such that all the vertices of B are on the boundary of the same face of
K, say F . We consider a planar embedding of the graph G in which all the vertices of the
block B are on the boundary of the outer face and in the reverse order as the corresponding
vertices in vG. For every vertex u of B except v, we place such a copy of G with vertex
set {u} × V (G) into face F close to the vertex (v, u). Then we connect vertices (v, u)
and (u, v) with an edge if u is a neighbour of v in G. The graphs uG, u ∈ V (B)\{v}
are now all in the same face of K. Moreover, the subgraphs uB, u ∈ V (B)\{v}, are
all in the same face of K with all their vertices on the boundary of the outer face (of the
12 Ars Math. Contemp. 23 (2023) #P1.01 / 1–25
embedding of uG in the plane) and the order of these vertices is reversed compared to the
order of the corresponding vertices in vB. Note that the cyclic order of the graphs uB,
u ∈ V (B)\{v} is the same as the order of the corresponding vertices in vB. Therefore, it
is possible to connect the vertices (u, u′) and (u′, u) for every edge {u, u′} of B without
crossings. Again we obtain a planar embedding of the graph G ⊗ G. This completes the
proof.
Theorem 2.17. Let G,H be connected graphs and let f : V (G) → V (H) be a function.
Assume that G is planar, ∆(G) ≤ 3 and H is outerplanar. Then G⊗f H is planar.
Proof. Denote K = G ⊗f H for convenience. Suppose K is not planar. Then it contains
a subdivision of K3,3 or K5 as a subgraph. First assume that K contains a subdivision of
K3,3. Denote the set vertices of degree 3 of the subdivision of K3,3 in K by X . There are
four cases to consider, depending on how many vertices from X are in the same copy of H
in K.
1. Every vertex from X is in a separate copy of H . If no path connecting two vertices
from X passes through any other copies of H containing vertices from X , and at
most one such path passes through every copy of H not containing vertices from X ,
then by contracting gH to a single vertex, for every g ∈ V (G), we see that K3,3 is
a minor in G, so G is not planar. Otherwise we need at least four edges connecting
some copy of H to the other copies of H in K. This is not possible, since the
maximal degree in G is at most three.
2. There are between two and four vertices from X in some gH . Then we need at least
four edges connecting gH to the other copies of H in K. This is again not possible,
since the maximal degree of G is at most 3.
3. There are five vertices from X in some gH and one vertex from X in some g′H for
g ̸= g′. Since H is outerplanar, gH cannot contain a subdivision of K2,3. Therefore,
we need at least two edges going out of gH to obtain a subdivision of K2,3 from the
five vertices in gH . We also need three edges going out of gH to connect gH to the
vertex of K3,3 in g′H . This is again not possible, since the maximal degree of G is
at most 3.
4. The only remaining possibility is that all six vertices from X are in the same copy gH
of H . Since H is outerplanar, there can be at most seven edges (or paths) between
pairs of vertices of K3,3 in gH . The remaining two paths must go through the other
copies of H , which means that we again need at least four edges connecting gH to
the other copies of H in K. A contradiction.
Therefore, K does not contain a subdivision of K3,3. Next assume that K contains a
subdivision of K5. Denote the set vertices of degree 4 of the subdivision of K5 in K by Y .
There are two cases to consider, depending on how many vertices from Y are in the same
copy of H .
1. There are between one and four vertices from Y in some gH . Then we need at least
four edges connecting gH to the other copies of H in K. This is not possible, since
the maximal degree in G is at most three.
J. Kovič et al.: The Sierpiński product of graphs 13
2. All five vertices from Y are in the same copy of H . Since H is outerplanar, it does
not contain a subdivision of K4 or K2,3. Therefore, there can be at most eight edges
(or paths) between pairs of these vertices in gH (in fact, there can be at most six
such paths). The remaining two paths must go through other copies of H , which
means that we need at least four edges connecting gH to other copies of H in K. A
contradiction.
It follows that K does not contain a subdivision of K3,3 or K5, so it is planar.
If a connected graph is not planar, it is natural to consider its genus. The genus of a
graph G is denoted by γ(G). Recall that by Lemma 2.5, if the graph H is connected, the
graph G is a minor of G⊗f H for any function f : V (G) → V (H), and the graph G⊗f H
contains |G| copies of the graph H as induced subgraphs. Suppose G,H are connected
and f is arbitrary. Then it is easy to see, cf. [21, Theorem 4.4.2], that
γ(G⊗f H) ≥ γ(G) + |G| · γ(H). (2.1)
Note that the bound is not sharp even if the factors are planar. In the case of a planar
Sierpiński product, we were able to settle the case in Theorem 2.13. It would be interest-
ing to find some sufficient condition for the equality in (2.1) to hold also for non-planar
Sierpiński products.
3 Symmetries of the Sierpiński product of graphs
Throughout this section let G,H be connected graphs and let f : V (G) → V (H) be a
function. Recall that the edge set of G⊗f H can be naturally partitioned into two subsets:
• inner edges {(g, h), (g, h′)} for every vertex g ∈ V (G) and every edge {h, h′} ∈
E(H), and
• connecting edges {(g, f(g′), (g′, f(g))} for every edge {g, g′} ∈ E(G).
We call this partition of the edge set the fundamental edge partition. We will say that an
automorphism of G⊗f H respects the fundamental edge partition if it takes inner edges to
inner edges and connecting edges to connecting edges. We denote the set of all automor-
phisms of G ⊗f H that respect the fundamental edge partition by Ã(G,H, f). This set is
a subgroup of the whole automorphism group of G⊗f H . For connected graphs G and H ,
the automorphisms that respect the fundamental edge partition have the following useful
property.
Proposition 3.1. Let G and H be connected graphs. Then every automorphism γ̃ ∈
Ã(G,H, f) permutes the subgraphs gH , g ∈ G. Moreover, the restriction γ̃|V (gH) :
V (gH) → V (g′H), where g′ ∈ V (G), is a graph isomorphism.
In this section we first show that every automorphism of G ⊗f H that respects the
fundamental edge partition induces automorphisms of G and H . And conversely, we define
two families of automorphisms of G ⊗f H that respect the fundamental edge partition
using automorphisms of G and H . As it turns out, one of them is a subfamily of the
other one. Then we show that in several cases all the automorphisms of G ⊗f H respect
the fundamental edge partition. Finally, we focus on the case when G = H and f is an
automorphism. In this case we can completely describe the group of automorphisms that
respect the fundamental edge partition.
14 Ars Math. Contemp. 23 (2023) #P1.01 / 1–25
3.1 Automorphisms that respect the fundamental edge partition
Let γ̃ be an automorphism of G⊗f H that respects the fundamental edge partition. Then,
it permutes the subgraphs gH , g ∈ G. Define the function γ : V (G) → V (G) such
that γ(g) = g′ if γ̃ maps gH to g′H . Obviously, γ is a bijection. Let {g, g1} be an
edge of G. Then {(g, f(g1)), (g1, f(g))} is a connecting edge of G ⊗f H . Since γ̃ re-
spects the fundamental edge partition, it maps this edge to another connecting edge, say
{(g′, f(g′1)), (g′1, f(g′))}, where g′ and g′1 are adjacent in G. But then γ maps the edge
{g, g1} to an edge (i.e. to {g′, g′1}) and γ is an automorphism. We will say that γ is the pro-
jection of γ̃ on G. Conversely, γ̃ is a lift of γ. Note that the projection of γ̃ ∈ Aut(G⊗f H)
on G is uniquely defined. However, given an automorphism of G, it can have a unique lift,
more than one lift or none at all.
On the other hand, the application of γ̃ on every copy of gH in G ⊗f H induces an
automorphism γg of H , defined by γg(h) = h′ if γ̃ sends (g, h) to (g1, h′) for some
g1 ∈ V (G) and h′ ∈ V (H).
We will now introduce the first family of automorphisms of G⊗fH that can be obtained
from automorphisms of G and H . All such automorphisms respect the fundamental edge
partition.
Definition 3.2. Let G,H be connected graphs and let f : V (G) → V (H) be a function.
Let α ∈ Aut(G) and let B : V (G) → Aut(H) be a function. For simplicity we will write
βg instead of B(g) for g ∈ V (G). Define the function Ψ(α,B) : V (G⊗fH) → V (G⊗fH)
by
Ψ(α,B) : (g, h) 7→ (α(g), βg(h)).
If B is a constant function, say βg = β for all g ∈ V (G), we denote Ψ(α,B) by Ψ(α, β).
By the discussion at the beginning of this section, we may conclude that the following
holds.
Theorem 3.3. Let G,H be connected graphs and let f : V (G) → V (H) be a function.
Then, every automorphism of G ⊗f H that respects the fundamental edge partition is of
form Ψ(α,B) for some α ∈ Aut(G) and some function B : V (G) → Aut(H).
We now determine when the function Ψ(α,B) from Definition 3.2 is an automorphism.
In Propositions 3.4, 3.5, 3.6 and Corollaries 3.7, 3.8, and 3.9, the assumptions from Defi-
nition 3.2 hold.
Proposition 3.4. The function Ψ(α,B) is a bijection.
Proof. It is enough to prove that Ψ(α,B) is injective. This is straightforward since α and
βg , g ∈ V (G), are all injective.
Proposition 3.5. The function Ψ(α,B) is an automorphism if and only if for every g ∈
V (G) we have f ◦ α = βg ◦ f on N(g). Moreover, in this case Ψ(α,B) respects the
fundamental edge partition.
Proof. We first show that Ψ(α,B) always maps an inner edge to an inner edge. To see
this, let e = {(g, h1), (g, h2)} be an inner edge. Then Ψ(α,B) maps the edge e to
{(α(g), βg(h1)),(α(g), βg(h2))}, which is an inner edge since βg is an automorphism of
H .
J. Kovič et al.: The Sierpiński product of graphs 15
Suppose now that Ψ(α,B) is an automorphism. Since Ψ(α,B) maps inner edges to
inner edges, it must map connecting edges to connecting edges. Let e = {(g, f(g1)),
(g1, f(g))} be a connecting edge. Then Ψ(α,B)(e) = {(α(g), βg(f(g1))), (α(g1),
βg1(f(g)))} is also a connecting edge. Therefore f(α(g1)) = βg(f(g1)). Since g1 can
be any neighbour of g in G, we have f ◦ α = βg ◦ f on N(g).
Conversely, let f ◦ α = βg ◦ f on N(g) for every g ∈ V (G). Let e = {(g, f(g1)),
(g1, f(g))} be a connecting edge in G ⊗f H . Then Ψ(α,B)(e) = {(α(g), βg(f(g1))),
(α(g1), βg1(f(g)))}. Since f(α(g)) = βg1(f(g)) and f(α(g1)) = βg(f(g1)), Ψ(α,B)(e)
is a connecting edge. Therefore, Ψ(α,B) is an automorphism.
Proposition 3.6. The function Ψ(α, β) is an automorphism if and only if f ◦ α = β ◦ f .
Proof. Let f ◦ α = β ◦ f on N(G) for every g ∈ V (G). Since G is connected, it has no
isolated vertices, and so f ◦ α = β ◦ f on V (G). The claim then follows from Proposi-
tion 3.5.
A few special cases now follow as simple corollaries. Recall that α, β,Ψ(α, β) are
defined in Definition 3.2.
Corollary 3.7. Suppose G = H and f is a bijective function. Then the function Ψ(α, β)
is an automorphism if and only if β = f ◦ α ◦ f−1.
Corollary 3.8. Suppose G = H and f is the identity function. Then the function Ψ(α, β)
is an automorphism if and only if α = β.
Corollary 3.9. Suppose V (G) ⊆ V (H), f is the identity function on its domain and
β|V (G) = α. Then the function Ψ(α, β) is an automorphism.
Remark 3.10. If f is injective and G ̸= H , we can always relabel the vertices of G,H
such that f is the identity on its domain.
We now give some examples showing that f need not be injective or surjective and we
can still have automorphisms of type Ψ(α,B).
Example 3.11. Let G = K3 and H = K3,3 with V (G) = {1, 2, 3} and V (H) =
{1, 2, . . . , 6}, with adjacencies as in Figure 6. Let f : V (G) → V (H) map 1 7→ 1, 2 7→ 3,
3 7→ 5. Let α = (1 2 3), β1 = (1 3 5)(2 4 6), β2 = (1 3 5)(2 6 4), β3 = (1 3 5) and let
B : V (G) → Aut(G) be defined by B(g) = βg . Then f ◦ α = β1 ◦ f = β2 ◦ f = β3 ◦ f
and
Ψ(α,B) = (11 23 35)(12 24 32)(13 25 31)(14 26 34)(15 21 33)(16 22 36)
is an automorphism of G⊗f H that cyclically permutes the subgraphs gH , see Figure 6.
Example 3.12. Let G = H = K1,3 with the edge set {{1, 2}, {2, 3}, {2, 4}}, and let
f : V (G) → V (G) be defined as f = (1 2 3 4). Note that f is a bijection that is not an
automorphism of G. If α = (3 4) and β = f ◦ α ◦ f−1 = (1 4), then f ◦ α = β ◦ f and
Ψ(α, β) = (11 14)(21 24)(31 44)(32 42)(33 43)(34 41)
is an automorphism of G⊗f G, that swaps the copies 3G and 4G, see Figure 7.
16 Ars Math. Contemp. 23 (2023) #P1.01 / 1–25
1
2 3 3 4
5
61
2
13 14
15
1611
12
23 24
25
2621
22 32
33 34
35
36
31
Figure 6: The graphs K3, K3,3 and their Sierpiński product with respect to f : V (K3) →
V (K3,3), f : 1 7→ 1, 2 7→ 3, 3 7→ 5.
1
2
3 4
22
13
12
24 21
23
3332 43
42
31
34 44
41
1411
Figure 7: The graph G = K1,3 and the Sierpiński product G ⊗f G with respect to f =
(1 2 3 4).
Example 3.13. Let G = C4 with V (G) = {1, 2, 3, 4} and adjacencies as in Figure 8, and
let H be a star K1,3, with the edge set {{1, 2}, {2, 3}, {2, 4}}. Let f : V (G) → V (H)
map 1 7→ 2, 2 7→ 2, 3 7→ 4 and 4 7→ 3. Note that the function f is neither injective nor
surjective. If α = (1 2)(3 4) and β = (3 4), then f ◦ α = β ◦ f and
Ψ(α, β) = (11 21)(12 22)(13 24)(14 23)(31 41)(32 42)(33 44)(34 43)
is a reflection automorphism of G ⊗f H , swapping the copies 1H, 2H and 3H, 4H , see
Figure 8.
Now let us introduce the second family of automorphisms of G ⊗f H . Let g ∈ V (G)
and β ∈ Aut(H). Define the function Φ(g, β) : V (G⊗f H) → V (G⊗f H) by
Φ(g, β) : (g1, h1) 7→
{
(g1, h1) if g1 ̸= g,
(g1, β(h1)) if g1 = g.
J. Kovič et al.: The Sierpiński product of graphs 17
1
2
3 4
4
1 2
3
32
3344
42
13
12 22
24
11
14
21
23
31
34
41
43
Figure 8: The graphs C4, K1,3 and their Sierpiński product with respect to f : V (C4) →
V (K1,3), f : 1 7→ 2, 2 7→ 2, 3 7→ 4, 4 7→ 3.
Proposition 3.14. Let g ∈ V (G) and let β ∈ Aut(H). The function Φ(g, β) is an auto-
morphism of G ⊗f H if and only if β is in the pointwise stabilizer of f(N(g)). Moreover,
in this case Φ(g, β) respects the fundamental edge partition.
Proof. The function Φ(g, β) is obviously a bijection since it fixes all the vertices of G⊗fH
not in gH and it permutes the vertices in gH . It also fixes the inner edges and connecting
edges that do not have any endvertices in gH and it permutes the inner edges in gH .
Take a vertex g′ ∈ N(g). Then {(g, f(g′)), (g′, f(g))} is a connecting edge. The
function Φ(g, β) maps {(g, f(g′)), (g′, f(g))} to the set {(g, β(f(g′)), (g′, f(g))}, which
is an edge if and only if β(f(g′)) = f(g′). So Φ(g, β) is an automorphism if and only if β
is in the stabilizer of f(g′) for every g′ ∈ N(G).
Remark 3.15. Note that by Theorem 3.3, the function Φ(g, β) is the same as Ψ(α,B)
for some α ∈ Aut(G) and B : V (G) → Aut(H). Indeed, it is easy to verify that for
α = id and B defined by the rules B : g1 → id if g1 ̸= g and B : g → β, we have
Φ(g, β) = Ψ(α,B).
Given a group X acting on a set Y , we denote by X(Y ) the pointwise stabilizer of Y ,
i.e. the subgroup of X that fixes every element of Y . For g ∈ G denote by B̂g(G,H, f) the
group generated by {Φ(g, βg)|βg ∈ Aut(H)(f(N(g)))}. Denote by B̂(G,H, f) the group
generated by {B̂g(G,H, f)| g ∈ V (G)}. We will now study the structure of the group
B̂(G,H, f).
Proposition 3.16. Let g, g′ be distinct vertices of G and let βg ∈ Aut(H)(f(N(g))),
βg′ ∈ Aut(H)(f(N(g′))). Then Φ(g, βg) and Φ(g′, βg′) commute.
18 Ars Math. Contemp. 23 (2023) #P1.01 / 1–25
14
15
16
12
11
13
21
26
25
23
24
22
32
36
35
34
33
31
43
45
46
41
42
44
1
2 3
4
6 5
4
3
1
2
Figure 9: The graphs C4, 2K3 + e and their Sierpiński product with respect to f = id.
Proof. The functions Φ(g, βg) and Φ(g′, βg′) commute since as permutations they have
disjoint supports.
Theorem 3.17. The group B̂(G,H, f) is a subgroup of the group Ã(G,H, f) and is a
direct product
B̂(G,H, f) =
∏
g∈V (G)
B̂g(G,H, f). (3.1)
Moreover, the group B̂(G,H, f) is isomorphic to the group
∏
g∈V (G) Aut(H)(f(N(g))).
Proof. The group B̂(G,H, f) is a subgroup of the group Ã(G,H, f) by definition and
Propositon 3.14. The groups B̂g(G,H, f), g ∈ V (G), have pairwise only the identity
in common, they generate the group B̂(G,H, f), and the elements of two distinct groups
B̂g(G,H, f) commute, therefore equation (3.1) holds. The last claim is true since for
every g ∈ G, the groups B̂g(G,H, f) and Aut(H)(f(N(g))) are isomorphic in the obvious
way.
3.2 When do all the automorphisms respect the fundamental edge partition
Given connected graphs G,H and a function f : V (G) → V (H), in general there can exist
automorphisms of G ⊗f H that do not respect the fundamental edge partition. Figure 9
shows such an example. There, G = C4, H = 2K3 + e and f : V (G) → V (H) is the
identity function on its domain. One can easily observe that by rotating the graph G⊗f H ,
the inner edge {16, 15} can be mapped to the connecting edge {14, 41}.
Note that in the example above, the graph H is not 2-connected. When two graphs
G and H are both 2-connected, we have so far not been able to find an automorphism of
J. Kovič et al.: The Sierpiński product of graphs 19
G ⊗f H that does not respect the fundamental edge partition. Therefore, we propose the
following conjecture.
Conjecture 3.18. Let G,H be 2-connected graphs and let f : V (G) → V (H) be a func-
tion. Then
Ã(G,H, f) = Aut(G⊗f H).
In this section we prove this conjecture for two special cases. In the first case G =
H, f ∈ Aut(G) and G is a regular triangle-free graph. In the second case every edge of
H is contained in a short cycle. Note that in these two cases the assumption that G,H are
2-connected is not needed.
Theorem 3.19. Let G be a connected regular triangle-free graph and let f : V (G) →
V (G) be an automorphism of G. Then every automorphism of G⊗f G respects the funda-
mental edge partition. In other words,
Ã(G,G, f) = Aut(G⊗f G).
Proof. Let k denote the valency of G. Then the endvertices of every connecting edge in
G ⊗f G have valency k + 1 by Lemma 2.8. An endvertex of an inner edge may have
valency k or k + 1. Clearly, if at least one endvertex of an inner edge has valency k, this
edge cannot be mapped to a connecting edge by any automorphism.
Suppose now that both endvertices of an inner edge {(g, g1), (g, g2)} have degree k+1.
This is only possible if (g, g1) and (g, g2) are the endvertices of some connecting edges,
say {(g, g1), (g′1, f(g))} and {(g, g2), (g′2, f(g))} where g1 = f(g′1) and g2 = f(g′2).
But then g′1 and g
′
2 are adjacent to g in G. Since g1 and g2 are adjacent in G and f is
an automorphism, g′1 and g
′
2 are also adjacent. But then g, g
′
1, g
′
2 form a triangle in G,
a contradiction. Therefore no inner edge can be mapped to a connecting edge, so every
automorphism of G⊗f G respects the fundamental edge partition.
Lemma 3.20. Let G and H be graphs and let f : V (G) → V (H) be a function. Let {g, g′}
be an edge of G.
(i) If {g, g′} is not contained in any cycle of G, then the edge {(g, f(g′)), (g′, f(g))} is
not contained in any cycle of G⊗f H .
(ii) Let c be the length of the shortest cycle that contains {g, g′}. Then the shortest cycle
that contains the edge {(g, f(g′)), (g′, f(g))} in G⊗f H has length at least c.
(iii) Suppose that f is locally injective and let c be the length of the shortest cycle that
contains {g, g′}. Then the shortest cycle that contains the edge {(g, f(g′)), (g′, f(g))}
in G⊗f H has length at least 2c.
Proof. Let C be a cycle in G ⊗f H that contains {(g, f(g′)), (g′, f(g))}. Suppose that
{(g, f(g′)), (g′, f(g))}, {(g′, f(g1)), (g1, f(g′))}, . . . , {(gk, f(g)), (g, f(gk))} are the con-
necting edges in C in that order. Then gg′g1g2 . . . gkg is a cycle of length k in G that
contains the edge {g, g′}, so k ≥ c. Furthermore, if {g, g′} is not contained in any cycle
of G, then the edge {(g, f(g′)), (g′, f(g))} cannot be contained in any cycle of G ⊗f H .
Recall that if f is locally injective, any vertex of G ⊗f H is an endvertex of at most one
connecting edge by Lemma 2.8. Therefore, in this case the shortest cycle that contains
{(g, f(g′)), (g′, f(g))} has length at least 2c.
20 Ars Math. Contemp. 23 (2023) #P1.01 / 1–25
Theorem 3.21. Let G and H be connected graphs, let f : V (G) → V (H) be a function
and let the girth of G be equal to c. In any of the following cases, every automorphism of
G⊗f H respects the fundamental edge partition, i.e.
Ã(G,G, f) = Aut(G⊗f G).
(i) G is a tree and H is a bridgeless graph;
(ii) every edge of H is contained in a cycle of length at most c− 1;
(iii) the function f is locally injective and every edge of H is contained in a cycle of
length at most 2c− 1.
Proof. By Lemma 3.20, the shortest cycle that contains a connecting edge has length at
least c in case (ii), length at least 2c in case (iii) and is not contained in any cycle in case (i).
Since every inner edge is contained in a cycle in case (i), in a cycle of length at most c− 1
in case (ii), and in a cycle of length at most 2c − 1 in case (iii), a connecting edge cannot
be mapped to an inner edge by any automorphism.
3.3 Group of automorphisms of G ⊗f G
We now consider the group of automorphisms that respect the fundamental edge partition
in the special case when G = H and f : V (G) → V (G) is an automorphism. Since in this
case G ⊗f G is isomorphic to G ⊗ G, we could restrict ourselves to the case where f is
the identity. Note that in that case, the structure of the automorphism group was sketched
in the paper [8], but the proofs were never published.
Recall that by Corollary 3.7, every automorphism α of G has a lift, Ψ(α, f ◦ α ◦ f−1).
We call this automorphism the diagonal automorphism of G⊗f G corresponding to α, and
denote it by ᾱ. Denote by Ā(G, f) the set of all diagonal automorphisms. The following
proposition is straightforward to prove.
Proposition 3.22. The set Ā(G, f) is a subgroup of Ã(G,G, f), isomorphic to Aut(G).
To determine the structure of the group Ã(G,G, f), we first show that every element
of Ã(G,G, f) can be written as a product of an element from B̂(G,G, f) and an element
of Ā(G, f). Furthermore, we show that B̂(G,G, f) is a normal subgroup of Ã(G,G, f).
Proposition 3.23. Let G be a connected graph and let f : V (G) → V (G) be an automor-
phism. Let γ̃ be an automorphism of G⊗f G that preserves the fundamental edge partition.
Then there exist α ∈ Aut(G) and βg ∈ Aut(G)(f(N(g))) for every g ∈ V (G) such that
γ̃ = ᾱ
(∏
g∈V (G) Φ(g, βg)
)
.
Proof. Let α be the projection of γ̃ to Aut(G). Then ᾱ = Ψ(α, f ◦ α ◦ f−1) permutes the
copies gG in the right way, such as γ̃ does. Observe that ᾱ already agrees with γ̃ on the
endvertices of all the connecting edges. To obtain γ̃ from ᾱ, we only need to adjust, for
every g ∈ V (G), the action of ᾱ on the vertices from f(N(g)) that are not endvertices of
connecting edges. We can do this on every copy gG separately, by acting with Φ(g, βg),
where βg ∈ Aut(G) is induced by ᾱ−1γ̃. Also, βg ∈ Aut(G)(f(N(g))) since the vertices
from f(N(g)) have the right image already and are fixed by βg .
J. Kovič et al.: The Sierpiński product of graphs 21
Proposition 3.24. Let G be a connected graph and let f : V (G) → V (G) be an automor-
phism. Then the group B̂(G,G, f) is a normal subgroup of the group Ã(G,G, f).
Proof. Observe that the function λ : Ã(G,G, f) → Ā(G, f) defined by λ : Ψ(α,B) →
Ψ(α, f ◦α ◦ f−1) is a homomorphism of groups, with B̂(G,G, f) being its kernel. There-
fore, B̂(G,G, f) is a normal subgroup of Ã(G,G, f).
Theorem 3.25. Let G be a connected graph and let f : V (G) → V (G) be an automor-
phism. Then the group Ã(G,G, f) is a semidirect product,
Ã(G,G, f) = Ā(G, f)⋉ B̂(G,G, f).
Proof. The group B̂(G,G, f) is a normal subgroup of Ã(G,G, f) by Proposition 3.24. By
Proposition 3.23, every element of Ã(G,G, f) can be written as a product of a diagonal
automorphism and an element from B̂(G,G, f). Moreover, only the identity is in both
Ā(G, f) and B̂(G,G, f). This proves that Ã(G,G, f) is a semidirect product of Ā(G, f)
and B̂(G,G, f).
4 Sierpiński product with multiple factors
To form a Sierpiński product G3 ⊗f (G2 ⊗f1 G1) of graphs G3, G2 and G1, one needs
functions f1 : V (G2) → V (G1) and f : V (G3) → G2 ⊗f1 G1, which is rather imprac-
tical. Suppose a function f2 : V (G3) → V (G2) is given. Then a function f : V (G3) →
V (G2⊗f1G1) can be defined in a natural way as f(g) = (f2(g), f1(f2(g)) for g ∈ V (G3).
In other words, let φ : V (G2) → V (G2 ⊗f1 G1) be the function that maps every vertex
g ∈ V (G2) to the vertex (g, f1(g)) ∈ V (G2 ⊗f1 G1). Then f = φ ◦ f2. Now we can
define the Sierpiński product of the graphs G3, G2 and G1 with respect to f2 and f1 in the
following way:
G3 ⊗f2 G2 ⊗f1 G1 = G3 ⊗φ◦f2 (G2 ⊗f1 G1).
Note that with given functions f2 and f1, we cannot form this product in any other way,
therefore, the Sierpiński product is not associative.
In Figure 10 it is shown how the product C3⊗f2 C4⊗f1 C3 is formed in two steps (with
f1 : V (C4) → V (C3), f1 : i 7→ i (mod 3) and f2 : V (C3) → V (C4) being the identity
function on its domain).
It is now easy to see that Sierpiński products possess a nice recursive structure, similar
to Sierpiński graphs and generalized Sierpiński graphs. By the same reasoning as above,
the product Gm ⊗fm−1 · · · ⊗f2 G2 ⊗f1 G1, where V (Gℓ) = {0, 1, . . . , |Gℓ| − 1}, and
fℓ : V (Gℓ+1) → V (Gℓ), ℓ = 1, . . . ,m − 1, are arbitrary functions, can be constructed as
follows.
• First, take |G2| copies of the graph G1 and label them iG1, i ∈ {0, . . . , |G2| − 1}.
Vertices of these graphs have labels of form g2g1, where g2 ∈ V (G2) and g1 ∈
V (G1).
• Connect any two copies iG1 and jG1 if there is an edge {i, j} in G2. More precisely,
if {i, j} ∈ E(G2), we add an edge {if1(j), jf1(i)} between iG1 and jG1. The
resulting graph is then indeed the Sierpiński product G2 ⊗f1 G1.
22 Ars Math. Contemp. 23 (2023) #P1.01 / 1–25
⊗f1 =
0
1 2
3
G2 = C4
0
1 2
G1 = C3
00
01
02
10
12
11 22
21
20
30
32
31
K
⊗φ◦f2 K =
0
1 2
G3 = C3
000
011 022
100
111 122
200
211 222
Figure 10: Construction of the graph C3 ⊗f2 C4 ⊗f1 C3, where f1 : i 7→ i (mod 3) and
f2 = id.
• Next, we form the Sierpiński product of the graphs G3 and K(2) := G2 ⊗f1 G1.
To do so we take |G3| copies of the graph K(2), label them iK(2), i ∈ {0, . . . ,
|G3| − 1}, and connect iK(2) and jK(2) whenever {i, j} is an edge in G3. Such an
edge then has the form {if2(j)f1(f2(j)), jf2(i)f1(f2(i))}.
• The final step is to form the Sierpiński product of the graphs Gm and K(m − 1) in
the same way as we formed all the products so far: make |Gm| copies of K(m− 1)
and label them iK(m−1); then for every edge {i, j} in Gm we add an edge between
copies iK(m− 1) and jK(m− 1). Such an edge has then the following form
{ifm−1(j) . . . f1(f2 . . . (fm−1(j)) . . . ) , jfm−1(i) . . . f1(f2 . . . (fm−1(i)) . . . )}.
J. Kovič et al.: The Sierpiński product of graphs 23
The resulting graph is the product Gm ⊗fm−1 · · · ⊗f2 G2 ⊗f1 G1.
If G1 = · · · = Gm = G and functions f1, . . . , fm−1 are all the identity function, then
Gm ⊗fm−1 · · · ⊗f2 G2 ⊗f1 G1 is the generalized Sierpiński graph SnG; see also [8].
We can calculate the order and the size of the Sierpiński product of multiple factors
directly from the above construction.
Proposition 4.1. Let m ≥ 2, and let G1, . . . , Gm be arbitrary graphs. Further, let
f1 : V (G2) → V (G1), . . . , fm−1 : V (Gm) → V (Gm−1) be arbitrary functions. Then
the order and the size of the Sierpiński product Gm ⊗fm−1 · · · ⊗f1 G1 are as follows
|Gm ⊗fm−1 · · · ⊗f1 G1| =
m∏
ℓ=1
|Gℓ| ,
||Gm ⊗fm−1 · · · ⊗f1 G1|| =
m∑
ℓ=1
 m∏
j=ℓ+1
|Gj |
 ||Gℓ|| .
Note that neither the order nor the size of the Sierpiński product depends on the func-
tions fℓ. It would also be interesting to study some properties of the Sierpiński product
with multiple factors, such as diameter and girth.
5 Conclusion
This paper generalizes Sierpiński graphs even further than generalized Sierpiński graphs,
where the whole structure is based only on one graph. Here we create a product-like struc-
ture of two (or more) factors. Some basic graph theoretical properties are studied in detail,
and planar Sierpiński products are completely characterized. Apart from this, the symme-
tries of Sierpiński products are studied as well. In general, these are not fully understood.
In several cases we are able to determine the automorphism group of the Sierpiński product
of two graphs exactly.
In [14] an algorithm is given for recognizing generalized Sierpiński graphs. Given a
graph it is also natural to ask whether it can be represented as a Sierpiński product of
two or more graphs. Moreover, one can ask if such a representation is unique. The latter
question has a negative answer. Consider the Sierpiński product of C4 and 2K3 + e with
function f as in Figure 9. It can be easily verified that it is isomorphic to C8 ⊗f ′ K3
where f ′ : V (C8) → V (K3) is defined by f ′(1) = f ′(2) = f ′(5) = f ′(6) = 1 and
f ′(3) = f ′(4) = f ′(7) = f ′(8) = 2. However, in this case not all the factors are prime
with respect to the Sierpiński product: C8 can be represented as a Sierpiński product of C4
and K2 while 2K3 + e can be represented as a Sierpiński product of K2 and K3. It would
be interesting to see whether there exist prime graphs with respect to the Sierpiński product
G,H,G′, H ′ and functions f : V (G) → V (H), f ′ : V (G′) → V (H ′) such that G,H are
not isomorphic to G′, H ′ while G⊗f H is isomorphic to G′ ⊗f ′ H ′.
The Sierpiński product can also be defined in a similar way for graphs with loops
and multiple edges. In this case, a loop in G, say {g, g}, would correspond to a loop
{(g, f(g)), (g, f(g))} in G ⊗f H and a multiple edge in G would correspond to a multi-
ple edge in G ⊗f H . Finally, as with other products, one could also study the Sierpiński
product of infinite graphs.
24 Ars Math. Contemp. 23 (2023) #P1.01 / 1–25
ORCID iDs
Jurij Kovič https://orcid.org/0000-0003-0567-2626
Tomaž Pisanski https://orcid.org/0000-0002-1257-5376
Sara Sabrina Zemljič https://orcid.org/0000-0003-4026-0854
Arjana Žitnik https://orcid.org/0000-0001-7737-1836
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.02 / 27–43
https://doi.org/10.26493/1855-3974.2842.6b5
(Also available at http://amc-journal.eu)
Total graph of a signed graph
Francesco Belardo
Department of Mathematics and Applications, University of Naples Federico II,
I-80126 Naples, Italy
Zoran Stanić *
Faculty of Mathematics, University of Belgrade, Studentski trg 16,
11 000 Belgrade, Serbia
Thomas Zaslavsky
Department of Mathematical Sciences, Binghamton University, Binghamton, NY
13902-6000, United States
Received 10 March 2022, accepted 24 March 2022, published online 8 September 2022
Abstract
The total graph is built by joining the graph to its line graph by means of the incidences.
We introduce a similar construction for signed graphs. Under two similar definitions of the
line signed graph, we define the corresponding total signed graph and we show that it is
stable under switching. We consider balance, the frustration index and frustration number,
and the largest eigenvalue. In the regular case we compute the spectrum of the adjacency
matrix of the total graph and the spectra of certain compositions, and we determine some
with exactly two main eigenvalues.
Keywords: Bidirected graph, signed line graph, signed total graph, graph eigenvalues, regular signed
graph, Cartesian product graph.
Math. Subj. Class. (2020): 05C22, 05C50, 05C76.
*Corresponding author. The research is partially supported by the Science Fund of the Republic of Serbia;
grant number 7749676: Spectrally Constrained Signed Graphs with Applications in Coding Theory and Control
Theory – SCSG-ctct.
E-mail addresses: fbelardo@unina.it (Francesco Belardo), zstanic@matf.bg.ac.rs (Zoran Stanić),
zaslav@math.binghamton.edu (Thomas Zaslavsky)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
28 Ars Math. Contemp. 23 (2023) #P1.02 / 27–43
1 Introduction
We define the total graph of a signed graph in a way that extends to signed graphs the
spectral theory of ordinary total graphs of graphs. The usual total graph is built by joining
the graph to its line graph by means of its vertex-edge incidences; this construction coordi-
nates well with the adjacency matrix. When we consider signed graphs, there is a similar
definition which extends the notion of line graph of a signed graph and which coordinates
combinatorial and matrix constructions. Working from two similar definitions of the line
signed graph we define the corresponding total graphs, and we show they are stable under
switching. We examine fundamental properties of the signed total graph including bal-
ance, the degree of imbalance as measured by the frustration index and frustration number,
and the largest eigenvalue. In the regular case we compute the spectrum of the adjacency
matrix and the spectra of certain compositions and determine some with exactly two main
eigenvalues.
A signed graph Σ is a pair (G, σ) = Gσ , where G = (V,E) is an ordinary (unsigned)
graph, called the underlying graph, and σ : E −→ {−1,+1} is the sign function (the
signature). The edge set of a signed graph is composed of subsets of positive and negative
edges. We interpret an unsigned graph G as the all-positive signed graph (G,+) = +G,
whose signature gives +1 to all the edges. Similarly, by (G,−) = −G we denote a graph G
with the all-negative signature. In general, we have −Σ = (G,−σ).
Many familiar notions about unsigned graphs extend directly to signed graphs. For
example, the degree d(v) of a vertex v in Gσ is simply its degree in G. On the other hand,
there also are notions exclusive to signed graphs, most importantly the sign of a cycle,
namely the product of its edge signs. A signed graph or its subgraph is called balanced if
every cycle in it, if any, is positive. Oppositely, Σ is antibalanced if −Σ is balanced, i.e.,
every odd (resp., even) cycle of Σ is negative (resp., positive), e.g., if Σ = −G. Balance
is a fundamental concept of signed graphs and measuring how far a signed graph deviates
from it is valuable information. The frustration index l (resp., the frustration number ν) of
a signed graph is the minimum number of edges (resp., vertices) whose removal results in a
balanced signed graph. These numbers generalize the edge biparticity and vertex biparticity
of a graph G, which equal l(−G) and ν(−G), respectively.
Also important is the operation of switching. If U is a set of vertices of Σ, the switched
signed graph ΣU (in which the vertices in U are switched) is obtained from Σ by reversing
the signs of the edges in the cut [U, V \ U ]. The signed graphs Σ and ΣU are said to be
switching equivalent, written Σ ∼ ΣU , and the same is said for their signatures. Notably,
a signed graph is balanced if and only if it is switching equivalent to the all-positive signa-
ture [15] and it is antibalanced if and only if it switches to the all-negative signature. For
basic notions and notation on signed graphs not given here we refer the reader to [15, 18].
The adjacency matrix AΣ of Σ = Gσ is obtained from the standard (0, 1)-adjacency
matrix of G by reversing the sign of all 1’s which correspond to negative edges. The
eigenvalues of Σ are identified to be the eigenvalues of AΣ; they form the spectrum of Σ.
The Laplacian matrix of Σ is defined by LΣ = DG−AΣ, where DG is the diagonal matrix
of vertex degrees of G. Analogously, the Laplacian eigenvalues of Σ are the eigenvalues
of LΣ.
It is well known that the signed graphs Σ = Gσ and Σ′ = Gσ′ are switching equivalent
if and only if there exists a diagonal matrix S of ±1’s, called the switching matrix, such that
AΣ′ = S
−1AΣS, and we say that the corresponding matrices are switching similar. More
generally, the signed graphs Σ and Σ′ are switching isomorphic if there exist a permutation
F. Belardo et al.: Total graph of a signed graph 29
matrix P and a switching matrix S such that AΣ′ = (PS)−1AΣ(PS); in fact, PS can be
seen as a signed permutation matrix (or a {1, 0,−1}-monomial matrix).
The frustration index and the frustration number are among the most investigated in-
variants – for more details one can consult [18]. Similarly, the largest eigenvalue of the
adjacency matrix is the most investigated spectral invariant of graphs. Evidently, switch-
ing preserves the eigenvalues of AΣ and LΣ, and it also preserves the signs of cycles, so
that switching equivalent signed graphs share the same set of positive (and negative) cy-
cles and have the same frustration index and frustration number. For the above reasons,
when we consider a signed graph Σ perspective, we are considering its switching isomor-
phism class [Σ], and we focus our attention to the properties of Σ which are invariant under
switching isomorphism.
Here is the remainder of the paper. In Section 2 we discuss the concept of line graph
of a signed graph. The total graph is presented in Section 3. In Section 4 we consider
regular underlying graphs and, similarly to what is known for unsigned graphs [6], we give
the eigenvalues of a total graph of a signed graph by means of the eigenvalues of its root
signed graph.
We stress that a line (total) graph of a signed graph is always signed, so “line (total)
graph” of Σ means the same as “signed line (total) graph” of Σ. For brevity we also call
these graphs “line (total) signed graphs” and “signed line (total) graphs” (although literally
the latter can mean any signature on an unsigned line or total graph; indeed an entirely
different signed total graph has been defined by Sinha and Garg [10]).
2 Line graph(s)
The line graph is a well-known concept in graph theory: given a graph G = (V (G), E(G)),
the line graph L(G) has E(G) as vertex set, and two vertices of L(G) are adjacent if and
only if the corresponding edges are adjacent in G. If we consider signed graphs Σ = Gσ ,
then a (signed) line graph L(Gσ) should have L(G) as its underlying graph. However, what
signature should we associate to it? The answer to this question is a matter of discussion
because it is possible to have several very different signatures. In this section we shall
consider the two relevant ones defined in the literature.
2.1 Definitions of a line graph
Zaslavsky gave the first definition of incidence matrix of signed graphs [15], which is a
necessary step in a spectrally consistent definition of a line graph. For a signed graph
Σ = Gσ , we introduce the vertex-edge orientation η : V (G) × E(G) −→ {1, 0,−1}
formed by obeying the following rules:
(O1) η(i, jk) = 0 if i /∈ {j, k};
(O2) η(i, ij) = 1 or η(i, ij) = −1;
(O3) η(i, ij)η(j, ij) = −σ(ij).
(The minus sign in (O3) is necessary for several purposes, such as with signed-graph ori-
entations [14, 17] and geometry [17].) The incidence matrix Bη = (ηij) is a vertex-edge
incidence matrix derived from Gσ , such that its (i, e)-entry is equal to η(i, e). However, it
is not uniquely determined by Σ alone. As in the definition of the oriented incidence matrix
for unsigned graphs, one can randomly choose an entry η(i, ij) to be either +1 or −1, but
30 Ars Math. Contemp. 23 (2023) #P1.02 / 27–43
the entry η(j, ij) is then determined by σ(ij), so η is called an orientation of Gσ (and a
biorientation of G, the unsigned underlying graph). Zaslavsky later interpreted Bη as the
incidence matrix of an oriented signed graph [17] and recognized that the same was an
alternate definition of bidirected graphs as in [7]. From a signed graph Σ we get many
bidirected graphs Ση , but each of them leads back to the same signed graph Σ.
Let A⊺ denote the transpose of the matrix A. The incidence matrix has an important
role in spectral theory. The Laplacian matrix can be derived as the row-by-row product of
the matrix Bη with itself:
BηB
⊺
η = LΣ.
Notably, regardless of the orientation η chosen, we get the same LΣ. It is well known that
the column-by-column product of Bη with itself is a matrix sharing the nonzero eigenvalues
with the row-by-row product. This was one motive for Zaslavsky [16, 18] to define the line
graph of a signed graph as the signed graph LC(Σ) = (L(G), σC) whose signature σC is
determined by the adjacency matrix is ALC(Σ) defined here:
ALC(Σ) = 2I −B⊺ηBη. (2.1)
Unlike in the case of the Laplacian matrix of Σ, the matrix ALC(Σ) does depend on the
orientation η. On the other hand, choosing a different orientation η′ of Σ leads to a matrix
A(L(G),σ′) that is switching similar to A(LC(G),σ) (cf. [18]). Hence, A(L(G),σ) defines
a line graph up to switching similarity, so it can be used for spectral purposes. One of
the benefits of this definition is that the line graph of a signed graph with an all-negative
signature is a line graph with an all-negative signature. In other words, if −G is a graph G
whose edges are taken negatively, we get
LC(−G) = −L(G).
The above fact has two evident consequences. The first one is that the iteration of the
(Zaslavsky) line graph operator always gives a signed graph with all-negative signature,
namely L(k)C (−G) = −L(k)(G). The second one is that if we map simple unsigned graphs
to the theory of signed graphs as signed graphs with the all-negative signature (instead
of the all-positive, as stated in the introduction), then Zaslavsky’s line graph is a direct
generalization of the usual line graph defined for unsigned graphs. We shall call this line
graph the combinatorial line graph of Σ.
However, from a spectral viewpoint, the fact that the matrix ALC(Σ) has spectrum in
the real interval (−∞, 2], is in contrast with the usual concept of spectral graph theory for
which a line graph has spectrum in the real interval [−2,+∞]. Hence, the authors of [3]
decided to modify Zaslavsky’s definition to
A(LS(G),σ) = B
⊺
ηBη − 2I. (2.2)
In fact, the two definitions are virtually equivalent, as LC(Σ) = −LS(Σ). Moreover,
they can be used for different purposes. The latter definition is tailored for those spectral
investigations in which an unsigned graph is considered as a signed one with all-positive
signature. Clearly, in this case its adjacency (and Laplacian) matrix remains unchanged and
the spectral theory of unsigned graphs can easily be encapsulated into the spectral theory
of signed graphs. For example, in this case LS(Σ) is coherent with the usual Laplacian and
signless Laplacian spectral theories of unsigned graphs, and it can be used to investigate
F. Belardo et al.: Total graph of a signed graph 31
their spectra (cf. [2]). For these reasons, we shall call LS(Σ) the spectral line graph. We
note that Hoffmann’s theory of generalized line graphs [9] fits well with both signatures.
In Figure 1 we illustrate an example of a signed graph Σ, an orientation Ση , and the
consequent line graph LC(Σ). Here and later, positive edges are represented by solid lines
and negative edges are represented by dotted lines.
t
t
t
t t
t
t
t
t t
t
t
t
t
t
qqqqq
qqqqq
q q q q q q q q q q q q q q q q q q q
q q q q q q qqqqq
qqqqq
✲✲
✻
✻
✲ ✲
✻
❄
✲✛
−→ −→
v1 v2 v5
v4 v3
e1
e2e4
e3
e5 e4 e1
e5
e3 e2
Σ Ση LC(Σ)
Figure 1: A signed graph, an orientation and the combinatorial line graph.
The matrix definitions of line graphs have combinatorial analogs; in fact, Zaslavsky’s
original definition of the line graph of a signed graph (even prior to [16]) was combinatorial.
For Σ = Gσ the underlying graph of L∗(Σ) is the line graph L(G), while the sign of the
edge ef (e, f being the edges of Σ with a common vertex v) is
σ(ef) =
{
−η(ve)η(vf) for ∗ = C,
η(ve)η(vf) for ∗ = S. (2.3)
Indeed, we may orient the line graph by defining ηL(e, ef) = η(v, e) for two edges e, f
with common vertex v in Σ [18]. Then the combinatorial definition of line graph signs
follows the rule (O3).
Remark 2.1. Which is the best definition of a line graph of a signed graph? Zaslavsky
prefers the one defined in (2.1) because it is consistent with the basic relationship between
signs and orientation stated in (O3). Belardo and Stanić instead prefer (2.2) because it
is the one coherent with existing spectral graph theory and it is more prominent in the
literature. This led to a long discussion among the three authors of this manuscript and the
late Slobodan Simić. In the end, we recognize the validity of each definition, since either
variant can be used and, after all, they are easily equivalent.
Remark 2.2. The identities (2.1) and (2.2) remain valid even if Σ contains multiple edges.
A pair of edges located between the same pair of vertices form a digon, i.e., a 2-vertex
cycle, which is positive if and only if the edges share the same sign. Here the matrix defi-
nition diverges from the combinatorial definition. In the combinatorial definition, a digon
in Σ with edges e and f gives rise to a digon in the line graph having the same sign as the
original digon; aside from signs, this is as in the unsigned line graph. In the matrix defini-
tions of L∗(Σ) it gives rise to a pair of non-adjacent vertices if the corresponding digon is
negative and a pair joined by two parallel edges of the same sign if the corresponding digon
is positive; this is consistent with the fact that a negative digon in Σ disappears in the adja-
cency matrix AΣ. Zaslavsky calls this kind of line graph, where parallel edges of opposite
sign cancel each other, reduced. Therefore, an unreduced line graph has no multiple edges
if and only if Σ has no digons, and a reduced line graph has no multiple edges if and only
if Σ has no positive digons.
32 Ars Math. Contemp. 23 (2023) #P1.02 / 27–43
2.2 Properties of line graphs
Here are some observations that follow directly from (2.1) and (2.2). All triangles that arise
from a star of Σ are negative (resp., positive) in LC(Σ) (resp., LS(Σ)). Every cycle of Σ
keeps its signature in LC(Σ). Every even cycle of Σ keeps its signature in LS(Σ) and every
odd cycle of Σ reverses its signature in LS(Σ).
Theorem 2.3. Let G be an unsigned graph. Then −LC(−G) and LS(−G) are balanced
signed graphs, and therefore switching equivalent to +L(G).
First Proof. Recall that a balanced graph has no negative cycles. If we consider the line
graph L(G), we distinguish three types of cycle:
(i) those that arise from the cycles of G,
(ii) those that arise from induced stars in G (forming cliques),
(iii) those obtained by combining the types (i) and (ii).
Let us consider −LC(−G) and LS(−G). We have to prove that LC(−G) is antibalanced,
or equivalently that LS(−G) is balanced. In LS(Σ) the signed cycles of type (i), originat-
ing from cycles Ck of Σ, get the sign (−1)kσ(Ck). Hence, they are all transformed into
positive cycles of LS(Σ) if and only if Σ ∼ −G. Consider next the cycles of type (ii). The
cliques (Kt, σ) of LS(Σ), originating from an induced K1,t of G, are switching equivalent
to +Kt. To see the latter, without loss of generality one can choose the biorientation of
K1,t for which the vertex-edge incidence at the center of the star is positive (the arrows are
inward directed), so the obtained clique is indeed +Kt. These cycles are always positive,
regardless of the signature of Σ. Finally, for the cycles of type (iii), we know from [13]
that the signs of a set of cycles that span the cycle space determine all the signs. Hence,
the cycles of type (iii) are positive if and only if the cycles of type (i) are positive, that is,
Σ = −G.
Second Proof. Choose the orientation for −G in which η(v, e) = +1 for every incident
vertex and edge. Then LC(−G) is easily seen to be all negative by the combinatorial
definition (2.3) of edge signs, thus LC(−G) = −L(G), which is antibalanced. Then,
LS(−G) = −LC(−G) = +L(G), which is balanced. Choosing a different orientation for
−G has the effect of switching both line graphs, so it does not change the state of balance
or antibalance.
We conclude this section by analysing balance and the degree of imbalance of line
graphs. Because LS(Σ) = −LC(Σ), balance of the combinatorial line graph LC(Σ) is
equivalent to antibalance of the spectral line graph LS(Σ), and balance of the latter is
equivalent to antibalance of the former.
Theorem 2.4. Let Σ = Gσ be a signed graph of order n and size m. The following hold
true:
(i) LC(Σ) is balanced (and LS(Σ) is antibalanced) if and only if Σ is a disjoint union
of paths and positive cycles.
(ii) LS(Σ) is balanced (and LC(Σ) is antibalanced) if and only if Σ is antibalanced.
F. Belardo et al.: Total graph of a signed graph 33
(iii) l(LS(Σ)) ≥ ν(LS(Σ)) = l(−Σ).
(iv) l(LC(Σ)) ≥
∑
v∈V (Σ)
⌊ (d(v)−1)2
4
⌋
.
Proof. (i): Balance of LC(Σ) means that Σ does not contain a vertex of degree 3 or greater,
as the corresponding edges produce negative triangles. Evidently, Σ cannot contain nega-
tive cycles, because this leads to negative cycles in LC(Σ). If Σ is a disjoint union of paths
and positive cycles, then LC(Σ) is again a disjoint union of paths and positive cycles.
(ii): A line graph L(G) has three kinds of cycle. A vertex triangle corresponds to three
edges incident with a single vertex of G; all vertex triangles are negative in LC(Σ). A
line cycle is derived from a cycle C in G and has the same sign in LC(Σ). The remaining
cycles are obtained by concatenation of cycles of the first two kinds. It follows that LC(Σ)
is antibalanced if and only if every line cycle is antibalanced, thus if and only if Σ is
antibalanced.
(iii): An edge set A in −Σ is a set A of vertices in LS(Σ); and LS(Σ)\A = LS(Σ\A).
By (ii), LS(Σ) \ A is balanced if and only if −Σ \ A is balanced. Thus, the smallest size
of an edge set A such that −Σ \A is balanced, which is l(−Σ), equals the smallest size of
a vertex set A such that LS(Σ \A) is balanced, which is ν(LS(Σ)).
The inequality follows from the general observation that l ≥ ν for every signed graph.
(iv): A vertex of degree d(v) in Σ generates in LC(Σ) an antibalanced vertex clique
−Kd(v). An edge set B in the line graph such that LC(Σ) \ B is balanced must contain
enough edges to make each such clique balanced; this number is l(−Kd(v)) =
⌊ (d(v)−1)2
4
⌋
,
obtained by dividing the vertices of Kd(v) into two nearly equal sets and deleting the edges
within each set. Each edge of the line graph is in only one vertex clique, so the minimum
number of edges required to balance every vertex clique is the sum of these quantities. That
proves the inequality.
We do not expect equality in part (iv) because deleting the edges as in the proof may
not eliminate all negative cycles in LC(Σ). The problem of finding an exact formula for
l(LC(Σ)) or l(LS(Σ)) in terms of Σ, or even a good lower bound that involves the signs
of Σ, seems difficult.
3 Total graph(s)
Recall (say, from [6, page 64]) that, for a given graph G, the total graph T (G) is the graph
obtained by combining the adjacency matrix of a graph with the adjacency matrix of its
line graph and its vertex-edge incidence matrix. Precisely, the adjacency matrix of T (G)
is given by
AT (G) =
(
AG B
B⊺ AL(G)
)
,
where L(G) denotes the line graph of G and B is the (unoriented) incidence matrix. Is it
possible to have an analogous concept for signed graphs?
We will give a positive answer to the latter question. However, since we have multiple
possibilities for line graphs of signed graphs, we build multiple total graphs.
34 Ars Math. Contemp. 23 (2023) #P1.02 / 27–43
3.1 Definitions of a total graph
Definition 3.1. The total graph of Σ = Gσ is the signed graph determined by
AT∗(Σ) =
(
AΣ Bη
B⊺η AL∗(Ση)
)
, (3.1)
where ∗ ∈ {C, S}.
In other words, in the combinatorial view, T∗(Σ) consists of two induced subgraphs, Σ
and L∗(Ση), along with edges joining a vertex v of Σ to all vertices of L∗(Ση) that arise
from the edges which are incident with v in Σ. The signature on these connecting edges
is given by η, i.e., for a root-graph vertex v and an incident line-graph vertex e, σT (ve) =
η(v, e). Note that this does not specify an orientation of the edge ve. One may adopt the
convention that ηT (v, ve) = +1, or any other convention, according to convenience. We
do not need to do that because we have not defined an incidence matrix for T∗(Σ).
To fix the notation, TC(Σ) and TS(Σ) denote the total graphs defined by the combina-
torial line graph (2.1) and the spectral line graph (2.2). Accordingly, we shall call them
respectively the combinatorial total graph and the spectral total graph. If we want to con-
sider both variants, we shall write T∗(Σ) instead. We need to show that our definition,
regardless of Σ ∈ [Σ] and of its chosen orientation η, gives rise to the same signed graph
up to switching equivalence.
In Figure 2 we illustrate an example of a total graph of a signed graph. The root graph
and the orientation are taken from Figure 1.
s
s s
s
s
s s
s s s
♣♣♣♣♣
♣♣♣♣♣
♣♣♣♣♣
♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣
♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣♣♣♣
♣♣♣♣♣
♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣❆❆❆
❆
✁
✁
✁
✁


❅
❅
❍❍
❍❍
✟✟
✟✟


❍❍❍❍
✁
✁
✁
✁
✟✟✟✟
v1 v2
v3v4
v5
e1
e2e4
e3
e5
TC(Σ)
s
s s
s
s
s s
s s s
♣♣♣♣♣
♣♣♣♣♣
♣♣♣♣♣
♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣
♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣♣♣♣
♣♣♣♣♣
♣♣♣♣♣♣
♣♣♣♣♣♣ ♣ ♣ ♣ ♣ ♣
❅
❅✏✏
✏✏
✏✏
❆
❆
❆
❆
✁
✁
✁
✁
❍❍
❍❍
✟✟
✟✟
❍❍❍❍
✁
✁
✁
✁
✟✟✟✟
v1 v2
v3v4
v5
e1
e2e4
e3
e5
TS(Σ)
Figure 2: The combinatorial and the spectral total graphs resulting from Ση depicted in
Figure 1.
We show that our definitions of a total graph are stable under reorientation and switch-
ing. For reorientation we have the first lemma.
Lemma 3.2. Let Σ = Gσ be a signed graph, and Ση and Ση′ two orientations of Σ. Then
T∗(Ση) and T∗(Ση′) are switching equivalent, for each ∗ ∈ {C, S}.
Proof. For the sake of readability, we will restrict the discussion to the combinatorial line
graph defined by Zaslavsky. Hence, hereafter L(Σ) := LC(Σ) and T (Ση) := TC(Ση).
Let G = (V,E), where |V | = n and |E| = m. Suppose that η and η′ differ on
some set F ⊆ E, and let Bη and Bη′ be the corresponding vertex-edge incidence matri-
ces, respectively. Let S = (sij) be the m×m diagonal matrix such that sii = −1 if ei ∈ F
F. Belardo et al.: Total graph of a signed graph 35
and sii = 1, otherwise. Then Bη′ = BηS. Since S = S⊺ = S−1, in view of (2.1), we
have
AL(Ση′ ) = 2I −B
⊺
η′Bη′ = S
⊺(2I −B⊺ηBη)S = S⊺AL(Ση)S.
Therefore,
AT (Ση′ ) =
(
AΣ Bη′
B⊺η′ AL(Ση′ )
)
=
(
AΣ BηS
S⊺B⊺η S
⊺AL(Ση)S
)
=
(
I O
O S⊺
)(
AΣ Bη
B⊺η AL(Ση)
)(
I O
O S
)
=
(
I O
O S
)−1
AT (Ση)
(
I O
O S
)
.
This completes the proof.
Next, we prove that switching equivalent signed graphs produce switching equivalent
total graphs.
Lemma 3.3. If Σ and Σ′ are switching equivalent, then T∗(Σ) and T∗(Σ′) are switching
equivalent as well, for each ∗ ∈ {C, S}.
Proof. The notation is the same as in Lemma 3.2. Since Σ and Σ′ are switching equivalent,
their adjacency matrices are switching similar. Hence, AΣ = S−1AΣ′S for some switching
matrix S. Observe that if B = Bη is a vertex-edge incidence matrix of Σ, then B′ = SBη
is a vertex-edge incidence matrix of Σ′. Additionally, in view of (2.1), we have AL(Σ) =
2I −B⊺B.
Therefore, (for Σ′) we have
AT (Σ′) =
(
AΣ′ B
′
B′⊺ AL(Σ′)
)
=
(
AΣ′ B
′
B′⊺ 2I −B′⊺B′
)
=
(
S−1AΣS SB
(SB)⊺ 2I −B⊺(S⊺S)B
)
=
(
S O
O I
)−1 (
AΣ B
B⊺ AL(Σ)
)(
S O
O I
)
=
(
S O
O I
)−1
AT (Σ)
(
S O
O I
)
.
Hence, T (Σ) is switching equivalent to T (Σ′), and we are done.
In view of Lemmas 3.2 and 3.3 the definition given by (3.1) can be used for spectral
investigations.
Remark 3.4. A careful reader has probably noticed that the switching matrix in Lemma 3.2
is the one realizing switching equivalence between the line graphs, while the switching
matrix in Lemma 3.3 is the one realizing switching equivalence between the root signed
graphs. In general, if we have two switching equivalent total graphs, then the switching
matrix will be obtained by combining the switching matrices of the corresponding root and
line graphs.
36 Ars Math. Contemp. 23 (2023) #P1.02 / 27–43
Remark 3.5. The spectral total graph TS(·) does not generalize the total graph of an un-
signed graph, because with the TS operator the total graph of an all-positive signed graph
does not have an all-positive signature, as the convention of treating unsigned graphs as
all positive and the definition of the unsigned total graph would imply. On the other
hand, if we consider unsigned graphs as signed graphs with the all-negative signature, then
TC(−G) = −T (G), so that TC can be considered as the generalization to signed graphs
of the unsigned total graph operator. This observation lends some support to using the line
graph operator LC in spectral graph theory and treating unsigned graphs as all negative,
though contrary to existing custom.
3.2 Properties of total graphs
Now we study some structural and spectral properties of TC(Σ) and TS(Σ). We begin by
computing the number of triangles of T∗(Σ).
Theorem 3.6. Let an unsigned graph G have order n, size m, degree sequence
(d1, d2, . . . , dn), and t triangles. Then the number of triangles of T∗(G) is 2t + m +∑n
i=1
(
di+1
3
)
.
Proof. Every triangle of T∗(G) is one of the following four types:
(a) belongs to G,
(b) belongs to L∗(G),
(c) has 1 vertex in G and 2 vertices in L∗(G),
(d) has 2 vertices in G and 1 vertex in L∗(G).
Every triangle of L∗(G) arises from a triplet of adjacent edges of G. Such a triplet ei-
ther forms a triangle or has a common vertex. Therefore, L∗(G) contains t +
∑n
i=1
(
di
3
)
triangles.
Every triangle of type (c) arises from a pair of adjacent edges of G, so their number
is
∑n
i=1
(
di
2
)
.
Every triangle of type (d) arises from an edge of G, so their number is m.
Altogether, the number of triangles is
t+ t+
n∑
i=1
(
di
3
)
+
n∑
i=1
(
di
2
)
+m = 2t+m+
n∑
i=1
(
di + 1
3
)
,
as claimed.
Moreover, we can establish which triangles are either positive or negative.
Theorem 3.7. Let Σ = Gσ be a signed graph of order n, size m, degree sequence
(d1, d2, . . . , dn), and t = t+ + t− triangles, where t+ (resp., t−) denotes the number of
positive (resp., negative) triangles. Then TC(Σ) has exactly 2t+ positive triangles, while
TS(Σ) has exactly t+m negative triangles.
Proof. The total number of triangles is computed in Theorem 3.6.
F. Belardo et al.: Total graph of a signed graph 37
We have the following facts that the reader can easily check. The triangles of type (a)
will keep their sign in the total signed graph. Hence, we have t+ positive triangles for
TC(Σ) and t− negative triangles for TS(Σ).
A positive (negative) triangle in Σ becomes a positive (negative) triangle in LC(G),
and a negative (positive) triangle in LS(G). A set of mutually adjacent edges in G will
give rise to a complete graph in L∗(G) whose signature is equivalent to the all-negative
(resp., all-positive) one for LC(Σ) (resp., LS(Σ)). Summing up, the triangles of type (b)
will be t+ positive for TC(Σ) and t+ negative for TS(Σ).
Next, let us consider a triangle of type (c). Such a triangle of T∗(Σ) is obtained from
two edges, say vu and vw, of Σ that are incident to the same vertex v. Regardless of
σ(vu) and σ(vw), we can assign an orientation such that the arrows from the side of v are
both inward (directed towards v). Hence, the edges {v, vw} and {v, uv} of T∗(Σ) will be
positive, while the edge {vu, vw} will be negative (resp., positive) in TC(Σ) (resp., TS(Σ)).
Finally, a triangle of type (d) comes from a pair of adjacent vertices u and v and the
joining edge uv. Again, regardless of σ(uv) and with a similar reasoning as above, the
resulting triangle will always be negative in T∗(Σ).
Now, the statement easily follows by counting the positive (negative) triangles of TC(Σ)
(resp., TS(Σ)).
Remark 3.8. From Theorem 3.7 we easily deduce that TC(Σ) and TS(Σ) have in general
switching inequivalent signatures which are not the opposite of each other. Hence, in con-
trast to the line graphs defined by (2.1) and (2.2), the total graphs derived from them have
unrelated signatures.
We conclude this section by analysing the degree of imbalance of these compound
graphs. A vertex cover of a graph is a set of vertices such that every edge has at least one
end in the cover. The smallest size of a vertex cover is the vertex cover number, τ .
Theorem 3.9. Let Σ = Gσ be a signed graph of order n, size m, and vertex cover num-
ber τ . The following hold true:
(i) T∗(Σ) is balanced if and only if G is totally disconnected.
(ii) T∗(Σ) is antibalanced if and only if either ∗ = S and Σ has no adjacent edges, or
∗ = C and Σ is antibalanced.
(iii) l(T∗(Σ)) ≥ m+ l(L∗(Σ)), with equality when ∗ = S, and also when ∗ = C and Σ
is a disjoint union of paths and cycles.
(iv) l(T∗(Σ)) = m if and only if either ∗ = S and Σ is antibalanced, or ∗ = C and Σ is
a disjoint union of paths and positive cycles.
(v) ν(T∗(Σ)) ≥ τ , with equality if ∗ = S and Σ is antibalanced.
(vi) ν(TS(Σ)) ≤ τ + ν(LS(Σ)).
(vii) The largest (adjacency) eigenvalue λ of T∗(Σ) satisfies
λ ≤ max
{
−di +
√
5d2i + 4(dimi − 4)
2
: 1 ≤ i ≤ n+m
}
,
where di and mi denote, respectively, the degree of a vertex i of T∗(Σ) and the
average degree of its neighbours.
38 Ars Math. Contemp. 23 (2023) #P1.02 / 27–43
Proof. (i): Each edge of Σ leads to a negative triangle of type (d), so T∗(Σ) is balanced if
and only if Σ has no edges.
(ii): Adjacent edges lead to a positive triangle of type (c) in TS(Σ), hence it cannot be
antibalanced. If there are no adjacent edges, TS(Σ) consists only of negative triangles of
type (d) and any isolated vertices of Σ, which is antibalanced.
There are four kinds of cycle to consider in TC(Σ): triangles of types (c) and (d) and
cycles in Σ and LC(Σ). The triangles are negative, hence antibalanced. If Σ is not an-
tibalanced, the total graph cannot be, but if Σ, thus also LC(Σ) by Theorem 2.4(ii), is
antibalanced, then it follows – from the fact that all cycles in the total graph are obtained
by combining cycles of those four kinds – that the total graph is antibalanced.
(iii): The m triangles of type (d) of the proof of Theorem 3.6 are negative and indepen-
dent, in the sense that no two of them share the same edge. Therefore, to eliminate each
of them it is necessary to delete m edges (for example, the edges of Σ in T∗(Σ)). Deleting
these edges does not change the frustration index of the subgraph L∗(Σ), so at least an
additional l(L∗(Σ)) edges must be deleted to attain balance of T∗(Σ). Hence, we have the
inequality.
Equality holds for TS(Σ) because the triangles of type (c) are positive. In TC(Σ) those
triangles are negative. When Σ is a disjoint union of paths and positive cycles, then the
negative cycles are those of type (c) and (d) which share a common edge. Deleting such
independent edges (there are m of them) leads to a balanced signed graph. When Σ has a
negative cycle, one edge in those triangles can be replaced by one edge each in the negative
cycle in Σ and in the corresponding negative cycle in LC(Σ) for a total of one extra edge
for each negative cycle of Σ.
(iv): The equality for TS(Σ) holds under the formulated conditions since there LS(Σ)
is balanced and then the entire TS(Σ) becomes balanced after deleting all m edges of Σ.
For TC(Σ) the result follows from (iii).
Conversely, if l(T∗(Σ)) = m, then L∗(Σ) must be balanced, i.e., it cannot contain a
negative cycle. For ∗ = S, this means that Σ is antibalanced. For ∗ = C, this means that Σ
does not contain a vertex of degree 3 or greater, as the corresponding edges produce nega-
tive triangles. Evidently, Σ cannot contain negative cycles, because this leads to additional
negative cycles in TC(Σ). If Σ is a disjoint union of paths and positive cycles, the equality
follows from (iii).
(v): Consider T∗(Σ). For both variants we need to eliminate (at least) the negative
triangles of type (d). Instead of deleting the edges of Σ, we can just delete a minimum
vertex cover of Σ and obtain the same effect. The equality is obtained for TS(−G).
(vi): If B is a minimum set of vertices of LS(Σ) such that deleting every vertex in B
leaves a balanced line graph LS(Σ), then deleting the same vertices from the line graph in
TS(Σ) while also deleting a minimum vertex cover from Σ in LS(Σ) as in the proof of (v)
eliminates all negative cycles in the total graph.
(vii): Note that, unless G is totally disconnected, every vertex of T∗(Σ) belongs to
at least one negative triangle – this triangle is again of type (d). Accordingly, the result
follows by the inequality of [11]:
λ ≤ max
{−di +√5d2i + 4(dimi − 4t−i )
2
: 1 ≤ i ≤ n+m
}
,
where t−i stands for the number of negative triangles passing through a vertex i.
F. Belardo et al.: Total graph of a signed graph 39
Remark 3.10. It is not the case that ν(TC(Σ)) ≤ τ + ν(LC(Σ)). A counterexample is a
sufficiently long cycle of either sign, for which τ ≈ 12m, ν(LC(Σ)) ≤ 1, and ν(T∗(Σ)) ≈
2
3m ≈ 43τ > τ + 1 due to the negative triangles of types (c) and (d).
4 Total graphs of regular signed graphs
A signed graph Σ = Gσ is said to be r-regular if its underlying graph G is an r-regular
graph.
4.1 Spectrum
We infer that the spectrum of Σ lies in the real interval [−r, r]. We compute the spectrum
of T∗(Σ) by means of the eigenvalues of the root (signed) graph Σ, when it is regular.
Theorem 4.1. Let Σ be an r-regular signed graph (r ≥ 2) with n vertices and eigenvalues
λ1, λ2, . . . , λn. Then:
(i) The eigenvalues of TC(Σ) are 2 with multiplicity ( r2 − 1)n and
1
2
(
2 + 2λi − r ±
√
r2 − 4λi + 4
)
, for 1 ≤ i ≤ n.
(ii) The eigenvalues of TS(Σ) are −2 with multiplicity ( r2 − 1)n and
1
2
(
r − 2±
√
(r − 2λi)2 + 4(λi + 1)
)
, for 1 ≤ i ≤ n.
Proof. The proof is inspired from Cvetković’s proof of the theorem concerning the total
graphs of regular unsigned graphs [6, Theorem 2.19]. Due to the inconsistency between
the concepts of line graphs of unsigned graphs and that of spectral line graphs of signed
graphs, our proof differs at some points, as do the final results.
Since Σ is r-regular, for some incidence matrix B, we have BB⊺ = LΣ = DG−AΣ =
rI −AΣ, A(LC(Σ)) = 2I −B⊺B, and A(LS(Σ)) = B⊺B − 2I .
The characteristic polynomial of TC(Σ) is given by
ΦTC(Σ)(x) =
∣∣∣∣ xI −AΣ −B−B⊺ xI − LC(Σ)
∣∣∣∣
=
∣∣∣∣ xI − rI +BB⊺ −B−B⊺ xI − 2I +BB⊺
∣∣∣∣ .
Multiplying the first row of the block determinant by B⊺ and adding to the second, and
then multiplying the second by 1x−2B and adding to the first one, we get
ΦTC(Σ)(x) =
∣∣∣∣ (x− r)I +BB⊺ + 1x−2((x− r − 1)BB⊺ +BB⊺BB⊺) O(x− r − 1)B⊺ +B⊺BB⊺ (x− 2)I
∣∣∣∣ .
40 Ars Math. Contemp. 23 (2023) #P1.02 / 27–43
Further, we compute
ΦTC(Σ)(x) = (x− 2)
r
2n
∣∣(x− r)I +BB⊺ + 1
x− 2
(
(x− r − 1)BB⊺ +BB⊺BB⊺
)∣∣
=(x− 2) r2n
∣∣xI −AΣ + 1
x− 2
(
(x− r − 1)(rI −AΣ) + (rI −AΣ)2
)∣∣
=(x− 2)( r2−1)n
∣∣A2Σ + (3− 2x− r)AΣ + (x2 + x(r − 2)− r)I∣∣
=(x− 2)( r2−1)n
n∏
i=1
(
λ2i + (3− 2x− r)λi + (x2 + x(r − 2)− r)
)
=(x− 2)( r2−1)n
n∏
i=1
(
x2 + (r − 2− 2λi)x+ λ2i + (3− r)λi − r
)
.
Since the roots of x2 +(r− 2− 2λi)x+λ2i +(3− r)λi − r = 0 are given by 12
(
2+2λi −
r ±
√
r2 − 4λi + 4
)
, (i) follows.
Item (ii) follows similarly, by taking the spectral variant of the line graph.
From the above theorem we can deduce the real interval containing the eigenvalues of
the total graph of a regular signed graph.
Corollary 4.2. Let Σ be an r-regular signed graph with n vertices and eigenvalues λ1 ≥
λ2 ≥ · · · ≥ λn. Then:
(i) The spectrum of TC(Σ), if r ≥ 4, lies in the interval[1
2
(2 + λn − r −
√
r2 − 4λn + 4),
1
2
(2 + λ1 − r +
√
r2 − 4λ1 + 4)
]
.
(ii) The spectrum of TS(Σ), if r ≥ 2, lies in the interval[1
2
(r − 2−
√
(r − 2λn)2 + 4(λn + 1)),
1
2
(r − 2 +
√
(r − 2λn)2 + 4(λn + 1))
]
.
Proof. Consider first TC(Σ). It is routine to check that the function f1(λ) = 12 (2+λ−r+√
r2 − 4λ+ 4) is increasing for λ ∈ [−r, r] when r ≥ 4. Hence, the maximum of f1 is
attained for λ1. The function f2(λ) = 12 (2 + λ− r−
√
r2 − 4λ+ 4) is always increasing,
therefore, its minimum is achieved by λn. Hence, the entire spectrum of TC(Σ) lies in
[f2(λn), f1(λ1)], and we get (i).
Consider next TS(Σ). Analysing the function
f3(λ) =
√
(r − 2λ)2 + 4(λ+ 1) =
√
(2λ− (r − 1))2 + 2r + 3,
we find that it is decreasing for λ ≤ r−12 , increasing for λ ≥ r−12 , and symmetric around
r−1
2 . Since λ1 ≤ r and λn ≤ −1 (this holds for every signed graph by induction on the
number of edges using eigenvalue interlacing), we have r−12 −λn ≥ | r−12 −λ1|. Since our
function is symmetric around r−12 , the last inequality leads to f3(λn) ≥ f3(λ1). Hence,
1
2 (r−2+f3(λn)) is the largest eigenvalue of TS(Σ). There are two candidates for the least
eigenvalue of the same signed graph: 12 (r−2−f3(λn)) and (according to Theorem 4.1(ii))
−2. Taking into account that λn ≤ −1, we get 12 (r− 2− f3(λn)) ≤ −2, and thus the least
eigenvalue is 12 (r − 2− f3(λn)), which completes (ii).
F. Belardo et al.: Total graph of a signed graph 41
4.2 A composition
We next consider a particular composition of spectral total graphs – those whose definition
is based on the spectral line graph. Of course, similar results can be obtained in case of the
combinatorial definition. Some further definitions and notation are needed. The Cartesian
product (see also [8]) of the signed graphs Σ1 = (G1, σ1) and Σ2 = (G2, σ2) is determined
in the following way: (1) Its underlying graph is the Cartesian product G1 ×G2; we state
for the sake of completeness that its vertex set is V (G1)×V (G2), and the vertices (u1, u2)
and (v1, v2) are adjacent if and only if either u1 = v1 and u2 is adjacent to v2 in G2 or
u2 = v2 and u1 is adjacent to v1 in G1. (2) The sign function is defined by
σ((u1, u2), (v1, v2)) =
{
σ1(u1, v1) if u2 = v2,
σ2(u2, v2) if u1 = v1.
For the real multisets S1 and S2, we denote by S1 + S2 the multiset containing all
possible sums of elements of S1 and S2 (taken with their repetition). Especially, if S2
consists of the additive identity repeated i times, then the previous sum is denoted by Si1
and consists of the elements of S1, each with multiplicity increased by the factor i. Let
further Spec(Gσ) denote the spectrum of Gσ .
Inspired by [4, 5], we consider the polynomial p(Gσ) =
∑k
i=0 ciG
i
σ , where c0, c1, . . . ,
ck are non-negative integers (ck ̸= 0), Gσ is a regular signed graph with a fixed orientation,
Giσ is the ith power of Gσ with respect to the spectral total graph operation (that is, G
i
σ =
T i−1S (Gσ), i > 0, and G0σ = K1), ciGiσ denotes the disjoint union of ci copies of Giσ , and
the sum of signed graphs is their Cartesian product.
Theorem 4.3. For an r-regular signed graph Gσ (r ≥ 2) with n vertices,
Spec(p(Gσ)) =
k∑
i=0
Spec(Giσ)
ci , (4.1)
where, for i ≥ 2, Spec(Giσ) is comprised of −2 with multiplicity ( ri−12 − 1)ni−1 and
1
2
(
ri−1 − 2±
√(
ri−1 − 2λ(i−1)j
)2
+ 4
(
λ
(i−1)
j + 1
) )
, for 1 ≤ j ≤ ni−1,
where ri = 2i−1r, n1 = n , ni = n
∏i
j=2(2
j−3r+1), and with λ(i−1)1 , λ
(i−1)
2 , . . . , λ
(i−1)
ni−1
being the eigenvalues of Gi−1σ .
Proof. Since, for a non-negative integer c, Spec(cΣ) = Spec(Σ)c and Spec(Σ1 + Σ2) =
Spec(Σ1) + Spec(Σ2) (for the latter, see [8]), we arrive at (4.1), and thus it remains to
compute Spec(Σi). The case i ≤ 1 is clear; clearly, for i ≥ 2, Spec(Σi) is as in the
theorem, where ri−1 and ni−1 are the vertex degree and the number of vertices of Σi−1.
By the definition of a total graph we have ri = 2ri−1, which along with r1 = r leads
to ri = 2i−1r.
By the same definition, we also have ni = ni−1 +mi−1, where mi−1 = 12ni−1ri−1 is
the number of edges of Σi−1. So, ni = ni−1 + 12ni−1ri−1 = ni−1
(
1
4ri + 1
)
. Solving, we
get
ni = n
i∏
j=2
(1
4
ri + 1
)
= n
i∏
j=2
(
2j−3r + 1
)
,
and we are done.
42 Ars Math. Contemp. 23 (2023) #P1.02 / 27–43
Regarding the last theorem, for r = 0 the resulting spectrum is trivial. For r = 1,
Spec(Σ2) is computed directly, not as in the theorem.
4.3 Eulerian regular digraph
We conclude the paper by offering a result which applies only to a signed graph that is
regular and with all edges positive. Note that, by the definition, an oriented all-positive
signed graph is precisely a directed graph. An eigenvalue of a signed graph Gσ is called
main if there is an associated eigenvector not orthogonal to the all-1 vector j. An orientation
of a (signed) graph is called Eulerian if the in-degree equals the out-degree at every vertex.
Theorem 4.4. Let Σ = Gσ be an r-regular signed graph with the all-positive signature
and an Eulerian orientation η. Then TS(Ση) has exactly two main eigenvalues: r and −2.
Proof. The assumptions of positive edges and Eulerian orientation mean that the row sums
of Bη are zero. Under our assumptions, every block of (3.1) has a constant row sum given
in the following (quotient) matrix
Q =
(
r 0
0 −2
)
.
The spectrum of Q contains the main part of the spectrum of TS(Ση). (The explicit proof
can be found in [12], but the reader can also consult [6, Chapter 4] or [1].) By definition,
every signed graph has at least one main eigenvalue. If TS(Ση) has exactly one main
eigenvalue, then the eigenspace of any other is orthogonal to j, which implies that j is
associated with the unique main eigenvalue, but this is impossible (for TS(Ση), observe Q).
Therefore, both eigenvalues of Q are the main eigenvalues of TS(Ση), and we are done.
ORCID iDs
Francesco Belardo https://orcid.org/0000-0003-4253-2905
Zoran Stanić https://orcid.org/0000-0002-4949-4203
Thomas Zaslavsky https://orcid.org/0000-0003-0851-7963
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.03 / 45–56
https://doi.org/10.26493/1855-3974.2637.f61
(Also available at http://amc-journal.eu)
Domination type parameters of Pell graphs*
Arda Buğra Özer
Middle East Technical University, Northern Cyprus Campus, Kalkanlı, Güzelyurt,
Mersin 10, Turkey
Elif Saygı
Department of Mathematics and Science Education, Hacettepe University, Ankara, Turkey
Zülfükar Saygı †
Department of Mathematics, TOBB University of Economics and Technology, Ankara,
Turkey
Received 20 May 2021, accepted 7 March 2022, published online 12 September 2022
Abstract
Pell graphs are defined on certain ternary strings as special subgraphs of Fibonacci
cubes of odd index. In this work the domination number, total domination number, 2-
packing number, connected domination number, paired domination number, and signed
domination number of Pell graphs are studied. Using integer linear programming, exact
values and some estimates for these numbers of small Pell graphs are obtained. Further-
more, some theoretical bounds are obtained for the domination numbers and total domina-
tion numbers of Pell graphs.
Keywords: Pell graphs, Fibonacci cube, domination number, integer linear programming.
Math. Subj. Class. (2020): 05C69, 68R10
1 Introduction
One of the basic models for interconnection networks is the n-dimensional hypercube
graph Qn. It has 2n vertices, represented by all binary strings of length n, and two ver-
tices in Qn are adjacent if they differ in exactly one coordinate. For convenience, we set
*This work is partially supported by TÜBİTAK under Grant No. 120F125. The authors are grateful to the
anonymous reviewers for their valuable comments and careful reading.
†Corresponding author.
E-mail addresses: abozer@metu.edu.tr (Arda Buğra Özer), esaygi@hacettepe.edu.tr (Elif Saygı),
zsaygi@etu.edu.tr (Zülfükar Saygı)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
46 Ars Math. Contemp. 23 (2023) #P1.03 / 45–56
Q0 = K1. The n dimensional Fibonacci cube Γn is defined as the subgraph of Qn induced
by the vertices whose string representations are Fibonacci strings. They were introduced
by Hsu [10] as an alternative model for interconnection networks and extensively studied
in the literature [13]. There are numerous subgraphs and variants of Fibonacci cubes in the
literature, such as Lucas cubes [15], generalized Fibonacci cubes [11], k-Fibonacci cubes
[5] and Pell graphs [14].
Let G = (V,E) be a graph with vertex set V = V (G) and edge set E = E(G). A set
D ⊆ V is called a dominating set of G if every vertex in V \D is adjacent to some vertex
in D. Then the domination number γ(G) of G is defined as the minimum cardinality of a
dominating set of G. Similarly, a set D ⊆ V is called a total dominating set of a graph G
without isolated vertices, if every vertex in V is adjacent to some vertex in D and the total
domination number γt(G) ofG is defined as the minimum cardinality of a total dominating
set of G.
The domination type parameters of Fibonacci and Lucas cubes are first considered in
[3, 17]. Using integer linear programming, domination and total domination numbers of
these cubes and some additional domination type parameters of these cubes [2, 12] and hy-
percubes [2] are considered in the literature. Furthermore, upper bounds and lower bounds
on domination and total domination numbers of Fibonacci and Lucas cubes are obtained in
[2, 3, 17, 18, 19, 20]. The domination and total domination number of k-Fibonacci cubes
are considered in [6]. In this work, we studied some domination type parameters of Pell
graphs.
2 Preliminaries
Let fn denote the Fibonacci numbers defined as f0 = 0, f1 = 1 and fn = fn−1 + fn−2
for n ≥ 2. Similarly, let pn denote the Pell numbers defined as p0 = 1, p1 = 2 and
pn = 2pn−1 + pn−2 for n ≥ 2. Here we remark that the generating function of pn (see,
for example [9]) is ∑
n≥0
pnx
n =
1
1− 2x− x2
. (2.1)
Binary strings of length n not containing two consecutive 1s constitute the set of Fi-
bonacci strings Fn of length n, that is, the binary strings b1b2 . . . bn such that bi · bi+1 = 0
for all i = 1, 2, . . . , n− 1.
Ternary strings over the alphabet {0, 1, 2} where there are no maximal blocks of 2s
of odd length constitute the set of Pell strings, Pn. Then the n dimensional Pell graph,
Πn, is defined as the simple graph where the vertices are represented by the Pell strings
of length n, and two vertices are adjacent whenever one of them can be obtained from the
other by replacing a 0 with a 1 (or vice versa), or by replacing a factor 11 with 22 (or vice
versa) [14]. The vertices of Πn can be partitioned into vertices that start with 0, vertices
that start with 1 and vertices that start with 22. The subgraphs induced by these vertices
are isomorphic to Πn−1, Πn−1, and Πn−2, respectively. This gives the following canonical
decomposition of Pell graphs for n ≥ 2
Πn = 0Πn−1 + 1Πn−1 + 22Πn−2, (2.2)
where Π0 = K1 and Π1 = K2. Here remark that we have also to add the edges of perfect
matchings between 0Πn−1 and 1Πn−1; and also between 22Πn−2 and 11Πn−1 (an induced
subgraph of 1Πn−1).
A. Buğra Özer et al.: Domination type parameters of Pell graphs 47
Every Pell string decomposes uniquely into the product of the factors 0, 1 and 22. Let
ψ : Pn → F2n where ψ(0) = 10, ψ(1) = 00 and ψ(22) = 0100. Hence, we know that
ψ maps any Pell string of length n to a unique Fibonacci string of length 2n with no 0101
factors and without a final 1, which are called Pell binary strings. For a graphG, we denote
a subgraphH ofG byH ⊆ G. Then using this notation and the ψ mapping it is shown that
Theorem 2.1 ([14, Theorem 7]). For n ≥ 1, we have the inclusion Πn ⊆ Γ2n−1.
Let Γ∗2n be the Hamming graph generated by the set of all Pell binary strings of length
2n then we have the following result showing that Πn is isomorphic to an induced subgraph
of Γ2n−10.
Theorem 2.2 ([14, Theorem 8]). The graphs Πn and Γ∗2n are isomorphic.
Let N(v) denote the open neighborhood of v ∈ V , that is, the set of vertices adjacent
to v, and N [v] = N(v) ∪ {v}. Using Theorem 2.1, we have the following Lemma.
Lemma 2.3. Let v ∈ Πn ⊆ Γ2n−10. For any u ∈ N(v) ⊆ Γ2n−10, the binary string
representation of u can not have two non-overlapping 0101 factors as a substring.
Proof. Assume that there is a vertex u ∈ N(v) of the form α10101α20101α30 ∈ F2n−10.
Then we know that the distance between u and v in Γ2n−1 is 1. Hence, v should have a
0101 factor, which is a contradiction.
Let α0(0101)0β ∈ F2n for some Fibonacci strings α and β which do not have a 0101
factor. Let us define the maps ϕ1, ϕ2 and ϕ from F2n into F2n by setting
ϕ1(α0(0101)0β) = α0(0001)0β,
ϕ2(α0(0101)0β) = α0(0100)0β,
ϕ(α0(0101)0β) = α0(0000)0β.
3 Main results
We first interrelate the domination and total domination numbers of Fibonacci cubes and
Pell graphs using Theorem 2.1 and Lemma 2.3.
Proposition 3.1. For any positive integer n, we have
(i) γ(Πn) ≤ γ(Γ2n−1)
(ii) γt(Πn) ≤ γt(Γ2n−1)
Proof. (i) Let D be a minimal dominating set of Γ2n−1 and set
D′ = {α | α is a Pell binary string from D0}∪
∪ {ϕ(β0) | β0 ∈ D0 has one 0101 factor}.
Note that |D′| ≤ |D|. Let u be a vertex of Πn. Then the vertex ψ(u) is dominated in
Γ2n−10 by some d0 ∈ D0. If d0 is a Pell binary string then d0 belongs to D′. If d0 is not
a Pell binary string then we know that it has only one 0101 factor and ψ(u) must be of the
form ϕ1(d0) or ϕ2(d0), which are also dominated by a Pell binary string ϕ(d0). Then we
obverse that D′ is a dominating set of Πn. Hence, we have γ(Πn) ≤ γ(Γ2n−1).
48 Ars Math. Contemp. 23 (2023) #P1.03 / 45–56
(ii) Using the same argument in the previous part, assume that D is a minimal total
dominating set of Γ2n−1. Then we merely need to show that D′ is a total dominating set.
Since D is a total dominating set in Γ2n−1, we know that every vertex v ∈ V (Πn) ⊆
V (Γ2n−10) must be adjacent to some vertex w ∈ D0. If w ∈ D′, there is nothing to show.
Otherwise, w must have one 0101 factor. Since Pell binary string representations of the
vertices in Πn do not have a 0101 factor, v ∈ V (Πn) must be of the form ϕ1(w) or ϕ2(w).
Hence, v is also adjacent to ϕ(w) ∈ D′.
Using the canonical decomposition (2.2) of Πn, we obtain the following results.
Proposition 3.2. For any integer n ≥ 3, we have
(i) γ(Πn) ≤ 2γ(Πn−1) + γ(Πn−2)
(ii) γt(Πn) ≤ 2γ(Πn−1) + γt(Πn−2)
(iii) γ(Πn) ≤ γt(Πn) ≤ 5γ(Πn−2) + 2γ(Πn−3)
Proof. (i) This follows directly from the canonical decomposition (2.2) of Pell graphs.
(ii) Let D1 be a dominating set for Πn−1 and D2 be a total dominating set for Πn−2.
From (2.2) we know that there is a perfect matching between 0Πn−1 and 1Πn−1. Using
this perfect matching, we conclude that the set 0D1 ∪ 1D1 ∪ 22D2 is a total dominating set
for Πn, which gives the desired result.
(iii) This follows from using the canonical decomposition (2.2) of Pell graphs recur-
sively and the perfect matchings between the induced subgraphs, namely 5 copies of Πn−2
and 2 copies of Πn−3.
Considering the vertices of high degrees, lower bounds on γ(Γn) and γ(Λn) are ob-
tained in [17, Theorem 3.2] and [3, Theorem 3.5.], respectively. Using the same argument,
we obtain the lower bound for γ(Πn) in Proposition 3.4. Before we introduce this lower
bound, we have the following remark on the degree distribution of the vertices of Πn.
Remark 3.3. We know that Πn is an induced subgraph of Γ2n−10, which means that the
degrees of the vertices of Πn is at most 2n − 1. It is shown in [14, Proposition 27] that
1n is the unique vertex having degree 2n − 1 for n ≥ 2. Using the recursive relation in
[14, Theorem 29], which gives the number of all vertices of Πn having fixed degree, it is
easy to show that for n ≥ 3, there are only 2 vertices having degree 2n− 2 (namely, 01n−1
and 1n−10), and for n ≥ 4 there are exactly n+ 1 vertices having degree 2n− 3. The rest
of the vertices of Πn have degree at most 2n− 4 for n ≥ 4.
Proposition 3.4. For any n ≥ 7, we have γt(Πn) ≥ γ(Πn) ≥
⌈
pn − n− 8
2n− 3
⌉
.
Proof. Let D be a minimum dominating set of Πn and define the over domination of Πn
with respect to D as
OD(Πn) =
(∑
v∈D
(
deg(v) + 1
))
− |V (Πn)| .
A. Buğra Özer et al.: Domination type parameters of Pell graphs 49
Let S = {v ∈ V (Πn) | deg(v) ≥ 2n− 3}. Using Remark 3.3, we have
0 ≤ OD(Πn) = 2n+ 2(2n− 1) + (n+ 1)(2n− 2)− pn +
∑
v∈D\S
(
deg(v) + 1
)
≤ 2n2 + 6n− 4− pn + (|D| − |S|)(2n− 3)
= n+ 8− pn + |D|(2n− 3)
which gives the desired result.
3.1 Integer linear programming for domination numbers
Suppose each vertex v ∈ V (Πn) is associated with a binary variable xv . The problems of
determining γ(Πn) and γt(Πn) can be expressed as problems of minimizing the objective
function ∑
v∈V (Πn)
xv (3.1)
subject to the following constraints for every v ∈ V (Λn):∑
a∈N [v]
xa ≥ 1 (for domination number),
∑
a∈N(v)
xa ≥ 1 (for total domination number).
The value of the objective function (3.1) gives γ(Πn) and γt(Πn), respectively. Note that
this problem has pn binary variables and pn constraints.
We implemented the integer linear programming problem (3.1) on Intel Core i7-10875H
CPU @ 2.30GHz with 32GB RAM running the Ubuntu 20.04 LTS Linux operating system
and using Gurobi Optimizer [8]. We obtain the exact values of γ(Πn) for n ≤ 6 and γt(Πn)
for n ≤ 7. Furthermore, we obtain the estimates 60 ≤ γ(Π7) ≤ 64 (takes approximately
1 hour) and 137 ≤ γ(Π7) ≤ 162 (takes approximately 1 hour) . We collect the values of
γ(Πn) and γt(Πn) that we obtained from (3.1) in Table 1. In Tables 2 and 3 we present
examples of a minimal dominating and total dominating sets that were obtained during the
computation of these values. We also present an example of a dominating set of Π7 having
cardinality 64 in Appendix (see, Table 11).
Table 1: Domination and total domination numbers for small Pell graphs.
n 1 2 3 4 5 6 7 8
|V (Πn)| 2 5 12 29 70 169 408 985
γ(Πn) 1 2 4 7 14 30 60−64
γt(Πn) 2 2 4 9 16 34 72 137−162
Using the computation results presented in Table 1, Proposition 3.2 and a simple induc-
tion argument we obtain the following results.
Theorem 3.5. For n ≥ 6, we have γ(Πn) ≤ 22pn−4 − 40pn−5 ; and for n ≥ 9, we have
γt(Πn) ≤ 22pn−4 − 40pn−5 .
50 Ars Math. Contemp. 23 (2023) #P1.03 / 45–56
Proof. From Proposition 3.2 and Table 1, we know that
γ(Πn) ≤ 2γ(Πn−1) + γ(Πn−2) (3.2)
and γ(Π6) = 30, γ(Π7) ≤ 64. We set s6 = 30, s7 = 64 and sn = 2sn−1 + sn−2 for
n ≥ 8. Using (3.2), one can easily see that γ(Πn) ≤ sn for n ≥ 6. Let S =
∑
n≥0 sn+6x
n
be the generating function of the sequence sn+6. Therefore, S satisfies
S − 30− 64x = 2x(S − 30) + x2S
which gives
S =
30 + 4x
1− 2x− x2
.
Then using (2.1), we obtain sn+7 = 30pn+1 + 4pn for n ≥ 0 and s6 = 30p0. This is
equivalent to sn = 22pn−4 − 40pn−5 for all n ≥ 6. Using a similar argument, we obtain
the desired result for the total domination number.
Remark 3.6. For any graph G of minimum degree δ, a general upper bound due to Arnau-
tov [1] and Payan [16] is
γ(G) ≤ |V (G)|
δ + 1
δ+1∑
j=1
1
j
. (3.3)
We know that δ(Πn) = ⌈n2 ⌉ (cf. [14, Proposition 27]). Computing the upper bound in
Theorem 3.5 and the right-hand side of the bound (3.3) for γ(Πn), we observe that our
bound from Theorem 3.5 is better than the bound from (3.3) for n ≤ 44.
Table 2: Example of a minimal dominating set for Π6.
000000, 000221, 001022, 001101, 001110, 001122,
010011, 010110, 012200, 022000, 022111, 022220,
100011, 100220, 101100, 102211, 110101, 110122,
111001, 111010, 111221, 112211, 112222, 122022,
122100, 220000, 220022, 220220, 221111, 222200.
Table 3: Example of a minimal total dominating set for Π7.
0000000, 0001022, 0001122, 0001220, 0001221, 0002211, 0010000, 0010011,
0010111, 0011100, 0012211, 0022011, 0022100, 0100101, 0100111, 0101010,
0101101, 0110220, 0111220, 0122011, 0122111, 0122220, 0220111, 0220122,
0221000, 0221001, 0221122, 0222200, 0222210, 1000110, 1000111, 1001001,
1002200, 1010111, 1011001, 1011010, 1011110, 1022111, 1022122, 1022221,
1100022, 1100220, 1101010, 1102200, 1102210, 1102222, 1110001, 1110010,
1110022, 1110100, 1110220, 1111022, 1112201, 1112222, 1122000, 1122001,
1220010, 1220100, 1221111, 1221221, 2200000, 2200111, 2201000, 2201111,
2201221, 2210001, 2210111, 2211022, 2211110, 2212201, 2222110, 2222111.
A. Buğra Özer et al.: Domination type parameters of Pell graphs 51
3.2 Additional domination type parameters of small Pell graphs
By using the integer linear programming approach several additional parameters of small
Fibonacci cubes, Lucas cubes and k-Fibonacci cubes are obtained in [2, 6, 12, 20]. In
this section we use a similar approach to obtain domination type parameters of small Pell
graphs. For completeness of the paper, we first give the definition of these parameters and
corresponding linear optimization problems similar to (3.1).
A set X ⊆ V is a 2-packing if the distance d(u, v) ≥ 3 for any u, v ∈ X , u ̸= v. The
maximum size of a 2-packing of G is the 2-packing number of G denoted ρ(G). It can be
determined using the following optimization problem:
ρ(G) =max
∑
v∈V
xv
subject to
∑
u∈N [v]
xu ≤ 1, for all v ∈ V.
The independent domination number i(G) is the minimum size of a dominating set that
induces no edges (or, equivalently, the size of the smallest maximal independent set), which
can be determined using the following optimization problem:
i(G) =min
∑
v∈V
xv
subject to
∑
u∈N [v]
xu ≥ 1, for all v ∈ V
(|V | − 1)xv +
∑
u∈N(v)
xu ≤ |V | − 1, for all v ∈ V.
A set X ⊆ V is a k-tuple dominating set of G if for every vertex v ∈ V we have
|N [v]∩X| ≥ k, that is, v ∈ X and has at least k−1 neighbors in S or v ∈ V \X has at least
k neighbors inX . The k-tuple domination number γ×k(G) is the minimum cardinality of a
k-tuple dominating set of G. Clearly, γ(G) = γ×1(G) ≤ γ×k(G), while γt(G) ≤ γ×2(G)
and γ×k(G) can be determined using the following optimization problem:
γ×k(G) =min
∑
v∈V
xv
subject to
∑
u∈N [v]
xu ≥ k, for all v ∈ V.
Specifically, a k-tuple dominating set where k = 2 is called a double dominating set and in
this work we determine double domination number γ×2(Πn) of small Pell graphs.
A function f : V → {−1, 1} is called a signed dominating function if
∑
u∈N [v] f(u) ≥
1 holds for every v ∈ V [4]. The signed domination number γs(G) of G is the minimum
of
∑
v∈V f(v) taken over all signed dominating functions f of G and it can be determined
using the following optimization problem [2]:
γs(G) =min
∑
v∈V
(2xv − 1)
subject to
∑
u∈N [v]
(2xu − 1) ≥ 1, for all v ∈ V.
52 Ars Math. Contemp. 23 (2023) #P1.03 / 45–56
Here we note that binary variables xv associated with every vertex v ∈ V indicates whether
v is assigned weight 1 (xv = 1) or −1 (xv = 0).
The connected domination number γc(G) is the order of a smallest dominating set that
induces a connected graph. We used the Miller-Tucker-Zemlin constraints to find a minimal
connected domination set for Pell graphs [7].
The paired domination number γp(G) is the order of a smallest dominating set S ⊆ V
s.t. the graph induced by S contains a perfect matching. We associate to each edge e =
uv ∈ E a binary variable xe = xuv indicating whether e is present in the graph induced by
a paired dominating set. Then the following optimization problem determines γp(G) [2]:
γp(G) = 2 ·min
∑
e∈E
xe
subject to
∑
u∈N(v)
xuv ≤ 1, for all v ∈ V
∑
u∈N(v)
∑
w∈N(u)
xuw ≥ 1, for all v ∈ V.
Using the integer linear programming approaches described in this section, we obtain
the values and estimates of ρ(Πn), i(Πn), γ×2(Πn), γs(Πn), γc(Πn), γp(Πn) for some
small values of n and collect these results in Table 4. Furthermore, in Tables 5, 6, 7, 9
and 10 in Appendix, we present example of a set of vertices giving ρ(Πn) and γp(Πn) for
n = 7, γc(Πn) for n = 5, and i(Πn) and γ×2(Πn) for n = 6 that were obtained during the
computation of these values. In Table 8, we also present the set of vertices v ∈ V (Π6) for
which f(v) = −1, where f is a signed dominating function giving γs(Π6) = 45.
Table 4: Values of additional domination type parameters for small Pell graphs.
n 1 2 3 4 5 6 7
|V (Πn)| 2 5 12 29 70 169 408
ρ(Πn) 1 2 3 6 11 22 46
i(Πn) 1 2 4 7 15 31 60−69
γ×2(Πn) 2 4 7 13 27 56 113−121
γs(Πn) 2 3 4 11 20 45 88−102
γc(Πn) 1 2 4 9 18 35−38 66−82
γp(Πn) 2 2 4 10 16 34 72
ORCID iDs
Arda Buğra Özer https://orcid.org/0000-0002-6505-7038
Elif Saygı https://orcid.org/0000-0001-8811-4747
Zülfükar Saygı https://orcid.org/0000-0002-7575-3272
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A. Buğra Özer et al.: Domination type parameters of Pell graphs 55
Appendix
Table 5: Example of a 2-packing set for Π7.
0000110, 0001001, 0010000, 0010122, 0011221, 0012201, 0022022, 0022110,
0100022, 0100221, 0101100, 0102222, 0110101, 0111011, 0122000, 0220010,
0221122, 0221220, 0222200, 1000101, 1001022, 1002210, 1010011, 1010220,
1011100, 1012222, 1022001, 1100010, 1101111, 1122122, 1122220, 1220022,
1220100, 1220221, 1221001, 1222211, 2200122, 2200220, 2201000, 2202201,
2210001, 2211022, 2211221, 2212210, 2222010, 2222101.
Table 6: Example of a minimal independent dominating set for Π6.
000000, 000221, 001011, 001122, 001220, 002200, 010011, 010110,
010122, 011101, 012211, 022000, 022022, 022220, 100022, 100101,
100110, 101000, 102211, 111001, 111010, 111100, 111221, 112222,
122111, 220000, 220111, 220220, 221022, 222201, 222210.
Table 7: Example of a minimal double dominating set for Π6.
000000, 000022, 000111, 000220, 001022, 001100, 001101, 001110,
002211, 010000, 010010, 010101, 010220, 011011, 011101, 011122,
011221, 012210, 012211, 022000, 022011, 022022, 022110, 022220,
100001, 100110, 100111, 101001, 101010, 101221, 102200, 102211,
102222, 110022, 110100, 110122, 111010, 111022, 111122, 111220,
111221, 112200, 122000, 122101, 122111, 220001, 220011, 220100,
220220, 220221, 221001, 221010, 221110, 221122, 222201, 222211.
56 Ars Math. Contemp. 23 (2023) #P1.03 / 45–56
Table 8: The set of vertices v ∈ V (Π6) for which f(v) = −1, where f is a signed
dominating function giving γs(Π6) = 45.
000001, 000010, 000101, 000110, 000122, 001010, 001022, 001101,
001110, 001221, 002222, 010001, 010100, 010122, 010220, 011001,
011011, 011100, 011111, 011221, 012200, 012211, 022010, 022022,
022100, 022122, 022220, 100000, 100011, 100111, 100221, 101000,
101022, 101101, 101110, 102201, 102210, 102222, 110000, 110011,
110100, 110111, 110220, 111011, 111110, 111111, 122001, 122010,
122101, 122220, 220010, 220022, 220101, 220122, 220220, 221001,
221010, 221101, 221221, 222200, 222210, 222222.
Table 9: Example of a minimal connected dominating set for Π5.
00011, 00101, 00111, 00221, 01000, 01111,
01122, 02201, 02211, 10011, 11000, 11100,
11110, 11111, 11122, 11220, 22011, 22111.
Table 10: Example of a minimal paired dominating set for Π7.
0000000, 0000100, 0000220, 0001022, 0001122, 0001220, 0010111, 0011001,
0011010, 0011100, 0011111, 0012200, 0022001, 0022220, 0100011, 0100111,
0101101, 0102201, 0110022, 0111110, 0112222, 0122011, 0122111, 0220000,
0220111, 0221000, 0221110, 0221111, 1000011, 1000111, 1001101, 1002210,
1002211, 1010010, 1011010, 1011101, 1022022, 1022122, 1022220, 1101000,
1101010, 1101221, 1102210, 1110001, 1110010, 1110022, 1110100, 1110101,
1110122, 1110220, 1110221, 1111221, 1112222, 1122000, 1122100, 1220220,
1221011, 1221022, 1222200, 1222201, 2200110, 2200111, 2201000, 2201022,
2201122, 2202201, 2210001, 2211110, 2211220, 2212201, 2222011, 2222111.
Table 11: Example of a dominating set having 64 vertices for Π7.
0000010, 0001001, 0002211, 0010022, 0010101, 0010221, 0011100, 0011110,
0022010, 0022122, 0100000, 0100111, 0101022, 0101220, 0102200, 0110000,
0111100, 0111110, 0112222, 0122001, 0122221, 0220000, 0220122, 0220220,
0221011, 0222201, 1000001, 1000100, 1000122, 1000220, 1001122, 1001221,
1002210, 1011000, 1011011, 1012201, 1022101, 1022220, 1101010, 1101101,
1110010, 1110011, 1110110, 1110111, 1112222, 1122000, 1122022, 1220101,
1221000, 1221022, 1221221, 1222210, 2200022, 2200100, 2200221, 2201010,
2202211, 2210001, 2211001, 2211122, 2211220, 2212200, 2222110, 2222111.
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.04 / 57–79
https://doi.org/10.26493/1855-3974.2704.31a
(Also available at http://amc-journal.eu)
Finitizable set of reductions for polyhedral
quadrangulations of closed surfaces
Yusuke Suzuki *
Department of Mathematics, Niigata University, 8050 Ikarashi 2-no-cho, Nishi-ku,
Niigata, 950-2181, Japan
Received 27 September 2021, accepted 19 March 2022, published online 22 September 2022
Abstract
In this paper, we discuss generating theorems of polyhedral quadrangulations of closed
surfaces. We prove that the set of the eight reductional operations {R1, . . . , R8} de-
fined for polyhedral quadrangulations is finitizable for any closed surface F 2, that is,
there exist finitely many minimal polyhedral quadrangulations of F 2 using such opera-
tions R1, . . . , R7 and R8. Furthermore, we show that any proper subset of {R1, . . . , R8}
is not finitizable for polyhedral quadrangulations of the torus.
Keywords: Generating theorem, reduction, finitizable set, polyhedral quadrangulation.
Math. Subj. Class. (2020): 05C10
1 Introduction
In this paper, we consider simple connected graphs embedded on closed surfaces. Al-
though we follow the standard graph theory terminology, for some technical terms without
description here, refer to Section 2. Sometimes, such an embedded graph is expected to
be a “good” one, that is, every facial walk is a cycle, and any two of them are disjoint,
intersect in one vertex, or intersect in one edge. It is known that a graph G embedded on
the sphere satisfies the above good conditions if and only if G is 3-connected. However,
if G is embedded on a non-spherical closed surface, then G is required to be polyhedral,
i.e., 3-connected and 3-representative; note that 3-connected graphs on the sphere are also
polyhedral.
For example, a simple graph G cellularly embedded on a closed surface F 2 each of
whose face is bounded by a cycle of length 3 is polyhedral if G is not a 3-cycle on the
sphere. Such a graph triangulating a closed surface F 2 is known as a triangulation of
*This work was supported by JSPS KAKENHI Grant Number 20K03714.
E-mail address: y-suzuki@math.sc.niigata-u.ac.jp (Yusuke Suzuki)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
58 Ars Math. Contemp. 23 (2023) #P1.04 / 57–79
F 2. On the other hand, following the convention in topological graph theory, a 4-cycle
embedded on the sphere is regarded as a quadrangulation, which is a graph cellularly
embedded on a closed surface F 2 so that each face is bounded by a cycle of length 4. In
this paper, our main subject is the set of polyhedral quadrangulations of closed surfaces.
In topological graph theory, we sometimes discuss generating theorems of graphs em-
bedded on closed surfaces (i.e., constructing all graphs in a certain class C from C0 ⊂ C
by a repeated applications of certain expanding operations only through C). This notion is
equivalent to that every graph in C can be reduced to one in C0 by a repeated applications of
the reductional operations (or reductions, simply), which are inverses of the above expand-
ing operations; we denote the set of such reductions by X here. In a generating theorem
of graphs, |X| and |C0| are expected to be small. In particular, X is called finitizable for C
if |C0| is finite. If X ′ is not finitizable for any proper subset X ′ ⊂ X , then the finitizable
set X is minimal. For example, if C is the set of simple triangulations of the sphere, then
X = {contraction} is finitizable and C0 = {tetrahedron}. (See [19]. A contraction of e in
a triangulation G is to remove e, identify the two ends of e and replace two pairs of multiple
edges by two single edges respectively.) In fact, it was proved in [2, 3, 7, 16] that for every
closed surface F 2, {contraction} is finitizable for the set of simple triangulations of F 2.
Furthermore, see [1, 9, 10, 20, 21] for the complete lists of minimal triangulations on fixed
non-spherical closed surfaces with low genera. Moreover, finitizable sets of reductions for
even triangulations, i.e., triangulations such that each vertex has even degree, are discussed
in literatures; e.g., see [6, 18].
As mentioned above, in this paper, we focus on quadrangulations of closed surfaces.
Figure 1 shows the eight reductions, denoted by R1, . . . , R7 and R8 simply for our pur-
pose, defined for quadrangulations of closed surfaces. In fact, R1, R2 and R3 are typical
ones which were first given by Batagelj [4] (see e.g., [23] for the formal definition); es-
pecially, R1 and R2 are called a face-contraction and a 4-cycle removal, respectively, in
the literature. Further, the fourth reduction R4 was defined and discussed in [22]; which is
called a cube-contraction in the paper. The other four reductions will be defined in the next
section.
Let C be a set of quadrangulations of a closed surface F 2 with some certain conditions,
and let G ∈ C. For a subset X ⊆ {R1, . . . , R8}, G is X-irreducible if we cannot apply any
reduction in X without violating the condition of C; i.e., the resulting graph is no longer in
C. In particular, an {R1}-irreducible quadrangulation in the set of simple quadrangulations
of a closed surface F 2 is known as just a irreducible quadrangulation of F 2. In [16],
it was proved that for any closed surface F 2 there exist only finitely many irreducible
quadrangulations of F 2, that is, {R1} is finitizable for the set of simple quadrangulations
of every closed surface. Actually, the complete lists of irreducible quadrangulations of the
sphere, the projective plane, the torus and the Klein bottle were obtained in [4, 5, 14, 17]
and [13], respectively; for example, a 4-cycle is the unique irreducible quadrangulation
of the sphere, and the unique quadrangular embeddings of K4 and K3,4 are irreducible
quadrangulations of the projective plane. (Note that a restricted R1 was used in [5].)
The situation for 3-connected (and simple) quadrangulations of closed surfaces is a
little bit complicated in comparison with the above case of irreducible quadrangulations.
Throughout the researches in [4, 5, 12, 15], it had been proved that for any closed surface
F 2, {R1, R2, R3} is finitizable for 3-connected quadrangulations of F 2; note that the min-
imal one on the sphere is the cube, and for any non-spherical closed surface F 2, the set of
the minimal graphs coincides with the set of irreducible quadrangulations of F 2. Further-
Y. Suzuki: Finitizable set of reductions for polyhedral quadrangulations of closed surfaces 59
v0
v0
R6
v6
v5
v4v3
v2
v1
u
v0
v0
v0 R4
v3
v2
v1
u2
R3
v0
v5
v4
v3
v2
v1 u0
u1v0
v1
v2
v3
R2
v0
v1
v2
v3
R1
v0 [v0v3]
[v0v3]
v3
v5
v4
v3
v2
v1
v0
R7
v0
v0 v0
v0
u2
u1v1
R5
v4
v2
v3
v0
R8
v0
v1
v2
v4
v6
v7
v4
[v0v4]
[v0v4]
v3
v5
Figure 1: Reductional operations for quadrangulations.
more, it was shown that {R1, R2, R3} is minimal for those graphs on the sphere and the
projective plane while it is not minimal on the other closed surfaces; in fact, R3 is unneces-
sary and hence {R1, R2} is minimal and finitizable for those closed surfaces. Moreover, it
was proved in [22] that {R1, R3, R4} is minimal and finitizable for 3-connected quadran-
gulations of the sphere and the projective plane, and {R1, R4} is minimal and finitizable
for those graphs on the other closed surfaces.
As mentioned above, in this paper, we deal with polyhedral quadrangulations of closed
surfaces. Recently in [23], the generating theorem for such polyhedral quadrangulations
of the projective plane was discussed using three reductions R1, R2 and R3, and they ob-
tained 26 families of {R1, R2, R3}-irreducible quadrangulations of the projective plane.
However, such families contains infinite series of graphs; i.e., unfortunately, {R1, R2, R3}
is not finitizable for those graphs. The following is our main result in the paper:
Theorem 1.1. For every closed surface F 2, {R1, . . . , R8} is finitizable for polyhedral
quadrangulations of F 2.
Since every reduction in the above theorem preserves bipartiteness of quadrangulations
and each of R5 and R7 requires an essential cycle of length 3, we obtain the following
corollary.
Corollary 1.2. For every closed surface F 2, {R1, R2, R3, R4, R6, R8} is finitizable for
bipartite polyhedral quadrangulations of F 2.
One might think that the eight reductions in Theorem 1.1 are a little bit too many.
However, at least those on the torus, we can show the necessity of such eight reductions as
follows.
Theorem 1.3. For polyhedral quadrangulations of the torus, {R1, . . . , R8} is minimal
finitizable.
60 Ars Math. Contemp. 23 (2023) #P1.04 / 57–79
Furthermore, R7 (resp., R8) requires an annular region on the closed surface which is
bounded by two 2-sided 3-cycles (resp., 4-cycles). Therefore, in particular on the projective
plane, {R1, . . . , R6} is finitizable by Theorem 1.1. As well as the previous case on the
torus, we can show the following.
Theorem 1.4. For polyhedral quadrangulations of the projective plane, {R1, . . . , R6} is
minimal finitizable.
This paper is organized as follows. In the next section, we define terminology and
the remaining four new reductions for our argument in the paper. Next, we show some
propositions and lemmas holding for polyhedral quadrangulations for our purpose, some
of which are quoted from [23]. Section 4 is devoted to prove our main result in the paper.
In Section 5, we discuss the minimality of the set of eight reductions by showing some
infinite series of polyhedral quadrangulations.
2 Basic definitions
We denote the vertex set and the edge set of a graph G embedded on a closed surface F 2
by V (G) and E(G), respectively. A k-path (resp., k-cycle) in a graph G is a path (resp.,
cycle) of length k. (The length of a path (or cycle) is the number of its edges in this paper.)
We denote the set of vertices of degree 3 by V3 in our argument, and ⟨V3⟩G represents the
subgraph induced by V3 in G.
Let G be a graph embedded on a closed surface F 2. Then, a connected component
of F 2 − G is a face of G, and we denote the face set of G by F (G). If every face of
G is homeomorphic to an open 2-cell (or an open disc), then, G is a 2-cell embedding or
2-cell embedded graph on F 2. Clearly, every quadrangulation (or triangulation) of a closed
surface is a 2-cell embedded graph. A facial cycle C of a face f is a cycle bounding f in G;
i.e., C = ∂f . Then, f̄ denotes a closure of f , i.e., f̄ = f ∪ ∂f . For brevity, we sometimes
denote like f = v0v1v2v3 where v0v1v2v3 is a facial cycle of f ∈ F (G). Furthermore
in our argument, we often discuss the interior of a 2-cell region D bounded by a closed
walk W of G, i.e., W = ∂D, which contains some vertices and edges. (Note that a 2-cell
region implies an “open” 2-cell region in this paper.) Similarly, D̄ denotes a closure of D,
i.e., D̄ = D ∪ ∂D. Let f1, . . . , fk denote the faces of G incident to v ∈ V (G) where
deg(v) = k. Then, the boundary walk of f̄1 ∪ · · · ∪ f̄k is the link walk of v and denoted by
lw(v). Clearly, lw(v) bounds a 2-cell region containing a unique vertex v.
A simple closed curve γ on a closed surface F 2 is trivial if γ bounds a 2-cell region on
F 2, and essential otherwise. Among essential simple closed curves, one with an annular
neighborhood is called 2-sided while one whose tubular neighborhood forms a Möbius
band is called 1-sided. Since cycles in graphs embedded on surfaces can be regarded as
simple closed curves, we use the above terminology for them; e.g., we say that a cycle is
essential and 2-sided.
The representativity of G, denoted by r(G), is the minimum number of intersecting
points of G and γ, where γ ranges over all essential simple closed curves on the surface. A
graph G embedded on F 2 is r-representative if r(G) ≥ r. Note that the “representativity”
is also called the “face-width” in the literature; see e.g., [11] for the details. A graph G
embedded on a non-spherical closed surface F 2 is polyhedral if G is 3-connected and 3-
representative. Observe that for every vertex v of a polyhedral graph, the link walk of v
forms a cycle.
Y. Suzuki: Finitizable set of reductions for polyhedral quadrangulations of closed surfaces 61
Let G be a quadrangulation of a closed surface F 2 and let f = v0v1v2v3 be a face
of G. Then a pair {vi, vi+2} is called a diagonal pair of f in G for each i ∈ {0, 1}.
A closed curve γ on F 2 is a diagonal k-curve for G if γ passes only through distinct k
faces f0, . . . , fk−1 and distinct k vertices x0, . . . , xk−1 of G such that for each i, fi and
fi+1 share xi, and that for each i, {xi−1, xi} forms a diagonal pair of fi of G, where the
subscripts are taken modulo k. Furthermore, we call a simple closed curve γ on F 2 a semi-
diagonal k-curve if in the above definition {xi−1, xi} is not a diagonal pair for exactly one
i; note that xi−1xi is an edge of ∂fi in this case. Each simple curve βi along γ joining
xi−1 and xi in fi is called a γ-segment; where
∪k−1
i=0 βi = γ.
For a simple closed curve ℓ on F 2, when ℓ intersects with G at only vertices of G, that
is, G ∩ ℓ is a subset S ⊂ V (G), then we say that ℓ passes S; observe that ℓ does not pass
through any vertex in V (G) \ S in this case. For example, in the above definition of a
diagonal (or semi-diagonal) k-curve, we say that γ passes {x0, . . . , xk−1}. On the other
hand, when we say that ℓ passes through a vertex v (or some vertices) of G, then ℓ probably
passes through other vertices of G.
Let G be a simple quadrangulation of a non-spherical closed surface F 2. Assume that G
has a hexagonal 2-cell region D bounded by a closed walk ∂D = v0v1v2v0v3v4 containing
exactly two vertices u1 and u2 such that v0v1u1v4, v1v2u2u1, v3v4u1u2 and v2v0v3u2 are
faces of G in D, and that v0v1v2 is an essential cycle of length 3. Furthermore, we assume
that v0, v1, v2, v3 and v4 are different vertices, and that each of v1, v2, v3 and v4 has degree
at least 4 ; otherwise, G would not be polyhedral under the condition. A reduction R5
of D is to eliminate u1 and u2, and identify v1 (resp., v2) and v4 (resp., v3), and replace
three pairs of multiple edges by three single edges, respectively, as shown in Figure 1.
Throughout the paper, the vertex obtained by the identification of two vertices a and b is
denoted by [ab]. That is, v0[v1v4][v2v3] is an essential 3-cycle in the resulting graph.
Secondly, assume that G has an octagonal 2-cell region D bounded by a closed walk
W = v0v1v2v3v0v4v5v6 containing exactly one vertex u such that v0v1uv6, v1v2v4u,
v4v5v6u and v2v3v0v4 are faces of G in D, and that v0v1v2v3 is an essential cycle of
length 4. Furthermore, we assume that v0, v1, v2, v3, v4, v5 and v6 are different vertices.
Note that v1 and v4 has degree at least 4 under the condition. (If deg(v1) = 3, then G is
representativity at most 2. On the other hand, deg(v4) = 3 implies that v0 = v5, a contra-
diction.) A reduction R6 of D is to eliminate u and an edge v2v4, and identify v1 (resp.,
v2, v3) and v6 (resp., v5, v4), and replace four pairs of multiple edges by four single edges,
respectively, as shown in Figure 1. Then, v0[v1v6][v2v5][v3v4] is an essential 4-cycle in the
resulting graph.
Thirdly, assume that G has an annular region A bounded by two essential cycles C =
v0v1v2 and C ′ = v3v4v5 such that f1 = v0v1v4v3, f2 = v1v2v5v4 and f3 = v2v0v3v5 are
faces of G in A. (Sometimes, f1f2f3(= WF ) is called a face walk of length 3 in G, which
corresponds to a 3-cycle in the dual of G.) Here, note that C1 and C2 are essential 2-sided
cycles of G on F 2; if C1 is trivial, then it contradicts Proposition 3.2 in the next section.
The seventh reduction R7 of A (or the above face walk WF ) is to contract edges v0v3, v1v4
and v2v5 simultaneously, and replace three pairs of multiple edges by three single edges,
respectively, as shown in Figure 1. Note that C = [v0v3][v1v4][v2v5] is also an essential
2-sided 3-cycle in the resulting graph.
Fourthly, assume that G has an annular region A bounded by two essential cycles
C1 = v0v1v2v3 and C2 = v4v5v6v7 such that f1 = v0v1v6v5, f2 = v1v2v7v6, f3 =
v2v3v0v7 and f4 = v0v5v4v7 are faces of G in A. (As well as the previous reduction,
62 Ars Math. Contemp. 23 (2023) #P1.04 / 57–79
f1f2f3f4(= WF ) is a face walk of length 4.) Furthermore, we assume that C1 and C2
are essential cycles of G on F 2; observe that they are 2-sided. The eighth reduction R8 of
A (or the face walk WF ) is to eliminate edges v0v5, v1v6, v2v7 and v0v7, and identify vi
and vi+4 for each i ∈ {0, 1, 2, 3}, and replace four pairs of multiple edges by four single
edges, respectively, as shown in Figure 1. Note that C = [v0v4][v1v5][v2v6][v3v7] is also
an essential 2-sided 4-cycle in the resulting graph.
As mentioned in the introduction, for R1, R2, R3 and R4, see e.g., [22, 23] for formal
definitions. Note that the boundary of the hexagon of the graph in R3 in the figure is a cycle.
Furthermore, every quadrangulation of a closed surface is locally bipartite, and hence we
color vertices of graphs in R1, R2, R3, R4, R6 and R8 by black and white; however, graphs
in the reductions R5 and R7 contain short odd cycles, and hence we cannot do so.
3 Lemmas
First of all, we introduce the following two propositions for quadrangulations of closed
surfaces; these are well-known in topological graph theory, and hence we omit the proofs.
Proposition 3.1. The length of two essential cycles in a quadrangulation of a closed sur-
face have the same parity if they are homotopic to each other on F 2.
Proposition 3.2. A quadrangulation of a closed surface has no separating odd cycle.
It was shown in [23] that many facts hold for {R1, R2, R3}-irreducible polyhedral
quadrangulations of non-spherical closed surfaces. First, we show some of them, which
will be used in our later argument in the paper. In the following lemmas, G represents
a {R1, R2, R3}-irreducible polyhedral quadrangulations of a non-spherical closed surface
F 2 otherwise specified. (The assertions are a little bit changed so as to suit for this paper.)
Lemma 3.3 (Lemmas 3.5, 3.13 and 3.15 in [23]). Every connected component of ⟨V3⟩G is
a 4-cycle bounding a face of G or a path of length at most 2.
Lemma 3.4 (Lemmas 3.8, 3.10 and 3.12 in [23]). Let f = v0v1v2v3 be a face of G with
deg(v0), deg(v2) ≥ 4. Then, there exists
(i) an essential 4-cycle v0v1xv3 for x /∈ {v0, v1, v2, v3},
(ii) an essential diagonal 3-curve passing through v1 and v3, or
(iii) an essential semi-diagonal 3-curve passing through v1 and v3.
Lemma 3.5. Let f = v0v1v2v3 be a face of G with deg(v0), deg(v2) ≥ 4. Then, there
exists an essential cycle passing through v0, v1 and v3 with length 4, 5 or 6.
Proof. It is clear by Lemma 3.4. (For example, if (ii) in the previous lemma holds, then
there exists an essential cycle of length 6 along the essential diagonal 3-curve.)
Lemma 3.6 (Lemma 3.14 in [23]). Let P = u0u1u2 be a 2-path in ⟨V3⟩G as shown in the
left-hand side of R3 in Figure 1 where deg(v4) ≥ 4. Then, there is an essential diagonal
3-curve or an essential semi-diagonal 3-curve passing {v1, u1, v5}.
Y. Suzuki: Finitizable set of reductions for polyhedral quadrangulations of closed surfaces 63
Assume that G has a 4-cycle C = u0u1u2u3 in ⟨V3⟩G bounding a face of G such that
ui is adjacent to a third vertex vi /∈ {u0, u1, u2, u3} for each i ∈ {0, 1, 2, 3}. Under the
situation, a 4-cycle v0v1v2v3 bounds a 2-cell region which contains exactly four vertices
u0, u1, u2 and u3. We call the subgraph H isomorphic to a cube with eight vertices ui, vi
for i ∈ {0, 1, 2, 3} an attached cube. We denote ∂(H) = v0v1v2v3, and we call C an
attached 4-cycle of H .
Lemma 3.7 (Lemma 3.16 in [23]). Assume that G has an attached cube H with ∂(H) =
v0v1v2v3, an attached 4-cycle C = u0u1u2u3 and uivi ∈ E(G) for each i ∈ {0, 1, 2, 3}.
Then there is an essential diagonal (or semi-diagonal) 3-curve γ passing {v0, u1, v2} or
{v1, u2, v3}.
Next, we show three lemmas holding for {R1, R2, R3, R4}-irreducible polyhedral quad-
rangulations of non-spherical closed surfaces.
Lemma 3.8. Let G be an {R1, R2, R3, R4}-irreducible polyhedral quadrangulation of
a non-spherical closed surface F 2 having an attached cube H with ∂(H) = v0v1v2v3,
an attached 4-cycle C = u0u1u2u3 and uivi ∈ E(G) for each i ∈ {0, 1, 2, 3}. By
Lemma 3.7, we may assume that there exists an essential simple closed curve γ1 pass-
ing {v0, u1, v2}. Then, there exists an essential simple closed curve γ2 passing either
{v1, u2, v3} or {v1, u2, v3, x} where x /∈ V (H). In particular, if γ1 is 2-sided, then γ2 is
not homotopic to γ1.
Proof. Let G′ denote the quadrangulation obtained from G by applying an R4 of H so as
to identify v1 and v3. We denote the 2-path v0[v1v3]v2 in G′ by P . By our assumption, G′
is not polyhedral. If G′ has a loop e, then e is incident to [v1v3] such that e and P cross
transversally at [v1v3]; otherwise, G would have a loop, a contradiction. Further, this e is
essential by Proposition 3.2. Thus in this case, we find an essential semi-diagonal 3-curve
γ2 passing {v1, u2, v3} in G, half of which is along e.
Secondly, we suppose that G′ has a pair of multiple edges. Similar to the previous
case, we may assume that such multiple edges join [v1v3] and another vertex x /∈ {v0, v2};
otherwise, G would have multiple edges. Then, the 2-cycle C = [v1v3]x formed by the
above multiple edges crosses P transversally, similar to the previous case. Thus, C cannot
be trivial by the above observation and the existence of γ1, and hence we have our desired
simple closed curve γ2 passing {v1, u2, v3, x} in G; note that if v1xv3 forms a corner
of a face of G, then we can take an essential diagonal 3-curve passing {v1, u2, v3}. In
the following argument, we assume that G′ is simple and hence G′ is 2-connected and
2-representative.
By the above argument, we may assume that G′ has a diagonal (or semi-diagonal)
2-curve γ′ passing {[v1v3], x} such that γ′ and P cross at [v1v3] transversally; note that
if G′ has a 2-cut, then G′ also has a surface separating diagonal 2-curve by Lemma 3.6
in [23]. Observe that at least one of two γ′-segments β0 and β1, say β0 without loss of
generality, joins the diagonal pair of f0 = [v1v3]sxt for s, t ∈ V (G′). Here, suppose that
x is either v0 or v2, say v0. Then, let β̃0 denote a simple closed curve obtained from β0
by joining [v1v3] and v0 by a simple curve along the edge [v1v3]v0. In this case, β̃0 must
be essential by Proposition 3.2. Under the situation, we can take an essential simple closed
curve intersecting with G at exactly two vertices v0 and either v1 or v3, which corresponds
to β̃0, a contradiction. Thus, we conclude that x is neither v0 nor v2.
64 Ars Math. Contemp. 23 (2023) #P1.04 / 57–79
Observe that even when γ1 is an essential diagonal 3-curve passing through a face
f = v0pv2q for p, q ∈ V (G), we have {v0, v2} ∩ {p, q} = ∅ since G is simple. This
implies that the γ1-segment in f and γ′ cannot cross transversally, and hence we conclude
that γ′ is essential. Therefore, we have an essential diagonal (or semi-diagonal) 4-curve
γ2 passing {v1, u2, v3, x} in the statement, half of which is along γ′, and the other half is
inside the quadrangular region bounded by ∂(H).
Finally, assume that γ1 is 2-sided. Suppose, for a contradiction, that γ2 is homotopic to
γ1. Under the condition, γ2 must cross γ1 even times, i.e., twice here. However, this is not
the case by the above argument.
Lemma 3.9. Let G be an {R1, R2, R3, R4}-irreducible polyhedral quadrangulation of
non-spherical closed surface. Then any 2-cell region bounded by a 4-cycle is either a face
of G or contains exactly four vertices which is of an attached cube.
Proof. Using the above Lemma 3.8 and Lemma 4.3 in [23], we immediately have the
conclusion of the lemma.
Furthermore in [23], Suzuki determined configurations in a 2-cell region bounded by a
6-cycle in {R1, R2, R3}-irreducible polyhedral quadrangulations of non-spherical closed
surfaces. By combining the results of Lemmas 3.7, 3.8 and 3.9, we can easily obtain the
following lemma; so, we omit the proof.
Lemma 3.10. Let G be an {R1, R2, R3, R4}-irreducible polyhedral quadrangulation of
a non-spherical closed surface F 2. Then the number of vertices inside a 2-cell region
bounded by a 6-cycle (resp., 4-cycle) is at most 16 (resp., 4).
In the latter half of the section, we discuss reductions R5, R6, R7 and R8 applied to
polyhedral quadrangulations in turn.
Lemma 3.11. Let G be a polyhedral quadrangulation of a closed surface F 2 having a
2-cell region D with ∂D = v0v1v2v0v3v4 containing two vertices u1 and u2 as shown in
the left-hand side of R5 in Figure 1, and let G′ denote a quadrangulation obtained from G
by an R5 of D. If G′ is not polyhedral, then there exists an essential simple closed curve
γ′ such that
(i) γ′ intersects exactly two vertices of G′,
(ii) γ′ passes through at least one vertex of [v1v4] and [v2v3], and
(iii) γ′ does not pass through v0.
In particular, if C = v0[v1v4][v2v3] is 2-sided, then γ′ is not homotopic to C.
Proof. Some similar arguments as in Lemma 3.8 will appear, and we omit the long ex-
planation at that time for brevity. If G′ has a loop e with a vertex u, then u must be one
of [v1v4] and [v2v3], say [v1v4] up to symmetry, such that e and C = v0[v1v4]v2v3 cross
transversally at [v1v4]. Clearly e is essential, and we can take an essential simple closed
curve intersecting G at only v1 and v4, a contradiction.
Next, assume that G′ has a pair of multiple edges, which joins [v1v4] and another vertex
x ̸= v0. If the 2-cycle C ′ = [v1v4]x formed by the multiple edges is essential, then we can
take our desired simple closed curve along C ′. Thus, we suppose that C ′ is trivial below.
Y. Suzuki: Finitizable set of reductions for polyhedral quadrangulations of closed surfaces 65
If x /∈ V (C), then G would have multiple edges joining x and either v1 or v4; observe that
C and C ′ do not cross transversally, otherwise x ∈ V (C) since C ′ is trivial. If x ∈ V (C),
then x must be [v2v3]. Also in this case, G would have multiple edges joining either v1
and v2 or v3 and v4, a contradiction. Therefore, we assume that G′ is 2-connected and
2-representative below.
Now, G′ has a diagonal (or semi-diagonal) 2-curve γ′ passing {[v1v4], x} such that γ′
and C cross at [v1v4] transversally. We consider the γ′-segment β0 and β̃0 which play the
same role as in the argument in Lemma 3.8. If x = v0, then β̃0 is essential by Propo-
sition 3.2, and hence G is not polyhedral as well, a contradiction. If γ′ is trivial, then x
must be [v2v3] since x ̸= v0. However, this contradicts Proposition 3.2 for β̃0. Therefore,
γ′ is essential and satisfying the conditions in the statement. Similar to the argument in
Lemma 3.8, if C is 2-sided, then C and γ′ are not homotopic.
Lemma 3.12. Let G be a polyhedral quadrangulation of a closed surface F 2 having a
2-cell region D with ∂D = v0v1v2v3v0v4v5v6 containing a unique vertex u as shown in
the left-hand side of R6 in Figure 1, and let G′ denote a quadrangulation obtained from G
by an R6 of D. If G′ is not polyhedral, then there exists an essential simple closed curve
γ′ such that
(i) γ′ intersects at most two vertices of G′,
(ii) γ′ passes through at least one vertex of [v1v6], [v2v5] and [v3v4], and
(iii) γ′ does not pass through v0.
In particular, if C = v0[v1v6][v2v5][v3v4] is 2-sided, then γ′ is not homotopic to C.
Proof. The most part is same as the argument in Lemma 3.11, and hence we implicitly omit
the argument which had already done before. First, observe that there does not exist a face
f /∈ D such that v0, v2 ∈ ∂f ; otherwise, we can find a simple closed curve intersecting
with G at exactly two vertices, which passes through the face v2v3v0v4 and f . Similarly,
there is no face f /∈ D of G such that v4, v6 ∈ ∂f . Further, in the case when G′ is not
simple, a loop of a vertex [v2v5] might exist, unlike the argument in Lemma 3.11, and then,
it is essential by Proposition 3.2.
Thus, we assume that G′ has a diagonal (or semi-diagonal) 2-curve γ′ passing {x, y},
and we may assume that y is one of [v1v6], [v2v5] and [v3v4] such that γ′ and C =
v0[v1v6][v2v5][v3v4] cross at y transversally. If x = v0, then y must be [v2v5] by the
same argument as in the previous lemma; recall the argument of β̃0. However, under the
condition, G would have a face f /∈ D such that v0, v2 ∈ ∂f , which is passed by a γ′-
segment, a contradiction. Thus, γ′ does not pass through v0 in the following argument. If
γ′ is trivial, then {x, y} = {[v1v6], [v3v4]}, and γ′ crosses C exactly twice by the former
argument. Similarly, there exists a face f /∈ D such that v4, v6 ∈ ∂f and f is passed by a
γ′-segment, a contradiction. Therefore, γ′ is essential. Further, it is not difficult to see that
γ′ is not homotopic to C when C is 2-sided.
Lemma 3.13. Let G be a polyhedral quadrangulation of a closed surface F 2 having an
annular region A formed by three faces v0v1v4v3, v1v2v5v4 and v2v0v3v5 as shown in the
left-hand side of R7 in Figure 1, and let G′ be a quadrangulation obtained from G by an
R7 of A. If G′ is not polyhedral, then there exists an essential simple closed curve γ′ such
that
66 Ars Math. Contemp. 23 (2023) #P1.04 / 57–79
(i) γ′ intersects exactly two vertices of G′,
(ii) γ′ passes through exactly one vertex of [v0v3], [v1v4] and [v2v5], and
(iii) C = [v0v3][v1v4][v2v5] and γ′ are not homotopic.
Proof. Almost the same argument as in the proofs of Lemmas 3.11 and 3.12 holds, and
hence we omit the proof. (This is easier than those proofs.) Since any two homotopic
2-sided simple closed curves on a closed surface cross even times, (iii) immediately holds
from (ii).
Lemma 3.14. Let G be a polyhedral quadrangulation of a closed surface F 2 having an
annular region A formed by four faces v0v1v6v5, v1v2v7v6, v2v3v0v7 and v0v5v4v7 as
shown in the left-hand side of R8 in Figure 1, and let G′ be a quadrangulation obtained
from G by an R8 of A. If G′ is not polyhedral, then there exists an essential simple closed
curve γ′ such that
(i) γ′ intersects exactly two vertices of G′,
(ii) γ′ passes through at least one vertex of [v0v4], [v1v5], [v2v6] and [v3v7], and
(iii) C = [v0v4][v1v5][v2v6][v3v7] and γ′ are not homotopic.
Proof. Note that there does not exist a face f /∈ A (resp., f ′ /∈ A) such that v0, v2 ∈ ∂f
(resp., v5, v7 ∈ ∂f ′), similar to the argument in the proof of Lemma 3.12. Furthermore, for
example, there might be an edge v2v5 in G such that 2-cycle C ′ = [v1v5][v2v6] formed by
a pair of multiple edges is essential in G′; this is different from the previous lemma. The
argument is almost same, and hence we omit it as well.
4 Main result
First, we refer to the following lemma, which plays an important role in the proof of our
main result.
Lemma 4.1 (Juvan, Malnič and Mohar [8]). For any closed surface F 2 and any non-
negative integer k, there exists a constant f(k, F 2) such that if L is a set of pairwise
non-homotopic simple closed curves on F 2 such that any two elements of L cross at most
k times, then |L| ≤ f(k, F 2).
In the next lemmas, we show that there is an upper bound of the maximum degree
(resp., the diameter) of {R1, . . . , R6}-irreducible (resp., {R1, . . . , R8}-irreducible) poly-
hedral quadrangulations of a non-spherical closed surface F 2.
Lemma 4.2. Let G be an {R1, . . . , R6}-irreducible polyhedral quadrangulation of a non-
spherical closed surface F 2. Then the maximum degree of G is bounded by a constant
depending only on F 2.
Proof. We prove that ∆(G) ≤ 640f(5, F 2) + 79, where f(·, F 2) is the function in
Lemma 4.1. Suppose, for a contradiction, that G has a vertex v with deg(v) ≥
Y. Suzuki: Finitizable set of reductions for polyhedral quadrangulations of closed surfaces 67
640f(5, F 2) + 80. Let Lv be the link walk of v in G. Give a direction to Lv and de-
note the directed cycle by
−→
L v . Let
a11, . . . , a
1
16, b
1
1, . . . , b
1
7, c
1
1, . . . , c
1
17, a
2
1, . . . , a
2
16, b
2
1, . . . , b
2
7, c
2
1, . . . , c
2
17, . . . ,
al1, . . . , a
l
16, b
l
1, . . . , b
l
7, c
l
1, . . . , c
l
17
be 40l consecutive vertices of Lv taken along
−→
L v , where l ≥ 16f(5, F 2)+2. Then, we may
assume that vb11b
1
2b
1
3 is a face of G; note that vb
i
1b
i
2b
i
3 is also a face for each i ∈ {2, . . . , l}
under the assumption. Let P (a, b) denote the path in Lv starting at a ∈ V (Lv) and ending
at b ∈ V (Lv) along
−→
L v .
In the former half of the proof, we show the following fact: For each i ∈ {1, . . . , l},
there exists either (A) a cycle of length at most 6 containing a path bisvb
i
t (1 ≤ s < t ≤ 6),
or (B) a cycle of length at most 4 containing a path bisvu (1 ≤ s ≤ 6) where u ∈ V (Lv).
We call the cycle having the above property (A) (resp., (B)) a type-A cycle (resp., type-B
cycle). Note that there might be a cycle having both properties (A) and (B); in that case,
we can classify it into either.
In the following argument, we discuss several cases around vertices bi1, . . . , b
i
6 and b
i
7.
To simplify notation, we put bij = bj for each j ∈ {1, . . . , 7} by omitting the upper sub-
script “i”. First of all, assume that deg(b2) ≥ 4. In this case, we apply an R1 of vb1b2b3
at {b1, b3}, i.e., identifying b1 and b3. By Lemma 3.4, we can easily find our desired cy-
cle containing a path b1vb3; take such a path using edges of faces passed by the diagonal
3-curve or the semi-diagonal 3-curve. The same fact holds for b4 and b6, and hence we
assume that deg(bh) = 3 for each h ∈ {2, 4, 6} below.
Next, assume deg(b3) = 3. Then, there exist faces b1b2xy, b2b3b4x and b4b5zx for
x, y, z ∈ V (G). If deg(x) ≥ 4, then we can find our desired cycle containing a path b1vb5
by Lemma 3.6 as a type-A cycle. On the other hand, if deg(x) = 3, i.e., y = z in this case,
then b2b3b4x is an attached 4-cycle. In this case, there exists either a type-A cycle or a type
B cycle, both of which contain vb1, by Lemma 3.7. Thus, we assume that deg(b3) ≥ 4 and
deg(b5) ≥ 4 in the following argument.
For the face vb3b4b5, there is
(i) an essential 4-cycle vb3b4x for x /∈ {v, b3, b4, b5},
(ii) an essential diagonal 3-curve γ passing through v and b4, or
(iii) an essential semi-diagonal 3-curve γ passing through v and b4, by Lemma 3.4.
First, we discuss (i). In this case, x is a vetex of Lv such that xv ∈ E(G), and hence there
exists our desired type-B cycle. Secondly, assume (ii), and let f1 = vb3b4b5, f2 = b4pqr
and f3 = vsqt be faces passed by γ where q, s, t ∈ V (Lv) (see the left-hand side of
Figure 2). Since deg(v4) = 3, we have |{b3, b5} ∩ {p, r}| = 1. Without loss of generality,
we may assume that p = b3, and we find our desired type-B 4-cycle vb3qs.
Thirdly, we discuss (iii). We further divide this case into the following two subcases:
(1) γ passes through f1 = vb3b4b5, f2 = b4pqr and f3 = vqst where q, s, t ∈ V (Lv),
and
(2) γ passes through f1 = vb3b4b5, f2 = b4pqr and f3 = vsrt where s, r, t ∈ V (Lv).
68 Ars Math. Contemp. 23 (2023) #P1.04 / 57–79
v
v
Lv
Lv
b3
b4
b5
p r
q
s t
v
v
Lv
Lv
b3
b4
b5
q
r
s t
v
v
Lv
Lv
b3
b4
b5
q
r
s t
y
γ′
Figure 2: Configurations around Lv .
First, assume the former case (iii)(1). Similar to the above argument, we have |{b3, b5} ∩
{p, r}| = 1 since deg(v4) = 3, and we may assume that p = b3 here. In this case, we find
a type-B cycle vb3q of length 3.
Next, suppose the latter case (iii)(2). Similarly, we have deg(v4) = 3, and hence we
may assume that p = b3 (see the center of Figure 2). Furthermore, if deg(r) = 3, then q
must be either s or t, and hence we find our desired type-B cycle vb3q of length 3. Thus,
we assume deg(r) ≥ 4 in the following argument. By applying Lemma 3.4 to f2 = b3b4rq
since deg(b3) ≥ 4 and deg(r) ≥ 4, we find either a 2-path P joining q and b4 such that
the cycle b4b3qP is essential, or an essential simple closed curve γ′ passing {q, b4, x} for
x ∈ V (G). If the former holds, then P = qb5b4 since deg(b4) = 3. In this case, there
exists our desired type-A cycle vb3qb5 of length 4. Next, we assume the latter, and suppose
that γ′ is an essential diagonal 3-curve. If γ′ passes through rb4b5y for y ∈ V (G), then
there exists a 2-path P ′ joining y and q along γ′ (see the right-hand side of Figure 2).
That is, there exists a type-A cycle vb3qP ′yb5 of length 6. If γ′ passes through b3b4b5v,
then q ∈ V (Lv) and γ′ passes {v, b4, q}. In this case, there exists a type-B cycle vb3qq′
of length 4 where qq′ ∈ E(Lv). When γ′ is an essential semi-diagonal 3-curve, similar
argument holds, and we have either a type-A cycle of length 5 or a type-B cycle of length
3.
In the latter half of the proof, we lead to a contradiction. For our purpose, let C lA denote
a type-A cycle containing blsvb
l
t where 1 ≤ s < t ≤ 6, and let C
i,j
B denote a type-B cy-
cle containing a 2-path bisvu where 1 ≤ s ≤ 6 such that u ∈ {a
j
1, . . . , a
j
16, b
j
1, . . . , b
j
7,
cj1, . . . , c
j
17}; i.e., C
i,j
B was obtained by the argument above when discussing vertices
bi1, . . . , b
i
7. (Note that C
i,i
B might exist for some i.) Then, any two type-A cycles cross
at most 5 times, since they cannot cross at a vertex v. Clearly, the number of crossing
points of a type-B cycle and another type-A or type-B cycle is at most 4.
First, assume that there exist at least 2f(5, F 2) + 1 type-A cycles. By the definition
of the function, F 2 admits at most f(5, F 2) simple closed curves which are pairwise non-
homotopic and cross at most 5 times, and hence there exist three such homotopic cycles
C iA , C
j
A and C
k
A (i < j < k) by the Pigeonhole Principle. Let D̃ denote the configuration
which is the union of the closed disk D̄ bounded by Lv and the three cycles C iA , C
j
A and
C kA . First, suppose that D̃ is an embedding on F
2 such that C iA , C
j
A and C
k
A are 2-sided.
Moreover, assume that C iA (resp., C
j
A ) contains b
i
svb
i
t with 1 ≤ s < t ≤ 6 (resp., b
j
s′vb
j
t′
with 1 ≤ s′ < t′ ≤ 6).
Y. Suzuki: Finitizable set of reductions for polyhedral quadrangulations of closed surfaces 69
bjt′
bjs′ b
i
t
bis
v
C kA
bjt′
bjs′ b
i
t
bis
v
C iAC
j
A
bjt′
bjs′ b
i
t
bis
v
C iAC
j
A C
j
A C
i
A
Figure 3: Type-A cycles around v.
Observe that in D̃, C iA and C
j
A bound a pinched annulus A (i.e., an annulus where the
two boundary components might touch several times) having a pinched point v (see the
left-hand side of Figure 3). If C iA and C
j
A have a common vertex other than v, then there
exists a 2-cell region R in A bounded by a cycle of length either 4 or 6 such that R̄ contains
P (bit, b
j
s′) or P (b
j
t′ , b
i
s). However, this contradicts Lemma 3.10 since P (b
i
t, b
j
s′) (resp.,
P (bjt′ , b
i
s)) contains vertices c
i
1, . . . , c
i
17, a
j
1, . . . , a
j
15, and a
j
16 (resp., c
j
1, . . . , c
j
17, a
i
1, . . . ,
ai15, and a
i
16). In the following argument, we call a region like the above R a dense quad-
rangle or a dense hexagon, which contains at least 5 or 17 inner vertices, respectively. Thus,
we conclude that C iA and C
j
A have the unique common vertex v. However, under the situa-
tion, the third type-A cycle C kA must cross transversally either C
i
A or C
j
A (see the center of
Figure 3), contradicting the same argument as above. In the case when each of C iA , C
j
A and
C kA is 1-sided, any two of them must cross, and hence there exists a dense quadrangle or a
dense hexagon, as well as the previous case (see the right-hand side of Figure 3).
Next, we discuss type-B cycles. Under our definition, for some i ̸= j, C i,jB and C
j,i
B
might exist; as an extreme example, C i,jB might coincide with C
j,i
B . If so, i.e., there exist
C i,jB and C
j,i
B , then we choose one from them. By the above argument, we may assume
that there exist at most 2f(5, F 2) type-A cycles. That is, there exist at least 7f(5, F 2) + 1,
which is the half of 14f(5, F 2) + 2, distinct type-B cycles around v, such that the set of
those cycles contains no pair of two cycles C i,jB and C
j,i
B for 1 ≤ i ≤ j ≤ l.
Similar to the argument for type-A cycles, there exist eight such homotopic cycles
simply denoted by Γ1,Γ2, . . . ,Γ8 having a common vertex v such that they are placed on
F 2 as shown in the left-hand side of Figure 4. Note that the lengths of those cycles are
same, which is either 3 or 4, by Proposition 3.1. Furthermore, note that if Γi and Γi+1 have
a common vertex other than v for some i ∈ {1, . . . , 7}, then we can easily find a dense
quadrangle or a dense hexagon, contradicting Lemma 3.10; only Γ1 and Γ8 might have a
common vertex other than v. Therefore, Γi∪Γi+1 bounds an octagonal (resp., a hexagonal)
2-cell region for each i ∈ {1, . . . , 7} if |Γi| = 4 (resp., if |Γi| = 3).
Let Di,j denote an octagonal (or a hexagonal) region bounded by Γi ∪ Γj for 1 ≤ i <
j ≤ 8. By Euler’s formula, Γ4,5 contains a vertex u of degree 3; e.g., see Lemma 4.1 in
[23]. By Lemma 3.3, u belongs to a connected component of ⟨V3⟩G which is
(i) a 4-cycle,
(ii) a 2-path,
70 Ars Math. Contemp. 23 (2023) #P1.04 / 57–79
v
v
Γ1
Γ2 Γ3 Γ4 Γ5
Γ6 Γ7
Γ8
v = v1
v = v1
Γ3 Γ6
v0
v3
v4
v5u2
v2
u1
u0
Figure 4: Type-B cycles around v.
(iii) a K2 or
(iv) an isolated vertex.
First, we assume that |Γi| = 4, and discuss the above four cases in order.
Case (i): In this case, an attached cube H with ∂(H) = v0v1v2v3 containing u as an
vertex of the attached 4-cycle is in D̄3,6. (Observe that faces incident to u are in D4,5, and
the other two faces in the 2-cell region bounded by ∂(H) are at least in D3,6.) Then, by
Lemma 3.7 and the existence of Γ2 and Γ7, one of v0, v1, v2 and v3, say v0 without loss
of generality, must be v; we call the above Γ2 and Γ7 obstructions throughout the proof.
However, Lemma 3.8 requires one more essential simple closed curve which does not pass
through v = v0, a contradiction; by the existence of obstructions again.
Case (ii): We assume that u belongs to a 2-path P = u0u1u2 and the configuration around
P is given by the left-hand side of R3 in Figure 1. Similarly, the hexagon bounded by
v0v1v2v3v4v5 is contained in D̄3,6, and hence the obstructions, which are Γ2 and Γ7, play
the same role in this argument. By Lemma 3.6, one of v1 and v5, say v1 without loss
of generality, must be v (see the right-hand side of Figure 4). Since deg(v3) ≥ 4 and
deg(v5) ≥ 4, there is an essential diagonal 3-curve passing {v4, u2, v0} or {v4, u2, u0}
by Lemma 3.4. However, in each case, such three vertices are inner vertices of D2,7, a
contradiction.
Case (iii): We assume that u0u1 ∈ E(G) is a connected component of ⟨V3⟩G, and there are
four faces v0v1u1v4, v1v2u2u1 and u1u2v3v4 and u2v2v′0v3 contained in D3,6. Here, we
locally color vertices in D̄3,6 by two colors black and white; we assume that v is colored
by black. Further, we may assume that v′0 is colored by black without loss of generality;
note that v0, v2 and v3 are white vertices. When considering a face v0v1u1v4, there is an
essential diagonal 3-curve passing either {v0, u1, v2} or {v0, u1, v3} by Lemma 3.4, since
we have deg(v1) ≥ 4 and deg(v4) ≥ 4. By the existence of obstructions, one of v0, v2 and
v3 must be v under the situation. However, it contradicts the above bipartition.
Case (iv): Assume that u is incident to three faces v0v1uv6, v1v2v4u and uv4v5v6, which
are in D4,5, and note that deg(vi) ≥ 4 for each i ∈ {1, 4, 6}. As well as the previous
case, we locally color vertices in D̄3,6; assume that v is colored by black. If u is a white
vertex, then it contradicts Lemma 3.4 by the existence of obstructions; note that there
should be a diagonal 3-curve passing three white vertices including u. Therefore, u is a
black vertex below. By Lemma 3.4 again, exactly one of v0, v2 and v5, say v0 without loss
Y. Suzuki: Finitizable set of reductions for polyhedral quadrangulations of closed surfaces 71
v = v0
v = v0
Γ3 Γ6
v1
v3
v4
v5
v2
u v6
p
v = v0
v = v0
Γ3 Γ6
v1
v3
v4
v2
u v5
p q
v = v0
v = v0
Γ3 Γ6
v1
v3
v4
v2
u v5
R
Figure 5: Configurations in the 2-cell region bounded by Type-B cycles.
of generality, coincides with v, and there exists a diagonal 3-curve passing through three
faces v0v1uv6, v1v2v4u and v2v3v0p for v3, p ∈ V (G), up to symmetry (see the left-hand
side of Figure 5). If deg(v2) ≥ 4, then Lemma 3.4 works for v2v3v0p, and it contradicts the
existence of the obstructions. Thus, we conclude that |{v1, v4}∩{v3, p}| = 1, and we may
suppose v4 = p since {v1, v4} ∩ {v3, p} ̸= {v1}; otherwise, G would have multiple edges.
Then, G has an octagonal region bounded by v0v1v2v3v0v4v5v6 satisfying the condition of
a reduction R6. However, it contradicts Lemma 3.12 by the existence of the obstructions.
Next, we assume that |Γi| = 3. We implicitly omit the same argument as in the case
assuming |Γi| = 4. (That is, we give only the different and important points below.)
Case (i): The same argument as in the case of |Γi| = 4 works.
Case (ii): We may assume that v1 = v, and there is an essential semi-diagonal 3-curve
passing {v4, u2, v0}, {v4, u2, u1} or {v4, u2, u0} by Lemma 3.4. However, in any case,
such three vertices are inner vertices of D2,7, a contradiction.
Case (iii): In this case, the similar argument (not using the bipartition) leads us to the
conclusion that v0 = v′0 = v such that the 3-cycle v0v1v2 is homotopic to Γi. However, it
contradicts Lemma 3.11 by the existence of the obstructions.
Case (iv): Assume that u is incident to three faces v0v1uv5, uv1v2v3 and uv3v4v5, which
are in D4,5, and note that deg(vi) ≥ 4 for each i ∈ {1, 3, 5}. For a face v0v1uv5, there
exists a semi-diagonal 3-curve passing either {v0, u, v3} or {v0, u, v4}, up to symmetry,
by Lemma 3.4. Fist assume the former case. If v = v0, then there is a face f = v3pvq
for p, q ∈ V (G) (see the center of Figure 5). For f , Lemma 3.4 works and we conclude a
contradiction by the existence of the obstructions since deg(v3) ≥ 4. On the other hand, if
v = v3, then there is a face vsv0t for s, t ∈ V (G). As well as the previous case, we can
apply Lemma 3.4 for vsv0t since deg(v0) ≥ 4; if {v1, v5} ∩ {s, t} ̸= ∅, then G would not
become 3-representative.
Next, we assume the latter case. In this case, v is either v0 or v4, say v0, up to sym-
metry. By the assumption, there exists an edge v4v0 such that v0v5v4 is homotopic to Γi.
Furthermore, applying Lemma 3.4 for a face v1v2v3u, there must be a semi-diagonal 3-
curve passing {v0, u, v2}; note that v2, u, v4 and v5 are vertices in D̄4,5, i.e., inner vertices
of D3,6. That is, we have v2v0 ∈ E(G) such that v2v0v4v3 bounds a 2-cell region R inside
D4,5 (see the right-hand side of Figure 5). By the above argument of (i), we may assume
that D4,5 does not contain a vertex of degree 3 belonging to an attached 4-cycle, and hence
72 Ars Math. Contemp. 23 (2023) #P1.04 / 57–79
R is a face of G by Lemma 3.9. However, v3 has degree 3, contrary to u being an isolated
vertex of ⟨V3⟩G. Therefore, we got our desired conclusion.
Lemma 4.3. Let G be an {R1, R2, R3}-irreducible polyhedral quadrangulation of a non-
spherical closed surface F 2. For any vertex v ∈ V (G), there exists an essential cycle of
length at most 6 either
(i) containing v, or
(ii) containing u ∈ V (G) such that uv ∈ E(G).
Proof. First, assume that deg(v) = 3, and let u0, u1 and u2 be vertices adjacent to v. If two
of u0, u1 and u2, say u0 and u1 without loss of generality, have degree at least 4, then we
can easily find our desired cycle by Lemma 3.5. Thus, by Lemma 3.3, we may assume that
deg(u0) = deg(u1) = 3 and deg(u2) ≥ 4 below. If v is contained in a 4-cycle of ⟨V3⟩G,
then there exists such a cycle by Lemma 3.7. On the other hand, if v is not contained in the
above 4-cycle in ⟨V3⟩G, that is, if a 2-path u0vu1 is a connected component of ⟨V3⟩G, then
G also has our desired cycle by Lemma 3.6.
Next, we assume deg(v) ≥ 4, and let u0 and u1 be vertices adjacent to v such that
u0vu1 forms a corner of a face of G. If one of u0 and u1, say u0 without loss of generality,
has degree 3, then G has a cycle of length at most 6 passing through u0 by the above
argument, and hence it satisfies (ii) of the statement in the lemma. If deg(u0) ≥ 4 and
deg(u1) ≥ 4, then there exists our desired cycle by Lemma 3.5 again.
Lemma 4.4. Let G be an {R1, . . . , R8}-irreducible polyhedral quadrangulation of a non-
spherical closed surface F 2. Then the diameter of G is bounded by a constant depending
only on F 2.
Proof. In this proof, we prove that diam(G) ≤ 50f(0, F 2)−1 where diam(G) is a diameter
of G and f(·, F 2) is the function in Lemma 4.1. Suppose, for a contradiction, that G has
two vertices x and y with distance at least 50f(0, F 2). Let P be a path from x to y attaining
the distance, and let x = v1, v2, . . . , vk be the vertices on P lying in this order, where
k ≥ 5f(0, F 2) + 1, so that the distance between vi and vi+1 is exactly 10 on P , for each
i ∈ {1, . . . , k− 1}. Then, there exists a cycle Ci of length at most 6 passing through either
vi or a vertex ui adjacent to vi for each i ∈ {1, . . . , k} by Lemma 4.3. Since the distance
between vi and vj is at least 10 for any i < j, two cycles Ci and Cj are mutually disjoint.
Since F 2 admits only f(0, F 2) pairwise non-crossing non-homotopic essential cycles, and
since we assumed k ≥ 5f(0, F 2) + 1, we can take six pairwise homotopic cycles from
{C1, . . . , Ck} by the Pigeonhole Principle. Let Γ1, . . . ,Γ6 be such six cycles of length at
most 6, which are mutually homotopic. Note that those cycles are 2-sided since any two of
them are disjoint, and that the parities of those cycles are pairwise same. We may assume
that these Γ1, . . . ,Γ6 lie on an annulus in this order.
Let Ai,j denote the annular region bounded by Γi and Γj for 1 ≤ i < j ≤ 6; similarly,
Āi,j contains its two boundaries Γi and Γj . Note that there is no edge joining vertices on Γi
and Γi+1 for each i ∈ {1, . . . , 5}; for otherwise, the distance between vi and vi+1 would
be at most 9, contradicting that P is a shortest path joining x and y in G. Similar to the
argument in Lemma 4.2, we call Γ1 and Γ6 obstructions for our purpose.
First, we discuss the case when G has a vertex u of degree 3 in Ā3,4. By Lemma 3.3, u
belongs to a connected component of ⟨V3⟩G which is
Y. Suzuki: Finitizable set of reductions for polyhedral quadrangulations of closed surfaces 73
v′0
v′0
v2
v2
v0
v1
v3
v4
u1
u2
v5
v5
v4
v4
v2
v3v0
v1
u1
u2
u0
γ
γ′
γγ′
v′0
v′0
v2
v2
v0
q
v1
v3
v4
u2
γγ′
u1
Figure 6: Configurations around connected components of ⟨V3⟩G.
(i) a 4-cycle,
(ii) a 2-path,
(iii) a K2 or
(iv) an isolated vertex.
We discuss the above four cases in order.
Case (i): Under the assumption, an attached cube containing u as an vertex of an attached
4-cycle is in Ā2,5. (For example, even if u is on Γ3, then there is no face f such that ∂f
contains both u and a vertex on Γ2, since there is no edge joining vertices on Γ2 and Γ3,
and since deg(u) = 3.) Similar argument in Case (i) in the proof of Lemma 4.2 works, and
we conclude that this is not the case; i.e, we cannot take two essential simple closed curves
γ1 and γ2 in Lemma 3.8 by the existence of the obstructions.
Case (ii): We assume that u belongs to a 2-path P = u0u1u2 and the configuration around
P is given by the left-hand side of R3 in Figure 1. Similarly, the hexagonal region R
bounded by v0v1v2v3v4v5 is contained in Ā2,5. By Lemma 3.6, there exists an essen-
tial diagonal (or a semi-diagonal) 3-curve γ passing {v1, u1, v5} (see the left-hand side of
Figure 6). On the other hand, since deg(v1) ≥ 4 and deg(v3) ≥ 4 hold, there exists an
essential diagonal (or semi-diagonal) 3-curve γ′ passing {v0, u0, v2} by Lemma 3.4. Ob-
serve that both γ and γ′ are homotopic to Γi by the existence of obstructions. Under the
situation, γ and γ′ cross transversally in R, and it must cross transversally one more time
since these two curves are 2-sided. This implies that there should be a face incident to four
vertices v0, v1, v2 and v5, in which γ and γ′ pass through. However, it contradicts that G is
simple.
Case (iii): Assume that u1u2 ∈ E(G) is a connected component of ⟨V3⟩G, and there are
four faces v0v1u1v4, v1v2u2u1, u1u2v3v4 and u2v2v′0v3 incident to u1 and u2. Note that
deg(vi) ≥ 4 for any i ∈ {1, 2, 3, 4}. When considering a face v0v1u1v4, there exists an
essential diagonal (or semi-diagonal) 3-curve γ passing either {v0, u1, u2} or {v0, u1, v2}
by Lemma 3.4, up to symmetry. Note that γ is homotopic to Γi. In the former case, we
have v0 = v′0, and hence we discuss an R5 to the hexagonal region containing u1 and u2.
However, it immediately contradicts that G is {R1, . . . , R8}-irreducible by the existence
of obstructions and by Lemma 3.11.
74 Ars Math. Contemp. 23 (2023) #P1.04 / 57–79
v0
v4
p
v5
v3
v2
u
v0
v1
v0
p
v3
u
v0
q
v2
v1
v4
v5
v0
v3
u
v2
v1
v4
v5
v3
u
v1
v5
p
pq
q
γ
γ
γ
v0
v2
v4
Figure 7: Configurations around connected components of ⟨V3⟩G.
Therefore, we assume the latter case. In this case, we may assume that there exists
an essential diagonal (or semi-diagonal) 3-curve γ′ passing {v′0, u2, v4} by the same argu-
ment as above. Note that γ′ is homotopic to γ under the condition. If γ and γ′ are both
essential semi-diagonal 3-curves (by Proposition 3.1) then, there exists a face v0v4v′0v2
by Lemma 3.9 and our former argument (see the center of Figure 6). However, since
deg(v2) ≥ 4 and deg(v4) ≥ 4, we apply Lemma 3.4, and conclude a contradiction.
Thus, we suppose that γ is an essential diagonal 3-curve, and there is a face f = v0pv2q
for p, q ∈ V (G) which is passed by γ. Here, observe that v1 /∈ {p, q} by the simplicity of
G, and hence we have deg(v2) ≥ 4. For f , if deg(v0) ≥ 4, then it is contrary to G being
{R1, . . . , R8}-irreducible by the existence of obstructions and by Lemma 3.4. Therefore,
we assume that deg(v0) = 3 below. Without loss of generality, we may assume that p = v4
(see the right-hand side of Figure 6). Under the situation, we can apply Lemma 3.12 to the
octagonal region bounded by v2v1v0qv2v4v3u2, and obtain a contradiction.
Case (iv): Assume that u is incident to three faces v0v1uv5, v1v2v3u and uv3v4v5. Note
that deg(vi) ≥ 4 for any i ∈ {1, 3, 5}. Hence, for a face v0v1uv5, we have
(a) an essential 4-cycle v0v1uv3, or
(b) an essential diagonal 3-curve or semi-diagonal 3-curve γ passing
(1) {v0, u, v3} or
(2) {v0, u, v2}
by Lemma 3.4, up to symmetry.
First, assume (a). In this case, for a face v1v2v3u, there must be an essential diagonal 3-
curve passing {v0, u, v2} by Lemma 3.4; it is not difficult to check that this is the unique
case by Proposition 3.1 and the existence of obstructions. Furthermore, by Lemma 3.9,
there exists a face v2pv0v3 for p ∈ V (G), and it contradicts Lemma 3.12 for an octagonal
region bounded by v0v1v2pv0v3v4v5 by the similar argument as above (see the first figure
of Figure 7).
Secondly, we assume (b)(1). In this case, γ is an essential semi-diagonal 3-curve, and
hence there exists a face v0pv3q for p, q ∈ V (G) which γ passes through (see the second
figure of Figure 7). Then, we have deg(v0) ≥ 4 since {p, q} ∩ {v1, v5} = ∅; otherwise, G
would become representativity at most 2. Therefore, for v0pv3q, we apply Lemma 3.4, and
obtain a contradiction as well as former cases.
Y. Suzuki: Finitizable set of reductions for polyhedral quadrangulations of closed surfaces 75
x
x
t
s
v3
v2
v1
v0
x
x
t
s
v3
v2
v1
v0
γ
γ′
x
x
t
s
v3
v2
v1
v0
γ′′
γ′′
x
xs = t
v3
v2
v1
v0
s = t
Figure 8: Configurations of Case (I) in Lemma 4.4.
Thirdly, we discuss the case (b)(2). First, assume that γ is an essential semi-diagonal 3-
curve; i.e., v0v2 ∈ E(G) which is along γ (see the third figure of Figure 7). Then, for a face
uv3v4v5, there exists either v4v0 or v4v2, say v4v0 without loss of generality, as an edge
of G such that v0v5v4 is homotopic to Γi. Under the situation, there exists a 2-cell region
R bounded by v0v4v3v2, which is a face of G by Lemma 3.9 and the former argument.
However, we obtain a contradiction since deg(v3) ≥ 4. Therefore, we suppose that γ is
an essential diagonal 3-curve; i.e., there exists a face bounded by v0pv2q for p, q ∈ V (G)
(see the last figure of Figure 7). If {p, q} ∩ {v3, v5} ̸= ∅, then it gives rise to the above
case (a), which had already discussed. On the other hand, if v1 ∈ {p, q}, then G would
have multiple edges, a contradiction. Thus, we have deg(v0) ≥ 4 and deg(v2) ≥ 4, and
conclude a contradiction by Lemma 3.4, similar to the former cases.
Therefore, in the following argument, we discuss the case when deg(u) ≥ 4 for any
vertex u in Ā3,4. In this case, we focus on a face f = v0v1v2v3 in Ā3,4 with deg(vi) ≥ 4
for each i ∈ {0, 1, 2, 3}. By Propositions 3.1 and 3.2, Lemma 3.4 and the existence of
obstructions, it suffices to discuss the following two cases (I) and (II), up to symmetry.
Case (I): There exist two essential semi-diagonal 3-curves γ and γ′ passing {v0, v2, x} and
{v1, v3, x}, respectively, for x ∈ V (G) such that γ and γ′ are homotopic to Γi (see the first
figure of Figure 8). Then, there are two faces f = v0v3xt and f ′ = v1sxv2 for s, t ∈ V (G)
by Lemma 3.9 (see the second figure of Figure 8). Under the situation, if s = t, then
there exists an annular region A bounded by two 3-cycles sv0v1 and xv3v2 which contains
exactly three edges dividing it into three faces (see the third figure of Figure 8). Then, we
apply Lemma 3.13 to A and obtain a contradiction by the existence of the obstructions.
Thus, we assume s ̸= t below, and hence s, t, v2 and v3 are distinct vertices; i.e., we
have deg(x) ≥ 4. Then, we apply Lemma 3.4 to f ′ and find an essential semi-diagonal
3-curve γ′′ passing {s, v2, z} for z ∈ V (G). By the existence of the obstructions, γ′ and
γ′′ should be homotopic. That is, γ′ and γ′′ cross even times (actually twice), and hence
we have z = v0 and sv0 ∈ E(G) (see the last figure of Figure 8). Then, there exists a
2-cell region bounded by sv0v3x, and it contradicts Lemma 3.9 since s ̸= t.
Case (II): There exists an essential diagonal 3-curve γ passing {v1, v3, x} for x ∈ V (G),
and v0x, v2x ∈ E(G) such that γ and the 4-cycle v0v1v2x are homotopic to Γi (see the
left-hand side of Figure 9). Then, there are two faces f = v2v1sx and f ′ = v0v3tx
for s, t ∈ V (G) by Lemma 3.9 (see the center of Figure 9). By the simplicity of G,
s, t /∈ {v0, v1, v2, v3}, and hence deg(x) ≥ 4. Thus, for f , there exists an essential diagonal
76 Ars Math. Contemp. 23 (2023) #P1.04 / 57–79
v3
v2
v1
v0
x
x
t
s
v3
v2
v1
v0
x
x
t
s
v3
v2
v1
v0
x
x
t
s
γ
p
p
γ′
Figure 9: Configurations of Case (II) in Lemma 4.4.
3-curve γ′ passing {v0, v2, s} by Lemma 3.4; this is a unique case by the same argument
as in Case (I). Then, by Lemma 3.9, there is a face f ′′ = spv0x for p ∈ V (G) which
γ′ passes through (see the right-hand side of Figure 9). Apply Lemma 3.14 to the annular
region bounded by two 4-cycles v0v1sp and v3v2xt, and obtain a contradiction.
Now, we prove our main result as follows.
Proof of Theorem 1.1. Let G be a graph with maximum degree ∆ and diameter d. Then,
the following inequality holds.
|V (G)| ≤ 1 +
d∑
k=1
∆(∆− 1)k−1 = 1 + ∆((∆− 1)
d − 1)
∆− 2
.
Therefore, every {R1, . . . , R8}-irreducible quadrangulation G of F 2 has a finite number
of vertices, since its maximum degree and diameter are bounded by Lemmas 4.2 and 4.4,
respectively. Thus, F 2 admits only finitely many {R1, . . . , R8}-irreducible quadrangula-
tions, up to homeomorphism.
5 Minimality of reductions
In the previous section, we proved that {R1, . . . , R8} is sufficient to finitize the number of
minimal quadrangulations of any closed surface. However, one might think that the eight
reductions are little too much. As mentioned in the introduction, Theorem 1.3 describes
more accurate facts for the torus.
Proof of Theorem 1.3. See Figure 10. Each Ji represents an infinite series of {R1, . . . ,
R8} \ {Ri}-irreducible quadrangulations of the torus. (To obtain the torus, identify two
horizontal segments and two vertical segments of the rectangle, respectively.) In each gray
colored quadrangular region in figures contains exactly four vertices which is of an attached
4-cycle. We can construct only J6 and J8 as bipartite quadrangulations since the others
require essential cycles of length 3. Observe that we cannot apply R8 to J6, since the dual
of J6 has no essential cycle of length at most 4. Moreover, each of J7 and J8 is an infinite
series of 4-regular quadrangulations of the torus.
Y. Suzuki: Finitizable set of reductions for polyhedral quadrangulations of closed surfaces 77
· · ·
J2J1
· · · · · ·
J3
J4
· · ·
J5
· · ·
J6
· · ·
J7
· · · · · ·
J8
Figure 10: Infinite series of quadrangulations of the torus.
Proof of Theorem 1.4. As mentioned in the introduction, the projective plane does not ad-
mit 2-sided essential simple closed curves and hence {R1, . . . , R6} is finitizable for poly-
hedral quadrangulations of the projective plane by Theorem 1.1. The infinite series of
minimal graphs can be obtained in a similar way as those of torus; we leave it for read-
ers. For example, an infinite series of polyhedral quadrangulations denoted by I26(2n+1)
(n ≥ 2), which can be found in [23], is {R1, . . . , R5}-irreducible quadrangulations of the
projective plane.
In the end of the paper, we pose the following problem.
Problem 5.1. For any closed surface F 2 other than the sphere, the projective plane and the
torus, is {R1, . . . , R8} a minimal finitizable set of reductions for polyhedral quadrangula-
tions of F 2?
ORCID iDs
Yusuke Suzuki https://orcid.org/0000-0001-6940-9777
78 Ars Math. Contemp. 23 (2023) #P1.04 / 57–79
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.05 / 81–96
https://doi.org/10.26493/1855-3974.2631.be0
(Also available at http://amc-journal.eu)
The fullerene graphs with a perfect star packing*
Lingjuan Shi †
School of Software, Northwestern Polytechnical University,
Xi’an, Shaanxi 710072, P. R. China
Received 15 May 2021, accepted 23 March 2022, published online 29 September 2022
Abstract
Fullerene graph G is a connected plane cubic graph with only pentagonal and hexagonal
faces, which is the molecular graph of carbon fullerene. A spanning subgraph of G is called
a perfect star packing in G if its each component is isomorphic to K1,3. For an independent
set D ⊆ V (G), if each vertex in V (G) \D has exactly one neighbor in D, then D is called
an efficient dominating set of G. In this paper we show that the number of vertices of a
fullerene graph admitting a perfect star packing must be divisible by 8. This answers an
open problem asked by Došlić et al. and also shows that a fullerene graph with an efficient
dominating set has 8n vertices. In addition, we find some counterexamples for the necessity
of Theorem 14 of paper of Došlić et al. from 2020 and list some subgraphs that preclude
the existence of a perfect star packing of type P0.
Keywords: Fullerene graph, perfect star packing, efficient dominating set.
Math. Subj. Class. (2020): 05C70, 05C92
1 Introduction
A chemical graph is a simple finite graph in which vertices denote the atoms and edges
denote the chemical bonds in underlying chemical structure. Perfect matchings of a chemi-
cal graph correspond to Kekulé structures of the molecule, which feature in the calculation
of molecular energies associated with benzenoid hydrocarbon molecules [20]. Alternat-
ing sextet faces (sextet patterns) also play a meaningful role in the prediction of molecular
stability, in particular, but not only, in benzenoid compounds. Although for fullerenes,
the above two structures do not play the same role as in benzenoid compounds, they have
received considerable attention in recent years, see [1, 4, 8, 13, 17, 21, 32, 33] etc..
*This work was supported in part by the National Natural Science Foundation of China (grant no. 11901458
and 11871256) and by the Fundamental Research Funds for the Central Universities (grant no. D5000200199).
†The author is very grateful to Wuyang Sun and the anonymous referees for their careful reading and valuable
comments, which greatly improved this paper.
E-mail address: shilj18@nwpu.edu.cn (Lingjuan Shi)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
82 Ars Math. Contemp. 23 (2023) #P1.05 / 81–96
A perfect matching in a graph G may be viewed as a collection of subgraphs of G,
each of which is isomorphic to K2, whose vertex sets partition the vertex set of G. This is
naturally generalized by replacing K2 by an arbitrary graph H . For a given graph H , an
H-packing of G is the set of some vertex disjoint subgraphs, each of which is isomorphic
to H . From the optimization point of view, the maximum H-packing problem is to find
the maximum number of vertex disjoint copies of H in G called the packing number. An
H-packing in G is called perfect if it covers all the vertices of G. If H is isomorphic to
K2, the maximum (perfect) H-packing problem becomes the familiar maximum (perfect)
matching problem. If H is the cycle C6 of length 6, for a fullerene or a hexagonal system
G, the packing number is related to the Clar number (the maximum number of mutually
disjoint sextet patterns) of G. If H is the star graph K1,3, it is the maximum star packing
problem. If a K1,3-packing covers all the vertices of G, we call it being a perfect star
packing. For a given family F of graphs, an H-packing concept can also be generalized to
an F-packing (we refer the reader to [29] for the definition).
Packing in graphs is an effective tool as it has lots of applications in applied sciences.
H-packing, is of practical interest in the areas of scheduling [5], wireless sensor tracking
[6], wiring-board design, code optimization [23] and many others. Packing problems were
already studied for carbon nanotubes [2]. Packing lines in a hypercube had been studied
in [15]. H-packing was determined for honeycomb [29] and hexagonal network [28]. For
representing chemical compounds or to problems of pattern recognition and image pro-
cessing, P3-packing has some applications in chemistry [30]. Packing stars in fullerene
graph have been investigated in [14] by Doslić et al.. For any integer n ≥ 5, they found
a fullerene graph of order 8n which has a perfect star packing. So they raised an open
problem “Is there a fullerene on 8n+ 4 vertices with a perfect star packing?”.
In the following section we introduce necessary preliminaries and characterize the clas-
sical fullerenes which have a perfect star packing. Section 3 gives a negative answer to the
open problem asked by Doslić et al. [14]. This implies that a fullerene graph with an effi-
cient dominating set must has 8n vertices. In Section 4, we generalize the Proposition 1 in
reference [14] and give three counterexamples for the necessity of Theorem 14 in the same
paper. We also list some subgraphs that preclude the existence of a perfect star packing of
type P0.
2 Characterization of fullerenes with a perfect star packing
A fullerene graph (simply fullerene) is a cubic 3-connected plane graph with only pen-
tagonal and hexagonal faces. By the Euler formula, each fullerene graph has exactly 12
pentagons. Such graphs are suitable models for carbon fullerene molecules: carbon atoms
are represented by vertices, whereas edges represent chemical bonds between two atoms
(see [16, 26]). For all even n ≥ 24 and n = 20, Grünbaum and Motzkin [19] showed that
there exists a fullerene graph with n vertices. Using a similar approach, Klein and Liu [24]
proved that a fullerene graph with isolated pentagons of order n exists for n = 60 and for
each even n ≥ 70. We refer the reader to the reference [16] for more details on fullerene
graphs.
A cycle of a fullerene graph G is a facial cycle if it is the boundary of a face in G,
otherwise, it is a non-facial cycle. Clearly, each pentagon and hexagon in G is a facial
cycle since G is 3-connected and any 3-edge-cut is trivial [31]. In paper [14], the authors
obtained the following basic conclusions.
L. Shi: The fullerene graphs with a perfect star packing 83
Proposition 2.1 ([14]). Let S be a perfect star packing of fullerene graph G. Then each
pentagon of G can contain at most one center of a star in S.
Lemma 2.2 ([14]). Let S be a perfect star packing of fullerene graph G. Then a vertex
shared by two pentagons of G cannot be the center of a star in S.
Recall that a vertex set X of a graph G is said to be independent if any two vertices in X
are not adjacent in G. A cycle C = v1v2 · · · vkv1 in G is called induced if vi has only two
adjacent vertices vi+1 and vi−1 around the k vertices v1, v2, · · · , vk (note that i + 1 := 1
if i = k, and i − 1 := k if i = 1). Otherwise, there exists some i and j /∈ {i − 1, i + 1}
such that vi and vj are adjacent in G, the edge vivj is a chord of C and C is not induced.
A subgraph R of a graph G is spanning if R covers all the vertices of G. For a vertex v of
a graph G, we call vertex u being a neighbor of v in G if u is adjacent to v in G.
Theorem 2.3. Let G be a fullerene graph. Then G has a perfect star packing if and only
if G has an independent vertex set S∗ such that each component of G − S∗ is an induced
cycle in G.
Proof. If G has a perfect star packing S, then S is a spanning subgraph of G and any
component in S is isomorphic to a star graph K1,3. Let S∗ be the set of all 3-degree
vertices in S. Clearly, S∗ is an independent vertex set in G and any vertex in G − S∗ has
degree 2. So each component of G− S∗ is an induced cycle in G.
Let S∗ be an independent vertex set of G such that each component of G − S∗ is an
induced cycle in G. Clearly, each vertex in S∗ and its three neighbors induce a star graph
K1,3. We collect all these star graphs and denote this set by H. For any vertex x on a cycle
C in G−S∗, x has exactly one neighbor in S∗ since G is 3-regular and induced cycle C is
a component of G− S∗. So H is a spanning subgraph of G and each component of H is a
star graph K1,3, that is, H is a perfect star packing of G.
We note that star graph K1,3 has exactly one center (the vertex of degree 3) and three
leaves. For a perfect star packing S of fullerene graph G, each 1-degree vertex in S is a
leaf. In the following, we denote by C(S) the set of all the centers of stars in S.
Remark 2.4. Let S be a perfect star packing of fullerene graph G. Then
(1) C(S) is an independent vertex set in G.
(2) Any leaf in S has exactly one neighbor belonging to C(S) and has exactly two neigh-
bors being leaves in S.
(3) Each cycle in G− C(S) does not have a chord.
Proposition 2.5. Each hexagon can contain at most two centers of a perfect star packing
of fullerene graph G. If a hexagon h contains two such centers, then they are antipodal
points on the hexagon h.
Proof. Let h be a hexagon in G. We denote the six vertices of h by v1, v2, . . . , v6 in the
clockwise direction. If vertex v1 is the center of a star H in a perfect star packing S of G,
then v2 and v6 are two leaves in H . Hence both v3 and v5 are leaves in S by Remark 2.4(2).
Clearly, v4 could be the center of a star in S. Hence h has at most two centers of S and if
h contains two such centers, then they are antipodal points on h.
84 Ars Math. Contemp. 23 (2023) #P1.05 / 81–96
3 The order of fullerenes with a perfect star packing
To show the main conclusion, we need to prepare as follows.
(a) ( )b ( )c
x
1x
2x
3x
1v
2v
3v
1u
2u
3u1w
2w
3w
x
1x
2x
3x
2f
1f1v
2v
3v
1u
2u
3u1w
2w
3w
1y
2y
x
1x
4x
2x
3x
2u
1u
1f 2f
1v
2v
3v
3u1
w
2w
3w
f
Figure 1: (a) Type 1; (b) Type 2; (c) Type 3.
Lemma 3.1. Let S be a perfect star packing of fullerene graph G. Then for any vertex
x ∈ C(S), all the vertices on the three faces sharing x are covered by S as Type 1, Type 2
or Type 3 (see Figure 1, S are depicted in bold lines).
Proof. By the Lemma 2.2, at most one of the three faces sharing x is a pentagon since
x ∈ C(S). There are two cases as follows.
Case 1: The three faces sharing x are all hexagons.
Clearly, x has three antipodal points on the three hexagons sharing x, denoted by x1,
x2 and x3 respectively as depicted in Figure 1(a). By Remark 2.4(2), the two neighbors v1
and v3 of v2 are leaves in S. Similarly, u1, u3, w1 and w3 are also leaves in S. We claim
that at least two of x1, x2 and x3 are centers of stars in S. If x1 is not the center of a star
in S, then x1 is a leaf in S. So the third neighbor of v1, say y1, is the center of a star in S
(see Figure 1(b)). Similarly, the third neighbor of w3, say y2, is also the center of a star in
S. Since the three vertices v1, v2 and v3 are leaves in S and y1 ∈ C(S), the face f1 has
only one center of S by Propositions 2.5 and 2.1. Hence the two neighbors of v3 on f1 are
leaves. By Remark 2.4(2), x3 is the center of a star in S, that is, x3 ∈ C(S). Similarly,
w1 is a leaf in S and the two neighbors of w1 on f2 are all leaves in S. Hence x2 ∈ C(S).
So at least two of x1, x2 and x3 belong to C(S). If exactly two of x1, x2 and x3 belong
to C(S), without loss of generality, we suppose that x2, x3 ∈ C(S), then all the vertices
on the three faces sharing x are covered by S as Type 2. If all the three vertices x1, x2 and
x3 belong to C(S) (see Figure 1(a)), then all the vertices on the three faces sharing x are
covered by S as Type 1.
Case 2: Exactly one of the three faces sharing x is a pentagon.
By Proposition 2.1, w1 and u3 are leaves in S (see Figure 1(c)). Hence x4, x3 ∈ C(S)
and f is a hexagon by Remark 2.4(2) and Proposition 2.5. By Remark 2.4(2), the neighbor
w3 of w2 is a leaf in S since the neighbor x of w2 belongs to C(S). Hence the other vertices
on f1 except for x4 are all leaves in S by Propositions 2.1 and 2.5. This follows that the
neighbor x1 of w3 is the center of a star in S by Remark 2.4(2). Similarly, we can show
L. Shi: The fullerene graphs with a perfect star packing 85
x2 ∈ C(S). Hence all the vertices on the three faces sharing x are covered by S as Type 3
(see Figure 1(c)).
Corollary 3.2. Let S be a perfect star packing of fullerene graph G. If a pentagon P of G
has a vertex x ∈ C(S), then G − C(S) has a non-facial cycle C of G such that the path
P − x is a subgraph of C.
Proof. By Proposition 2.2, x is shared by this pentagon P and two hexagons. So all the
vertices on the three faces sharing x are covered by S as Type 3 (see Figure 1(c)). Clearly,
the path P − x is a subgraph of a cycle C in G − C(S) and C is a non-facial cycle of
G.
We note that 3-connected graphs have only one embedding up to equivalence [12]. If
we embed a fullerene graph G in the plane, then any non-facial cycle C of G as a Jordan
curve separates the plane into two regions, denoted by R∗1 and R
∗
2, each of which has the
entire C as its frontier. We denote the subgraph of G induced by the vertices lying in the
interior of R∗i by Gi, i = 1, 2. Here we note that {V (G1), V (G2), V (C)} is a partition of
all the vertices of G. We say that C divide the graph G into two sides G1 and G2.
Theorem 3.3. Let S be a perfect star packing of fullerene graph G and C be a cycle in
G− C(S). Then C(S) does not have a vertex which has three neighbors on C.
x
1x
2x 3
x
1v
kv
1u tu
h
1C
2C
3C
(a) (b)
x
1x
2x 3
x
1v
kv
1u tu
h
1C
2C
3C
y
'y
1kv -
2kv -
2u
'h
f
Figure 2: x ∈ C(S) has three neighbors on C.
Proof. If C is a facial cycle of G, then C is a pentagon or a hexagon. The conclusion
clearly holds. Now, let C be a non-facial cycle of G. Then C divides G into two sides,
denoted by H1 and H2 respectively. We note that all vertices on C are leaves in S since
C is a cycle in G − C(S). On the contrary, we suppose that there is a vertex x ∈ C(S)
which has three neighbors on C, denoted by x1, x2 and x3 respectively. Without loss of
generality, we suppose that x ∈ V (H1) (see Figure 2(a)). The three vertices separate the
circle C into three sections, denoted by C1, C2 and C3 respectively, each of which is a
path with xi and xi+1 as two terminal ends, i = 1, 2, 3 (if i = 3, then i + 1 := 1). From
Lemma 3.1 we know that at most one of x1C1x2x, x2C2x3x and x3C3x1x is a facial cycle
of G since C is a cycle in G − C(S). Next, we suppose that x1C1x2x and x2C2x3x are
non-facial cycles of G. Let C1 = x1v1v2 · · · vkx2, C2 = x2u1u2 · · ·utx3. So k ≥ 5 and
86 Ars Math. Contemp. 23 (2023) #P1.05 / 81–96
t ≥ 5 since any non-facial cycle of G has length at least 8. By Remark 2.4(3), C does not
have a chord. So v1vk /∈ E(G) and u1ut /∈ E(G). This implies that h is a hexagon face
of G, and x1, x, x2 and v1, vk are five vertices on h. We denote the sixth vertex of h by y.
Clearly, y ∈ V (H1) by the planarity of G (see Figure 2(b)). Similarly, both u1 and ut have
a common neighbor in H1.
Since S is a perfect star packing of G and the two neighbors x1 and v2 of v1 are leaves
in S, y is the center of a star in S. If the third neighbor of y is on C, then it is on C1,
denoted it by vr. The three neighbors of y separate the circle C into three sections, two of
which are subgraphs of C1, denoted by C11 and C
2
1 respectively. As the above discussion,
we know that one of v1C11vry and vrC
2
1vky is a non-facial cycle of G. By the recursive
process and the finiteness of the order of G, we can suppose that the third neighbor of y is
not on C, and denoted it by y′.
See Figure 2(b), the five vertices vk−1, vk, x2, u1, u2 belong to a common facial cycle
h′ of G. Since C does not have a chord by Remark 2.4(3), vk−1 and u2 are not adjacent
in G. So h′ is a hexagon. By the planarity of G, vk−1 and u2 have a common neighbor
in H2. so vk−2, vk−1, vk, y and y′ are on a face of G, say f . If f is a pentagon, then
vk−2 is adjacent to y′. So all the three neighbors of vk−2 are leaves in S. This implies
a contradiction since vk−2 is also a leaf in S. If f is a hexagon, then vk−2 and y′ have
a common neighbor, denoted by z. Clearly, z is vk−3 or not. For z = vk−3, the three
neighbors of vk−3 are all leaves in S, a contradiction. For z ̸= vk−3, by Remark 2.4(2),
z is a leaf in S since y′ has a neighbor y ∈ C(S). So the three neighbors of vk−2 are
all leaves in S, a contradiction. All these contradictions imply that C(S) does not have a
vertex which has three neighbors on C.
Let S be a perfect star packing of fullerene graph G and C be a cycle in G − C(S)
which is a non-facial cycle of G. C divides G into two sides, denoted by H1 and H2
respectively. Set Ci be the set of all the vertices on C each of which has a neighbor in Hi,
i = 1, 2. Clearly, {C1, C2} is a partition of V (C). G[Ci] is a vertex induced subgraph of
G which has vertex set Ci and any two vertices of Ci are adjacent if and only if they are
adjacent in G. See Figure 4, G[C1] is depicted as red and G[C2] is depicted as blue. In the
following, we use these symbols no longer explaining.
Lemma 3.4. For i = 1, 2, if a vertex x on C has a neighbor in Hi, then the component of
the induced subgraph G[Ci] which contains x is a path with 2 or 3 vertices.
Proof. We suppose that x on C has a neighhbor in H1. For the convenience of the following
description, set C := xv1v2 · · · vkx. Since C is a cycle in G− C(S) which is a non-facial
cycle of G, the length of C is at least 8. So k ≥ 7. There are three cases for the two
neighbors v1 and vk of x on C.
Case 1: Both v1 and vk have neighbors in H2.
In this case, the three vertices v1, x and vk lie on the same face f of G (see Figure 3(a)).
Since all the vertices on C are leaves in S, the other neighbor of v1 (resp. vk) which is not
on C is the center of a star in S. So f has two vertices in C(S) which are the centers of
two stars in S covered v1 and vk, respectively. So f is a hexagon by Proposition 2.1. But
the case cannot hold by Propositions 2.5.
Case 2: Both v1 and vk have neighbors in H1.
In this case, the five vertices v2, v1, x, vk, vk−1 belong to a facial cycle h of G (see
Figure 3(b)). We claim that both v2 and vk−1 have neighbors in H2. Otherwise, at least
L. Shi: The fullerene graphs with a perfect star packing 87
x
1v
2v
3v 1kv -
kvg
4v
2kv -
1H
2H
x
1v
2v
3v
1kv -
kv
h
4v
2kv -
1H
2H
3kv -
( )c( )b(a)
x
1v
2v
1kv -k
v
f
2kv -
1H
2H
Figure 3: (a) v1 and vk have neighbors in H2; (b) v1 and vk have neighbors in H1; (c) v1
has a neighbor in H1, vk has one in H2.
one of v2 and vk−1 has a neighbor in H1. If v2 has a neighbor in H1 and vk−1 has a
neighbor in H2, then the six vertices v3, v2, v1, x, vk, vk−1 lie on a face h of G. So h is a
hexagon and C has a chord v3vk−1, a contradiction. For v2 having a neighbor in H2 and
vk−1 having a neighbor in H1, we can also obtain a chord of C, a contradiction. If both v2
and vk−1 have neighbors in H1, then the seven vertices v3, v2, v1, x, vk, vk−1, vk−2 belong
to a common face h of G. This implies that G has a facial cycle of length at least 7, a
contradiction. So both v2 and vk−1 have neighbors in H2, and v2, v1, x, vk, vk−1 lie on
a hexagon h of G (see Figure 3(b)). Since C does not have a chord, the path v1xvk is a
connected component of the induced subgraph G[C1].
Case 3: v1 has a neighbor in H1 and vk has a neighbor in H2, or v1 has a neighbor in
H2 and vk has a neighbor in H1.
By symmetry, it is sufficient to consider that v1 has a neighbor in H1 and vk has a
neighbor in H2. If v2 has a neighbor in H1, then v3 must have a neighbor in H2, otherwise,
C has a chord or G has a facial cycle of length at least seven, a contradiction. As the proof
of Case 2, v3, v2, v1, x, vk lie on a hexagonal facial cycle. So the path v2v1x is a connected
component of the induced subgraph G[C1]. Now, we suppose that v2 has a neighbor in H2.
Then the four vertices vk, x, v1, v2 lie on the same face g of G. Since vk, x, v1, v2 are all
leaves in S, g is a pentagon and v2, vk have a common neighbor in H2 which is the center
of a star in S (see Figure 3(c)). So the path xv1 is a connected component of the induced
subgraph G[C1].
In summary, the component of the induced subgraph G[C1] which contains x is a path
with 2 or 3 vertices since C does not have a chord.
In addition, we have the following Lemma.
Lemma 3.5. Each component of G[Ci] is a path with 2 or 3 vertices, i = 1, 2.
Proof. For any vertex x on C, x must have exactly one neighbor in H1 or H2 since G is
3-regular and C does not have a chord. Without loss of generality, we suppose that x has
exactly one neighbor in H1. By Lemma 3.4, the component of the induced subgraph G[C1]
which contains x is a path with 2 or 3 vertices. We note that the choice of x is arbitrary. So
the conclusion holds.
88 Ars Math. Contemp. 23 (2023) #P1.05 / 81–96
1f
2f
3f
4f
5f
(a) ( )b
1H
2H
1H
2H
Figure 4: (a) A cycle C of length 25; (b) A cycle C of length 30. (G[C1] is red and G[C2]
is blue.)
Proposition 3.6. Let C = v0v1 · · · vk−1 be a non-facial cycle in G−C(S) (In the follow-
ing, the subscript is modulo k).
(i) If both vi and vi+1 have neighbors in H1 (resp. H2) and vi−1 and vi+2 have neigh-
bors in H2 (resp. H1), then the four vertices vi−1, vi, vi+1 and vi+2 lie on a pen-
tagon of G.
(ii) If vi, vi+1, vi+2 have neighbors in H1 (resp. H2) and vi−1 and vi+3 have neighbors
in H2 (resp. H1), then the five vertices vi−1, vi, vi+1, vi+2 and vi+3 lie on a hexagon
of G.
(iii) For j = 1, 2, if both vi and vi+1 have neighbors in Hj (we denote the two edges
incident to vi and vi+1 not lie in C by ei and ei+1, respectively), then the facial
cycle containing both ei and ei+1 is a hexagon, and two antipodal points on this
hexagon are centers of two stars in the perfect star packing S.
Proof. Cases (i) and (ii) can be easily obtained from the proof of the Cases 2 and 3 of
Lemma 3.4 (see Figure 3). Since all the vertices on C are leaves in the perfect star packing
S, the other end of ei (resp. ei+1) which is not on C, denoted by ui (resp. ui+1), is the
center of a star in S. We know that any facial cycle of G is a pentagon or a hexagon. So ui
and ui+1 are distinct. By Lemmas 2.1 and 2.5, the facial cycle containing both ei and ei+1
is a hexagon, and ui and ui+1 are antipodal points on this hexagon.
For example, in Figure 4, except for fi, i ∈ {1, 2, 3, 4, 5} the other faces sharing edges
L. Shi: The fullerene graphs with a perfect star packing 89
with C are all hexagons. Moreover, how the vertices on C being covered by S is deter-
mined.
We recall that the union of two graphs G1 and G2 is denoted by G1 ∪ G2, which has
vertex set V (G1) ∪ V (G2) and edge set E(G1) ∪ E(G2). Let n3 be the number of the
components of G[C1]∪G[C2] each of which is isomorphic to a path with 3 vertices. Sim-
ilarly, n2 is the number of the components of G[C1] ∪G[C2] each of which is isomorphic
to a path with 2 vertices. For example, n3 = n2 = 5 in Figure 4(a) and n3 = 10, n2 = 0
in Figure 4(b).
Observation 1. n2 + n3 is even.
Proposition 3.7. Let S be a perfect star packing of fullerene graph G and C a cycle in
G − C(S) which is a non-facial cycle of G. Then the length of C is 3n3 + 2n2, and the
length of C has the the same parity with n2 and n3.
Proof. Clearly, the length of C is 3n3 + 2n2 by Lemma 3.5. So n3 is odd if and only if
the length of C is odd. Since n2 + n3 is even by Observation 1, the parity of n2 and n3 are
same. Then we are done.
Theorem 3.8. Let S be a perfect star packing of fullerene graph G. Then G − C(S) has
even number of odd cycles.
Proof. If G − C(S) does not have a non-facial cycle of G, then any pentagon of G does
not have a vertex in C(S) by Corollary 3.2. So all the vertices on pentagons are leaves in
S. It implies that G − C(S) has exactly twelve odd cycles, each of which is a pentagon.
Next, we suppose that G− C(S) has a non-facial cycle of G, denoted by C.
Claim 1: If C is an even cycle, then G has even number of pentagons which share
edges with C. If C is an odd cycle, then G has odd number of pentagons which share
edges with C.
By Proposition 3.6, the number of pentagons which share edges with C is equal to n2.
By Proposition 3.7, n2 and the length of C have the same parity. So the Claim holds.
Claim 2: Any pentagon of G shares edges with at most one non-facial cycle in
G− C(S).
Let P be a pentagon of G. By Proposition 2.1, P has at most one vertex which is the
center of a star in S. If P does not have a vertex in C(S), then P is a cycle in G − C(S).
By Theorem 2.3, each component of G − C(S) is an induced cycle of G. So P does not
share edges with any non-facial cycle in G − C(S). If P has a vertex x ∈ C(S), then by
Corollary 3.2 P − x is a subgraph of a non-facial cycle in G − C(S). So P shares edges
with exactly one non-facial cycle in G− C(S).
Now, we consider the following two cases for the non-facial cycles in G− C(S).
Case 1: G− C(S) does not have a non-facial cycle of odd length.
Then any non-facial cycle C in G−C(S) is of even length. By the above Claims, there
are even number of pentagons in G such that they share edges with C. Since G has exactly
twelve pentagons, there are even number of pentagons in G each of which does not share
edges with non-facial cycles in G − C(S). These pentagons must be cycles in G − C(S)
by Corollary 3.2. Hence G− C(S) has even number of odd cycles.
Case 2: G− C(S) has some non-facial cycle of odd length.
Suppose that G − C(S) has exactly k non-facial cycles of odd length. We denote the
number of pentagons in G each of which does not share edges with non-facial cycles in
90 Ars Math. Contemp. 23 (2023) #P1.05 / 81–96
G − C(S) by p. These p pentagons must be cycles in G − C(S) by Corollary 3.2. So
G − C(S) has p + k odd length cycles. Next, we show that p and k have the same parity.
If p is odd, then G has odd number of pentagons each of which share edges with exactly
one non-facial cycle in G−C(S) since G has exactly 12 pentagons. By the above Claims,
for each even length non-facial cycle in G−C(S), G has even number of pentagons which
share edges with the cycle, and for each odd length non-facial cycle in G − C(S), G has
odd number of pentagons which share edges with the cycle. So G−C(S) has odd number
of non-facial cycles of odd length. This means that k is odd. For p being even, we can
similarly show that k is even. So k and p have the same parity and p+ k is even.
Clearly, for a fullerene graph G with a perfect star packing, its order must be divisible
by 4. So the order of G is 8k or 8k+4 for some positive integer k. Now, we can obtain the
following main theorem which illustrates that the order of G can not be 8k + 4.
Theorem 3.9. If fullerene graph G has a perfect star packing, then the order of G is
divisible by 8.
Proof. We suppose that S is a perfect star packing of G and Co and Ce are the collections of
all the odd cycles and even cycles in G− C(S), respectively. Then we have the following
equation.
|V (G)| = |C(S)|+
∑
C∈Co
|C|+
∑
C∈Ce
|C|
=
|V (G)|
4
+
∑
C∈Co
|C|+ even.
(3.1)
By Theorem 3.8, Co has even number of elements. Combining the above equation, we
know that |V (G)|4 × 3 is even. Hence
|V (G)|
4 is even, that is, the order of G is divisible by
8.
This theorem is equivalent to the following corollary.
Corollary 3.10. A fullerene graph with order 8n+4 does not have a perfect star packing.
We recall that a dominating set of a graph G is a set D of vertices such that each vertex
in V (G)−D is adjacent to a vertex in D. Moreover, if each vertex in V (G)−D is adjacent
to exactly one vertex in D and D is an independent vertex set, then D is called efficient.
The problem of determining the existence of efficient dominating sets in some families of
graphs was first investigated by Biggs [7] and Kratochvil [25]. Later Livingston and Stout
[27] studied the existence and construction of efficient dominating sets in families of graphs
arising from the interconnection networks of parallel computers. It is algorithmically hard
to find an efficient dominating set [3]. For more results and some historical background
regarding efficient dominating set, we refer the reader to [9, 10, 11, 22] etc..
From the definitions of the efficient dominating set and the perfect star packing of a
fullerene graph, the following proposition is a natural result.
Proposition 3.11 ([14]). A fullerene graph G with n vertices has a perfect star packing if
and only if G has an efficient dominating set of cardinality n4 .
Combining Theorem 3.9 and Proposition 3.11, we get the following theorem.
Theorem 3.12. The order of a fullerene graph with an efficient dominating set is 8n.
L. Shi: The fullerene graphs with a perfect star packing 91
4 Some other conclusions
Došlić et al. gave the following necessary condition in terms of graph spectra.
Proposition 4.1 ([14]). If a fullerene graph G has a perfect star packing, then −1 must be
an eigenvalue of the adjacency matrix of G.
The proof of this Theorem can be translate to a simple r-regular graph. Here for com-
pleteness, we prove as follows. For the definition of eigenvalues of the adjacency matrix of
a graph, we refer the reader to [18].
Theorem 4.2. If a simple r-regular graph G has a perfect K1,r-packing S, then −1 must
be an eigenvalue of the adjacency matrix of G.
Proof. Let C(S) be the set of centers of stars K1,r in S. We define the characteristic vector−→c ∈ R|V (G)| of C(S) as follows: ci = 1 if i ∈ C(S), otherwise ci = 0. Since G is a
r-regular graph, we have A−→u = r−→u , where A is the adjacency matrix of G and −→u is the
all one vectors. Let −→w = −→u − (r + 1)−→c . As A−→c = −→u −−→c , we have
A−→w = A−→u − (r+1)A−→c = r−→u − (r+1)−→u +(r+1)−→c = (r+1)−→c −−→u = −−→w (4.1)
This means that −1 is an eigenvalue of A.
For a perfect star packing S of fullerene graph G, if for each center x ∈ C(S), all the
three faces of G sharing x are hexagons, then we call S being type P0. For such perfect
star packing, the following corollary holds.
Corollary 4.3. If a fullerene graph G has a perfect star packing S of type P0, then G −
C(S) does not have a non-facial cycle of odd length.
Proof. By the contrary, we suppose that G−C(S) has a non-facial cycle C of odd length.
By the Claim 1 of Theorem 3.8, G has a pentagon P which share edges with C. This
implies that P contains the center y of a star in S. So one of the three faces of G sharing
y is not a hexagon. This contradicts that S is of type P0. So G − C(S) does not have a
non-facial cycle of odd length.
In the above Corollary, we note that G − C(S) may have non-facial cycles of even
lengths (see Figure 5, the blue cycle in C120).
Now, we point out the flaw of the Theorem 14 in [14].
Theorem 4.4 ([14]). A fullerene graph on 8n vertices has a perfect star packing of type
P0 if and only if it arises from some other fullerene via the chamfer transformation.
Readers can consult reference [14] to see the chamfer transformation. Here for com-
pleteness, we introduce it as follows. Let F be a fullerene graph. In each face g of F , we
draw a polygon with the same number of sides as g. For each vertex v ∈ V (F ), we connect
v with three new vertices each of which is inside exactly one face of F incident with v (see
Figure 6, the vertices of original fullerene C20 are black, the new vertices are blue, each
black vertex are connected to three blue vertices). We notice that each new vertex must be
adjacent to exactly one vertex of F in this process, and the edges do not intersect inside.
Finally, we remove all the edges of F . The resulting graph is called arising from F via the
chamfer transformation. For example, (see Figure 6) the graph C80(Ih) arises from C20
92 Ars Math. Contemp. 23 (2023) #P1.05 / 81–96
120
C
144
C
384
C
Figure 5: Each of C120, C144, C384 has a unique perfect star packing of type P0 which is
depicted in bold edges.
L. Shi: The fullerene graphs with a perfect star packing 93
Figure 6: C20 is drawn in black line, C80(Ih) is drawn in red line.
via the chamfer transformation, and all the black vertices are the centers of stars in a perfect
star packing of type P0 of C80(Ih).
For a perfect star packing S of type P0 in fullerene graph G, we construct a new graph
with respect to S, and denoted it by GS . V (GS) := C(S) and any two vertices in V (GS)
are adjacent if and only if they belong to the same hexagon of G. In the proof of the
necessity of the Theorem 4.4, there exist the following problem. GS is planar, but does
not have to be 3-regular, 3-connected and have only pentagonal and hexagonal faces. For
example, it is easy to check that the fullerene graph C120 (resp. C144, C384) has a unique
perfect star packing S1 (resp. S2, S3) of type P0 (as depicted in bold edges in Figure 5).
CS1120, C
S2
144 and C
S3
384 are planar and not connected (the red dashed line in Figure 5 is the
CS1120, and here we omit the C
S2
144 and C
S3
384). In fact, we have Lemma 4.5.
I would like to thank Tomislav Došlić for conversations and email exchanges related to
the contents of this paragraph.
Lemma 4.5. The three fullerene graphs C120, C144 and C384 as depicted in Figure 5 can-
not arise from some other fullerene via the chamfer transformation.
Proof. On the contrary, we suppose that C120 can arise from some fullerene F via the
chamfer transformation. Then C120 has a perfect star packing S of type P0 which corre-
sponds to the chamfer transformation of F , that is, all the vertices of F are the centers of
stars in S. This means that CS120 = F .
We can check that C120 has a unique perfect star packing of type P0, denoted by S1
(as depicted in bold edges in Figure 5). So S1 = S. However, CS1120 is not connected (as
depicted by red dotted lines in Figure 5). So S1 ̸= S, a contradiction.
For the other two fullerenes C144 and C384, we can also check that each of them has a
unique perfect star packing of type P0 (as depicted in bold edges in Figure 5). As the above
proof, they also cannot arise from any fullerene graphs via the chamfer transformations.
From Lemma 4.5 we know that the necessity of Theorem 4.4 does not hold, however,
its sufficiency is right. So it can be corrected as follows.
94 Ars Math. Contemp. 23 (2023) #P1.05 / 81–96
Theorem 4.6. A fullerene graph that arises from some other fullerene via the chamfer
transformation must have a perfect star packing of type P0.
If fullerene graph G has two pentagons sharing an edge xy, then x (resp. y) can not be
center of a star in a perfect star packing of G by Lemma 2.2. Since all the three neighbors
of x belong to pentagons of G, G does not have a perfect star packing of type P0. Hence if
a fullerene graph has a perfect star packing of type P0, then all its pentagons are isolated.
Next we list some other forbidden subgraphs for guaranteeing a fullerene graph to own a
perfect star packing of type P0.
1PP 3PP 4PP
1
v1v 2v2v 1
v
2
v
1
x
2
x
3
x
Figure 7: Three forbidden configurations.
Proposition 4.7. If a fullerene graph G contains a subgraph PP1, PP3 or PP4 (see
Figure 7), then it cannot have a perfect star packing of type P0.
Proof. By the contrary, we suppose that G has a perfect star packing of type P0, denoted
by S. Clearly, the vertices v1 and v2 (see Figure 7) are leaves in S. If PP4 is a subgraph
of G, then x1 is the center of a star in S since all vertices on a pentagon are leaves in S. So
x2 is a leaf in S. By Remark 2.4(2), the neighbor x3 of x2 is also a leaf in S. This implies
that all the three neighbors of v2 are leaves in S, a contradiction. For subgraphs PP1 and
PP3, we can similarly show that v1 or v2 have all its three neighbors being leaves in S, a
contradiction.
ORCID iDs
Lingjuan Shi https://orcid.org/0000-0001-9440-4660
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.06 / 97–119
https://doi.org/10.26493/1855-3974.2697.43a
(Also available at http://amc-journal.eu)
On the Aα-spectral radius of connected graphs*
Abdollah Alhevaz , Maryam Baghipur
Faculty of Mathematical Sciences, Shahrood University of Technology,
P.O. Box: 316-3619995161, Shahrood, Iran
Hilal Ahmad Ganie
Department of School Education, JK Govt. Kashmir, India
Kinkar Chandra Das †
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea
Received 14 September 2021, accepted 23 June 2022, published online 5 October 2022
Abstract
For a simple graph G, the generalized adjacency matrix Aα(G) is defined as Aα(G) =
αD(G) + (1 − α)A(G), α ∈ [0, 1], where A(G) is the adjacency matrix and D(G) is the
diagonal matrix of the vertex degrees. It is clear that A0(G) = A(G) and 2A 1
2
(G) =
Q(G) implying that the matrix Aα(G) is a generalization of the adjacency matrix and the
signless Laplacian matrix. In this paper, we obtain some new upper and lower bounds for
the generalized adjacency spectral radius λ(Aα(G)), in terms of vertex degrees, average
vertex 2-degrees, the order, the size, etc. The extremal graphs attaining these bounds are
characterized. We will show that our bounds are better than some of the already known
bounds for some classes of graphs. We derive a general upper bound for λ(Aα(G)), in
terms of vertex degrees and positive real numbers bi. As application, we obtain some new
upper bounds for λ(Aα(G)). Further, we obtain some relations between clique number
ω(G), independence number γ(G) and the generalized adjacency eigenvalues of a graph
G.
Keywords: Adjacency matrix, signless Laplacian matrix, generalized adjacency matrix, spectral ra-
dius, degree sequence, clique number, independence number.
Math. Subj. Class. (2020): Primary: 05C50, 05C12; Secondary: 15A18.
*The authors would like to thank the handling editor and two anonymous referees for their detailed constructive
comments that helped improve the quality of the paper.
†Corresponding author. Partially supported by the National Research Foundation of the Korean government
with grant No. 2021R1F1A1050646.
E-mail addresses: a.alhevaz@shahroodut.ac.ir (Abdollah Alhevaz), maryamb8989@gmail.com (Maryam
Baghipur), hilahmad1119kt@gmail.com (Hilal Ahmad Ganie), kinkardas2003@gmail.com (Kinkar Chandra
Das)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
98 Ars Math. Contemp. 23 (2023) #P1.06 / 97–119
1 Introduction
Let G = (V (G), E(G)) be a simple connected graph with vertex set V (G) = {v1, v2, . . . ,
vn} and edge set E(G). The order of G is the number n = |V (G)| and its size is the
number m = |E(G)|. The set of vertices adjacent to v ∈ V (G), denoted by N(v), refers
to the neighborhood of v. The degree of v, denoted by dG(v) (we simply write dv if it
is clear from the context) means the cardinality of N(v). A graph is called regular if all
vertices have the same degree. The graph G is the complement of the graph G. Moreover,
the complete graph Kn, the complete bipartite graph Ks,t, the path Pn, the cycle Cn and
the star Sn are defined in the conventional way. The distance between two vertices u, v ∈
V (G), denoted by duv , is defined as the length of a shortest path between u and v in G.
The diameter of G is the maximum distance between any two vertices of G. Let mi be the
average degree of the adjacent vertices of vertex vi in G. If vi is an isolated vertex in G,
then we assume that mi = 0. Hence we can write
mi =
0 di = 0.1
di
∑
j:j∼i dj otherwise.
Let pi be the average degree of the vertices non-adjacent to vertex vi in G. If vi is adjacent
to all the remaining vertices, then we assume that pi = 0. Then we can write
pi =
0 di = n− 1.∑j:j≁i, j ̸=i dj
n−di−1 otherwise.
Let D(G) be the diagonal matrix of vertex degrees and A(G) be the adjacency matrix
of G. The signless Laplacian matrix of G is Q(G) = D(G) + A(G). Its eigenvalues can
be arranged as: q1(G) ≥ q2(G) ≥ · · · ≥ qn(G). In [20], Nikiforov proposed the following
matrix:
Aα(G) = αD(G) + (1− α)A(G), 0 ≤ α ≤ 1,
calling it the generalized adjacency matrix of G. Obviously, A0(G) = A(G), 2A 1
2
(G) =
Q(G), A1(G) = D(G) and Aα(G) − Aβ(G) = (α − β)L(G), where L(G) is the
well-studied Laplacian matrix of G, defined as L(G) = D(G) − A(G). Therefore, the
family Aα(G) can extend both A(G) and Q(G). The matrix Aα(G) is a real symmetric
matrix, therefore we can arrange its eigenvalues as λ1(Aα(G)) ≥ λ2(Aα(G)) ≥ · · · ≥
λn(Aα(G)), where λ1(Aα(G)) is called the generalized adjacency spectral radius of G.
Afterwards, we will denote λ1(Aα(G)) by λ(Aα(G)). If G is a connected graph and
α ̸= 1, then the matrix Aα(G) is non-negative and irreducible. Therefore by the Perron-
Frobenius theorem, λ(Aα(G)) is the simple eigenvalue and there is a unique positive unit
eigenvector x corresponding to λ(Aα(G)), which is called the generalized adjacency Per-
ron vector of G.
A column vector x = (x1, x2, . . . , xn)T ∈ Rn can be considered as a function defined
on V (G) which maps vertex vi to xi, i.e., x(vi) = xi for i = 1, 2, . . . , n. Then,
⟨x, Aαx⟩ = xTAα(G)x = α
n∑
i=1
dix
2
i + 2(1− α)
∑
i∼j
xixj ,
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 99
and λ is an eigenvalue of Aα(G) corresponding to the eigenvector x if and only if x ̸= 0
and
λxi = αdixi + (1− α)
∑
j∼i
xj , i = 1, 2, . . . , n.
These equations are called the (λ, x)-eigenequations of G. For a normalized column vector
x ∈ Rn, by the Rayleigh’s principle, we have
λ(Aα(G)) ≥ xTAα(G)x
with equality if and only if x is the generalized adjacency Perron vector of G.
The research on the (adjacency, signless Laplacian) spectrum is an intriguing topic
during past two decades [4, 10, 22]. At the same time, the adjacency or signless Laplacian
spectral radius have attracted many interests among the mathematical literature including
linear algebra and graph theory. An interesting problem in the spectral graph theory is
to obtain bounds for the (adjacency, signless Laplacian) spectral radius connecting it with
different parameters associated with the graph. Another interesting problem which is worth
to mention is to characterize the extremal graphs for the (adjacency, signless Laplacian)
spectral radius among all graphs of order n or among a special class of graphs of order
n. The spectral radius λ(G) of the adjacency matrix A(G), called the spectral radius (or
adjacency spectral radius) of the graph G and the spectral radius q1(G) of the signless
Laplacian matrix Q(G), called signless Laplacian spectral radius of the graph G, are both
well studied and their spectral theories are well developed. Various papers can be found in
the literature regarding the establishment of bounds for λ(G) and q1(G) connecting them
with different parameters associated with the structure of the graph G. Since the matrix
Aα(G) is a generalization of the matrices A(G) and Q(G), therefore it will be interesting
to see whether the results which already hold for the spectral radius of the matrices A(G)
and/or Q(G) can be extended to the spectral radius of the Aα(G). This is one of the
motivation to study the spectral radius of the matrix Aα(G).
Let A(G) be the adjacency matrix of the graph G and let B be a real diagonal matrix
of order n. In 2002, Bapat et al. [1] defined the matrix L
′
= B − A(G) and called it
the perturbed Laplacian matrix of the graph G. The aim of introducing this matrix was to
generalize the results that hold for the adjacency matrix and the Laplacian matrix L(G) of
the graph to some general class of matrices. For α ̸= 1, it is easy to see that
Aα(G) = αD(G) + (1− α)A(G) = (α− 1)
( α
α− 1
D(G)−A(G)
)
.
Clearly αα−1D(G) is a diagonal matrix with real entries, giving that the matrix Aα(G) is
a scaler multiple of a perturbed Laplacian matrix. This is another motivation to study the
spectral properties of the matrix Aα(G).
Although the generalized adjacency matrix Aα(G) of a graph G was introduced in
2017, but a large number of papers can be found in the literature regarding the spectral
properties of this matrix. Like other graph matrices, most of these papers are regarding the
generalized spectral radius λ(Aα(G)). In fact, various upper and lower bounds connecting
λ(Aα(G)) with different graph parameters and the graphs attaining these bounds can be
found in the literature. For some recent works regarding the spectral properties of Aα(G),
we refer to [8, 9, 11, 13, 14, 15, 16, 17, 21, 23, 24].
The rest of this paper is organized as follows. In Section 2, we obtain some new upper
and lower bounds for λ(Aα(G)), in terms of vertex degrees, average vertex 2-degrees, the
100 Ars Math. Contemp. 23 (2023) #P1.06 / 97–119
order, the size, etc. The extremal graphs attaining these bounds are characterized. We will
show that our bounds are better than some of the already known bounds for some classes
of graphs. In Section 3, we derive a general upper bound for λ(Aα(G)), in terms of vertex
degrees and positive real numbers bi. As application, we obtain some new upper bounds
for λ(Aα(G)). In Section 4, we obtain some relations between clique number ω(G), inde-
pendence number γ(G) and the generalized adjacency eigenvalues. We conclude this paper
by a remark in Section 5.
2 Bounds on generalized adjacency spectral radius
The average 2-degree of a vertex vi ∈ V (G) is denoted by mi = m(vi) and is defined as
mi =
∑
k:k∼i
dk
di
, where dk is the degree of the vertex vk.
The following gives an upper bound for the generalized adjacency spectral radius
λ(Aα(G)) of a graph in terms of the vertex degrees, the average vertex 2-degrees and
the parameter α.
Theorem 2.1. Let G be a graph of order n having vertex degrees di, vertex average 2-
degrees mi, 1 ≤ i ≤ n, and let α ∈ [0, 1]. Then
λ(Aα(G)) ≤ max
1≤i≤n
{
αdi + (1− α)
√
dimi
}
.
Moreover, the equality holds if G is a k-regular graph.
Proof. Let x = (x1, . . . , xn) be the generalized adjacency Perron vector of G and let
∥x∥ = 1. For any vi ∈ V (G), we have λ(Aα(G))xi = αdixi+(1−α)
∑
j:j∼i xj . Hence
λ2(Aα(G))x
2
i = α
2d2ix
2
i + 2α(1− α)dixi
∑
j:j∼i
xj + (1− α)2
∑
j:j∼i
xj
2 . (2.1)
By Cauchy-Schwarz inequality, we obtain∑
j:j∼i
xj
2 ≤ di ∑
j:j∼i
x2j . (2.2)
Therefore from (2.1) and (2.2), we get
λ2(Aα(G))x
2
i ≤ α2d2ix2i + 2αdixi
[
λ(Aα(G))xi − αdixi
]
+ (1− α)2di
∑
j:j∼i
x2j .
Thus, taking sum over all vi ∈ V (G), we get∑
vi∈V (G)
λ2(Aα(G))x
2
i
≤
∑
vi∈V (G)
[
2αdiλ(Aα(G))− α2d2i
]
x2i + (1− α)2
∑
vi∈V (G)
di
∑
j:j∼i
x2j
=
∑
vi∈V (G)
[
2αdiλ(Aα(G))− α2d2i
]
x2i + (1− α)2
∑
vi∈V (G)
dimix
2
i
=
∑
vi∈V (G)
[
2αdiλ(Aα(G))− α2d2i + (1− α)2dimi
]
x2i
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 101
as ∑
vi∈V (G)
di
∑
j:j∼i
x2j =
∑
vi∈V (G)
x2i
∑
j:j∼i
dj =
∑
vi∈V (G)
dimix
2
i .
From the above result, we obtain∑
vi∈V (G)
(
λ2(Aα(G))− 2αdiλ(Aα(G)) + α2d2i − (1− α)2dimi
)
x2i ≤ 0.
This is only true if there exist a vertex, say vj ∈ V (G), such that
λ2(Aα(G))− 2αdjλ(Aα(G)) + α2d2j − (1− α)2djmj ≤ 0,
hence, we get
λ(Aα(G)) ≤ αdj + (1− α)
√
djmj ≤ max
1≤i≤n
{
αdi + (1− α)
√
dimi
}
.
Now, suppose that G is a k-regular graph. So, for i = 1, . . . , n, we have di = mi = k,
then αdi + (1− α)
√
dimi = k and λ(Aα(G)) = k. This shows that equality occurs for a
regular graph.
For α = 0, the upper bound given by Theorem 2.1 reduces to the upper bound in the
following corollary.
Corollary 2.2 ([5]). Let G be a graph of order n having vertex degrees di, vertex average
2-degrees mi, 1 ≤ i ≤ n. Then
λ(A(G)) ≤ max
1≤i≤n
√
di mi.
The following upper bound for the generalized adjacency spectral radius λ(Aα(G)), in
terms of vertex degrees and average vertex 2-degrees was obtained in [20]:
Theorem 2.3. If G is a graph with no isolated vertices, then
λ(Aα(G)) ≤ max
vj∈V
{
αdj + (1− α)mj
}
.
If α ∈
(
1
2 , 1
)
and G is connected, equality holds if and only if G is regular.
Remark 2.4. For non-regular graphs the upper bound given by Theorem 2.1 and the upper
bound given by Theorem 2.3 are incomparable for different values of α. For example,
consider the graph G = K4 − e. For this graph we have d1 = 2, d2 = 3, d3 = 2, d4 =
3,m1 = 3,m2 =
7
3 ,m3 = 3 and m4 =
7
3 . By Theorem 2.3, we have
λ(Aα(G)) ≤ max
{
3− α, 7
3
+
2
3
α
}
.
It is easy to see that
max
{
3− α, 7
3
+
2
3
α
}
=
{
7
3 +
2
3α for α > 0.4,
3− α for α ≤ 0.4.
102 Ars Math. Contemp. 23 (2023) #P1.06 / 97–119
Also, by Theorem 2.1, we have
λ(Aα(G)) ≤ max
{√
6 + (2−
√
6)α,
√
7 + (3−
√
7)α
}
=
√
7 + (3−
√
7)α.
For α ≤ 0.4, we have 3 − α >
√
7 + (3 −
√
7)α giving that α < 5−
√
7
9 ≈ 0.2615. This
gives that for 0 ≤ α < 5−
√
7
9 , the upper bound given by Theorem 2.1 is better than the
upper bound given by Theorem 2.3; while as for 5−
√
7
9 ≤ α ≤ 0.4, the upper bound given
by Theorem 2.3 is better than the upper bound given by Theorem 2.1 for the graph K4 − e.
For the graph G = K1,3, we have d1 = 3, d2 = 1, d3 = 1, d4 = 1,m1 = 1,m2 = 3,m3 =
3 and m4 = 3. By Theorem 2.3, we have
λ(Aα(G)) ≤ max
{
1 + 2α, 3− 2α
}
.
It is easy to see that
max
{
1 + 2α, 3− 2α
}
=
{
3− 2α for α < 0.5,
1 + 2α for α ≥ 0.5.
Also, by Theorem 2.1, we have
λ(Aα(G)) ≤ max
{√
3 + (3−
√
3)α,
√
3 + (1−
√
3)α
}
=
√
3 + (3−
√
3)α.
For α < 0.5, we have 3− 2α >
√
3 + (3−
√
3)α giving that α < 3−
√
3
5−
√
3
≈ 0.38799. This
gives that for 0 ≤ α < 3−
√
3
5−
√
3
, the upper bound given by Theorem 2.1 is better than the
upper bound given by Theorem 2.3; while as for 3−
√
3
5−
√
3
≤ α < 0.5, the upper bound given
by Theorem 2.3 is better than the upper bound given by Theorem 2.1 for the graph K1,3.
The following gives another upper bound for the generalized adjacency spectral radius
λ(Aα(G)) of a graph G in terms of the vertex degrees, the average vertex 2-degrees and
the unknown parameter β.
Theorem 2.5. Let G be a connected graph of order n having vertex degrees di, average
vertex 2-degrees mi, 1 ≤ i ≤ n, and let α ∈ [0, 1). Then
λ(Aα(G)) ≤ max
1≤i≤n
{
−β +
√
β2 + 4di(αdi + (1− α)mi + β)
2
}
, (2.3)
where β ≥ 0 is an unknown parameter. Equality occurs if and only if G is a regular graph.
Proof. Let x = (x1, . . . , xn) be the generalized adjacency Perron vector of G and let
xi = max
1≤j≤n
xj .
Since
λ2(Aα(G))x = (Aα(G))
2
x = (αD + (1− α)A)2x
= α2D2x+ α(1− α)DAx+ α(1− α)ADx+ (1− α)2A2x,
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 103
we have
λ2(Aα(G))xi = α
2d2ixi + α(1− α)di
∑
j:j∼i
xj + α(1− α)
∑
j:j∼i
djxj
+ (1− α)2
∑
j:j∼i
∑
k:k∼j
xk.
Now, we consider a simple quadratic function of λ(Aα(G)):(
λ2(Aα(G)) + βλ(Aα(G))
)
x =
(
α2D2x+ α(1− α)DAx+ α(1− α)ADx
+ (1− α)2A2x
)
+ β(αDx+ (1− α)Ax).
Considering the i-th equation, we have(
λ2(Aα(G)) + βλ(Aα(G))
)
xi = α
2d2ixi + α(1− α)di
∑
j:j∼i
xj + α(1− α)
∑
j:j∼i
djxj
+ (1− α)2
∑
j:j∼i
∑
k:k∼j
xk + β
(
αdixi + (1− α)
∑
j:j∼i
xj
)
.
One can easily see that
α(1− α)di
∑
j:j∼i
xj ≤ α(1− α)d2ixi, α(1− α)
∑
j:j∼i
djxj ≤ α(1− α)dimixi,
(1− α)2
∑
j:j∼i
∑
k:k∼j
xk ≤ (1− α)2dimixi, (1− α)
∑
j:j∼i
xj ≤ (1− α)dixi.
Hence, we obtain(
λ2(Aα(G)) + βλ(Aα(G))
)
xi ≤ di(αdi + (1− α)mi)xi + βdixi,
that is, λ2(Aα(G)) + βλ(Aα(G))− di(αdi + (1− α)mi + β) ≤ 0,
that is, λ(Aα(G)) ≤
−β +
√
β2 + 4di(αdi + (1− α)mi + β)
2
.
From this the inequality (2.3) follows.
Suppose that equality occurs in (2.3). Then all the inequalities in the above argument
occur as equalities. Thus we obtain
α(1− α)di
∑
j:j∼i
xj = α(1− α)d2ixi, α(1− α)
∑
j:j∼i
djxj = α(1− α)dimixi,
(1− α)2
∑
j:j∼i
∑
k:k∼j
xk = (1− α)2dimixi, (1− α)
∑
j:j∼i
xj = (1− α)dixi.
Therefore we must have xj = xi for any j : j ∼ i and xk = xi for any k : k ∼ j, j ∼ i.
Let U = {vℓ : xℓ = xi}. Now we have to prove that U = V (G). Assume to the contrary
104 Ars Math. Contemp. 23 (2023) #P1.06 / 97–119
that U ̸= V (G). Then there exists a vertex r in U such that N(r) ⊆ U and t ∈ V (G)\U
with t ∼ s, where s ∈ N(r). Then xt < xi. One can easily see that
λ(Aα(G)) <
−β +
√
β2 + 4dr(αdr + (1− α)mr + β)
2
≤ max
1≤i≤n
{
−β +
√
β2 + 4di(αdi + (1− α)mi + β)
2
}
,
a contradiction as the equality holds in (2.3). Therefore U = V (G). Then x1 = x2 =
· · · = xn and λ(Aα(G)) = di, i = 1, 2, . . . , n. Hence G is a regular graph.
Conversely, let G be a r-regular graph. Then
λ(Aα(G)) = r = max
1≤i≤n
{
−β +
√
β2 + 4di(αdi + (1− α)mi + β)
2
}
.
This completes the proof of the theorem.
The following upper bound for the generalized adjacency spectral radius λ(Aα(G)), in
terms of vertex degrees and average vertex 2-degrees was obtained in [20]:
λ(Aα(G)) ≤ max
1≤i≤n
{√
αd2i + (1− α)wi
}
, (2.4)
where wi = dimi for i = 1, . . . , n. Also, equality holds if and only if αd2i + (1− α)wi is
same for all i.
Remark 2.6. For a connected graph G of order n, the upper bound given by Theorem 2.5
reduces to the upper bound given by (2.4) for β = 0. For β ̸= 0, the upper bound given by
Theorem 2.5 is incomparable with the upper bound given by (2.4). For example, consider
the graph G = K1,3. For this graph, the upper bound (2.4) gives
λ(Aα(G)) ≤ max
{√
3 + 6α,
√
3− 3α
}
=
√
3 + 6α.
While as the upper bound given by Theorem 2.5 gives
λ(Aα(G)) ≤max
{
−β +
√
β2 + 12β + 12α+ 12
2
,
−β +
√
β2 + 4β − 8α+ 12
2
}
=
−β +
√
β2 + 12β + 12α+ 12
2
.
Taking β = 1, we have
−β +
√
β2 + 12β + 12α+ 12
2
=
−1 +
√
25 + 12α
2
<
√
3 + 6α
giving that 3α2 − 8α + 2 < 0. This last inequality holds provided that α > 4−
√
10
3 ≈
0.279240. This shows that for β = 1, the upper bound given by Theorem 2.5 is better
than the upper bound given by (2.4) for α > 4−
√
10
3 . Taking β = 0.5, it can be seen that
the upper bound given by Theorem 2.5 is better than the upper bound given by (2.4) for
α > 0.177 and for β = 0.1, it can be seen that the upper bound given by Theorem 2.5 is
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 105
better than the upper bound given by (2.4) provided that α > 0.008.
Since for β = 0, the upper bounds given by Theorem 2.5 and inequality (2.4) are
same and for the graph K1,3, it follows from the above discussion that for small value of β
the upper bound given by Theorem 2.5 behaves well for all α, incomparable to the upper
bound given by (2.4). This gives that the choice of parameter β in the upper bound given
by Theorem 2.5 can be helpful to obtain a better upper bound.
Let xi = min{xj , j = 1, . . . , n} be the minimum among the entries of the generalized
distance Perron vector x = (x1, . . . , xn) of the graph G. Proceeding similar to Theorem
2.5, we obtain the following lower bound for λ(Aα(G)), in terms of the vertex degrees, the
average vertex 2-degrees and the unknown parameter β.
Theorem 2.7. Let G be a connected graph of order n having vertex degrees di, average
vertex 2-degrees mi, 1 ≤ i ≤ n, and let α ∈ [0, 1). Then
λ(Aα(G)) ≥ min
1≤i≤n
{
−β +
√
β2 + 4di(αdi + (1− α)mi + β)
2
}
,
where β ≥ 0 is an unknown parameter. Equality occurs if and only if G is a regular graph.
The following lower bound for the generalized adjacency spectral radius λ(Aα(G)), in
terms of vertex degrees and average vertex 2-degrees was obtained in [20]:
λ(Aα(G)) ≥ min
1≤i≤n
{√
αd2i + (1− α)wi
}
, (2.5)
where wi = dimi for i = 1, . . . , n. Equality occurs if and only if αd2i +(1−α)wi is same
for all i.
Remark 2.8. For a connected graph G of order n, the lower bound given by Theorem 2.7
reduces to the lower bound given by (2.5), for β = 0. For β ̸= 0, the lower bound given by
Theorem 2.7 is incomparable with the lower bound given by (2.5). For example, consider
the graph G = K1,3. For this graph, the lower bound (2.5) gives
λ(Aα(G)) ≥ min
{√
3 + 6α,
√
3− 3α
}
=
√
3− 3α.
While as the lower bound given by Theorem 2.7 gives
λ(Aα(G)) ≥ min
{
−β +
√
β2 + 12β + 12α+ 12
2
,
−β +
√
β2 + 4β − 8α+ 12
2
}
=
−β +
√
β2 + 4β − 8α+ 12
2
.
Taking β = 1, we have
−β +
√
β2 + 4β − 8α+ 12
2
=
−1 +
√
17− 8α
2
>
√
3− 3α
giving that 4α2 + 20α − 8 > 0. This last inequality holds provided that α >
√
33−5
2 ≈
0.372281. This shows that for β = 1, the lower bound given by Theorem 2.7 is better
than the lower bound given by (2.5) for α >
√
33−5
2 . Taking β = 0.1, it can be seen that
the lower bound given by Theorem 2.7 is better than the lower bound given by (2.5) for
106 Ars Math. Contemp. 23 (2023) #P1.06 / 97–119
α > 0.09 and for β = 0.01, it can be seen that the lower bound given by Theorem 2.7 is
better than the lower bound given by (2.5) provided that α > 0.023.
Again, it follows from the above discussion that for small value of β the lower bound
given by Theorem 2.7 behaves well for all α, incomparison to the lower bound given by
(2.5) for the graph K1,3. This gives that the choice of parameter β in the lower bound given
by Theorem 2.7 can be helpful to obtain a better lower bound.
We note that if, in particular we take the parameter β in Theorem 2.5/Theorem 2.7
equal to the vertex covering number, the edge covering number, the clique number, the
independence number, the domination number, the generalized adjacency rank, minimum
degree, maximum degree, etc., then Theorems 2.5/ Theorem 2.7 gives upper bound/lower
bound for λ(Aα(G)), in terms of the vertex covering number, the edge covering number,
the clique number, the independence number, the domination number, the generalized ad-
jacency rank, minimum degree, maximum degree, etc.
Let Sn be the class of graphs of order n with maximum degree n − 1. Clearly,
K1,n−1, Kn ∈ Sn. The following result gives an upper bound for maxvj∈V {dj +mj} in
terms of order n and size m.
Lemma 2.9 ([3]). Let G be a graph of order n with m edges. Then
max
1≤j≤n
{
dj +mj
}
≤ 2m
n− 1
+ n− 2, (2.6)
with equality if and only if G ∈ Sn or G ∼= Kn−1 ∪K1.
We now generalize the above result.
Theorem 2.10. Let G be a graph of order n with m edges and real numbers β, θ with
β ≥ θ > 0. Then
max
1≤j≤n
{
βdj + θmj
}
≤ 2mθ
n− 1
+ β (n− 1)− θ, (2.7)
with equality if and only if G ∈ Sn or G ∼= Kn−1 ∪K1 (β = θ).
Proof. If β = θ > 0, then by Lemma 2.9, we get the required result in (2.7). Moreover, the
equality holds if and only if G ∈ Sn or G ∼= Kn−1 ∪K1 (β = θ). Otherwise, β > θ > 0.
Let vi be the vertex in G such that
max
1≤j≤n
{
βdj + θmj
}
= βdi + θmi.
We have 2m = di + dimi + (n− di − 1) pi, where pi is the average of the degrees of the
vertices non-adjacent to vertex vi in G. We consider the following two cases:
Case 1: di = n− 1. One can easily see that
max
1≤j≤n
{
βdj + θmj
}
= βdi + θmi =
2mθ
n− 1
+ β (n− 1)− θ.
In this case G ∈ Sn.
Case 2: di ≤ n− 2. Now, to arrive at (2.7), we need to show that
βdi + θmi ≤
di + dimi + (n− di − 1)pi
n− 1
θ + β (n− 1)− θ,
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 107
that is,
(n− di − 1)
(
(n− 1)β + (pi − 1−mi) θ
)
≥ 0,
that is,
(n− 1)β + (pi − 1−mi) θ ≥ 0,
that is,
(n− 1)β − (∆− δ + 1) θ ≥ 0, (2.8)
as mi ≤ ∆ and pi ≥ δ. We consider the following two subcases:
Subcase 2.1: G is disconnected. Then ∆ ≤ n−2. From (2.8), we obtain (n−1) (β−θ) >
0, which is true always as β > θ > 0. This shows that the inequality (2.8) strictly holds in
this case.
Subcase 2.2: G is connected. In this case ∆− δ ≤ n− 2, again it follows from (2.8) that
(n−1) (β−θ) > 0, which is true always as β > θ > 0. This shows that the inequality (2.8)
strictly holds in this case as well.
As an immediate consequence of Theorem 2.10, we get the following corollary.
Corollary 2.11. Let G be a graph of order n with m edges and real number α ≥ 12 . Then
max
1≤j≤n
{
αdj + (1− α)mj
}
≤ 2m (1− α)
n− 1
+ αn− 1, (2.9)
with equality if and only if G ∈ Sn or G ∼= Kn−1 ∪K1 (α = 1/2).
Combining Theorem 2.3 with Corollary 2.11, we get the following result, which gives
an upper bound for the generalized adjacency spectral radius λ(Aα(G)), in terms of the
order n, the size m and the parameter α.
Theorem 2.12. Let G be a graph of order n with m edges, with no isolated vertices and
let α ∈
[
1
2 , 1
]
. Then
λ(Aα(G)) ≤
2m (1− α)
n− 1
+ αn− 1.
If α ∈
(
1
2 , 1
)
and G is connected, equality holds if and only if G = Kn.
Let Γ be the class of graphs G = (V,E) such that the maximum degree vertex (of
degree ∆) are adjacent to the vertices of degree ∆ and non-adjacent to the vertices of
degree δ. If m is the number of edges in G (∈ Γ), then
2m = ∆(∆+ 1) + (n−∆− 1) δ.
The following result gives an upper bound for di +mi in terms of the order n, the size m,
the maximum degree ∆ and the minimum degree δ.
Lemma 2.13 ([3]). Let G be a graph of order n with m edges having maximum degree ∆
and minimum degree δ. Then
di +mi ≤
2m
n− 1
+ ∆− δ + ∆
n− 1
[
n− 2− (∆− δ)
]
,
with equality if and only if G ∈ Sn or G ∈ Γ.
108 Ars Math. Contemp. 23 (2023) #P1.06 / 97–119
The following result gives an upper bound for maxvj∈V {β dj + θmj} in terms of the
order n, the size m, the maximum degree ∆, the minimum degree δ and the parameters β,
θ.
Theorem 2.14. Let G be a graph of order n with m edges and real numbers β, θ with
β ≥ θ > 0. Then
max
vj∈V
{βdj + θmj} ≤
2mθ
n− 1
+ θ (∆− δ) + ∆
n− 1
[
β (n− 1)− θ(∆− δ + 1)
]
(2.10)
with equality if and only if G ∈ Sn or G ∈ Γ.
Proof. Let vi be a vertex in G such that
max
vj∈V
{β dj + θmj} = β di + θmi.
First we assume that β = θ. Then by Lemma 2.13, we obtain
max
vj∈V
{βdj + θmj} = β (di +mi) ≤ β
[ 2m
n− 1
+ ∆− δ + ∆
n− 1
(
n− 2− (∆− δ)
)]
=
2mθ
n− 1
+ θ (∆− δ) + ∆
n− 1
[
β (n− 1)− θ(∆− δ + 1)
]
,
as β > 0. Moreover, the equality holds in (2.10) if and only if G ∈ Sn or G ∈ Γ.
Next, we assume that β > θ. We consider the following two cases:
Case 1: di = n− 1. In this case
β di + θmi = β (n− 1) + θ
2m− (n− 1)
n− 1
=
2mθ
n− 1
+ β (n− 1)− θ,
and so it is clear that the equality holds in (2.10) as ∆ = n− 1.
Case 2: di ≤ n− 2. Then there is at least one vertex non-adjacent to vi in G. Let G′ be the
graph obtained from the graph G by adding edges between vi and the vertices non-adjacent
to vi in G. Let d′i and m
′
i be the degree of the vertex vi and the average degree of the
vertices adjacent to the vertex vi in G′, respectively. Then d′i = n− 1 and hence G′ ∈ Sn.
Now,
βd′i + θm
′
i = β(n− 1) + θ
(2m+ 2(n− di − 1)− (n− 1)
n− 1
)
= β (n− 1)− θ + 2θ (m+ n− di − 1)
n− 1
. (2.11)
Let pi be the average degree of the vertices non-adjacent to vertex vi in the graph G. Hence
β d′i + θm
′
i − (β di + θmi)
= β (d′i − di) + θ (m′i −mi)
= β (n− di − 1) + θ
(
2m+ (n− di − 1)− (n− 1)
n− 1
−mi
)
= β(n− di − 1) + θ
(
dimi + (n− di − 1)(pi + 1)
n− 1
−mi
)
.
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 109
Since β ≥ θ > 0 and ∆− δ ≤ n− 2, we have β (n− 1) ≥ θ (∆− δ + 1). Moreover, we
have mi ≤ ∆ and pi ≥ δ for any vertex vi ∈ V (G). Using these results, we obtain
βdi + θmi
= β di − θ +
2θ (m+ n− di − 1)
n− 1
+ θ
(
mi −
dimi + (n− di − 1)(pi + 1)
n− 1
)
=
2mθ
n− 1
+
di
n− 1
(
β (n− 1)− θ
)
+ θ
(
1− di
n− 1
)
(mi − pi)
≤ 2mθ
n− 1
+
di
n− 1
(
β (n− 1)− θ
)
+ θ
(
1− di
n− 1
)
(∆− δ) (2.12)
=
2mθ
n− 1
+ θ (∆− δ) + di
n− 1
(
β (n− 1)− θ − θ (∆− δ)
)
≤ 2mθ
n− 1
+ θ (∆− δ) + ∆
n− 1
(
β (n− 1)− θ (∆− δ + 1)
)
(2.13)
as di ≤ ∆. The first part of the proof is done.
Now, suppose that equality in (2.10) holds with β > θ. Then all the above inequalities
must be equalities. If di = n− 1, then G ∈ Sn. Otherwise, di ≤ n− 2. From the equality
in (2.12), we have mi = ∆ and pi = δ. Since β > θ, we have β (n− 1) > θ (∆− δ + 1).
From the equality in (2.13), we have di = ∆. Therefore all the vertices those are adjacent
to the vertex vi are of degree ∆ and those are non-adjacent to the vertex vi are of degree δ.
Hence G ∈ Γ.
Conversely, let G ∈ Sn. Then ∆ = n− 1 and hence
max
vj∈V
{βdj + θmj} =
2mθ
n− 1
+ β (n− 1)− θ
=
2mθ
n− 1
+ θ (∆− δ) + ∆
n− 1
[
β (n− 1)− θ(∆− δ + 1)
]
.
Let G ∈ Γ. Then 2m = ∆(∆+ 1) + (n−∆− 1) δ and hence
max
vj∈V
{βdj +θmj} = (β+θ)∆ =
2mθ
n− 1
+θ (∆−δ)+ ∆
n− 1
[
β (n−1)−θ(∆−δ+1)
]
.
This completes the proof.
Corollary 2.15. Let G be a graph of order n with m edges and let α ≥ 12 . Let ∆ and δ are
respectively, the maximum degree and the minimum degree of G. Then
max
1≤j≤n
{
αdj + (1− α)mj
}
≤
2m (1− α)
n− 1
+
αn− 1
n− 1
∆ + (1− α)
(
1− ∆
n− 1
)
(∆− δ) (2.14)
with equality if and only if G ∈ Sn or G ∈ Γ.
110 Ars Math. Contemp. 23 (2023) #P1.06 / 97–119
Combining Theorem 2.3 with Corollary 2.15, we get the following result, which gives
an upper bound for the generalized adjacency spectral radius λ(Aα(G)), in terms of the
order n, the size m, the maximum degree ∆, the minimum degree δ and the parameter α.
Theorem 2.16. Let G be a graph of order n, with m edges and let α ≥ 12 . Let ∆ and δ are
respectively, the maximum degree and the minimum degree of G. Then
λ(Aα(G)) ≤
2m (1− α)
n− 1
+
αn− 1
n− 1
∆ + (1− α)
(
1− ∆
n− 1
)
(∆− δ) .
If α ∈
(
1
2 , 1
)
and G is connected, equality holds if and only if G ∼= Kn.
The following result gives a Nordhaus–Gaddum type upper bound for the generalized
adjacency spectral radius λ(Aα(G)), in terms of the order n, the size m, the minimum
degree δ, the maximum degree ∆ and the parameter α.
Theorem 2.17. Let G be a graph of order n, with m edges and let α ≥ 12 . Let ∆ and δ are
respectively, the maximum degree and the minimum degree of G. Then
λ(Aα(G)) + λ(Aα(Ḡ)) ≤ n− 1 +
(1− α)(∆− δ)
n− 1
(
n+ δ −∆− 1 + αn− 1
1− α
)
.(2.15)
If α ∈
(
1
2 , 1
)
and G is connected, equality holds if and only if G = Kn.
Proof. Following Theorem 2.16, we have
λ(Aα(G)) + λ(Aα(Ḡ)) ≤ (1− α)
2m+ 2m̄
n− 1
+
αn− 1
n− 1
(∆ + ∆̄)
+ (1− α)
(
1− ∆
n− 1
)
(∆− δ) + (1− α)
(
1− ∆̄
n− 1
)(
∆̄− δ̄
)
= (1− α)n+ αn− 1
n− 1
(∆− δ + n− 1)
+ (1− α)(∆− δ)
(
1− ∆
n− 1
+
δ
n− 1
)
= n− 1 + (1− α)(∆− δ)
n− 1
(
n+ δ −∆− 1 + αn− 1
1− α
)
,
since m+ m̄ = n(n−1)2 , ∆̄ = n− 1− δ and δ̄ = n− 1−∆.
Now, we consider the equality case in (2.15). If G is regular, then both sides of (2.15)
are equal to n − 1. Now, assume that equality occurs in (2.15) for G. Then the equalities
must hold in (2.15) for both G and Ḡ. Hence G ∼= Kn.
3 A general upper bound for the generalized adjacency spectral ra-
dius
In this section, we obtain a general upper bound for the generalized adjacency spectral
radius in terms of vertex degrees and arbitrary positive real numbers bi. If we replace bi by
some graph parameters, then we can derive some upper bounds for λ(Aα(G)), in terms of
vertex degrees. For this we need the following result:
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 111
Lemma 3.1 ([7]). Let D = (di,j) be an n×n irreducible non-negative matrix with spectral
radius σ and let Ri(D) =
n∑
j=1
di,j be the i-th row sum of D. Then
min{Ri(D) : 1 ≤ i ≤ n} ≤ σ ≤ max{Ri(D) : 1 ≤ i ≤ n} . (3.1)
Moreover, if the row sums of D are not all equal, then the both inequalities in (3.1) are
strict.
The following result gives an upper bound for λ(Aα(G)), in terms of vertex degrees
and the arbitrary positive real numbers bi.
Theorem 3.2. Let G be a connected graph of order n and 0 < α < 1. Let d1 ≥ d2 ≥
· · · ≥ dn be the vertex degrees of G. Then
λ(Aα(G)) ≤ max
1≤i≤n

αdi +
√
α2d2i +
4
bi
∑
j:j∼i
bj(1− α)(αdj + (1− α)b′j)
2
 , (3.2)
where bi ∈ R+ and b′i = 1bi
∑
j:j∼i
bj . Moreover, the equality holds if and only if αd1 +(1−
α)b′1 = αd2 + (1− α)b′2 = · · · = αdn + (1− α)b′n.
Proof. Let B = diag(b1, b2, . . . , bn), where bi ∈ R+ are positive real number. Since the
matrices Aα(G) and B−1Aα(G)B are similar and similar matrices have same spectrum,
it follows that if λ(Aα(G)) is the largest eigenvalue of Aα(G), then it is also the largest
eigenvalue of B−1Aα(G)B. Let x = (x1, x2, . . . , xn)T be an eigenvector corresponding
to the eigenvalue λ(Aα(G)) of B−1Aα(G)B. We assume that one eigencomponent xi is
equal to 1 and the other eigencomponents are less than or equal to 1. The (i, j)-th entry of
B−1Aα(G)B is 
αdi if i = j,
(1− α) bj
bi
if j ∼ i,
0 otherwise.
We have
B−1Aα(G)Bx = λ(Aα(G))x. (3.3)
From the i-th equation of (3.3), we have
λ(Aα(G))xi = αdixi + (1− α)
∑
j:j∼i
bj
bi
xj ,
i.e., λ(Aα(G)) = αdi + (1− α)
∑
j:j∼i
bj
bi
xj . (3.4)
Again from the j-th equation of (3.3),
λ(Aα(G))xj = αdjxj + (1− α)
∑
k:k∼j
bk
bj
xk.
112 Ars Math. Contemp. 23 (2023) #P1.06 / 97–119
Multiplying both sides of (3.4) by λ(Aα(G)) and substituting this value λ(Aα(G))xj , we
get
λ2(Aα(G)) = αdiλ(Aα(G)) + (1− α)
∑
j:j∼i
bj
bi
αdjxj + (1− α) ∑
k:k∼j
bk
bj
xk

= αdiλ(Aα(G)) + α(1− α)
∑
j:j∼i
bjdj
bi
xj + (1− α)2
∑
j:j∼i
∑
k:k∼j
bk
bi
xk
≤ αdiλ(Aα(G)) + α(1− α)
∑
j:j∼i
bjdj
bi
+ (1− α)2
∑
j:j∼i
bjb
′
j
bi
(3.5)
= αdiλ(Aα(G)) +
∑
j:j∼i
bj(1− α)(αdj + (1− α)b′j)
bi
,
as bi b′i =
∑
j:j∼i bj . Hence we get the upper bound.
Suppose that the equality holds in (3.2). Then all inequalities in the above argument
must be equalities. Since 0 < α < 1, from equality in (3.5), we get xj = 1 for all j such
that j ∼ i, and xk = 1 for all k such that k ∼ j and j ∼ i. From the above, one can easily
prove that xi = 1 for all i ∈ V (G), that is, αd1 + (1− α)b′1 = αd2 + (1− α)b′2 = · · · =
αdn + (1− α)b′n.
Conversely, let G be a connected graph such that αd1+(1−α)b′1 = αd2+(1−α)b′2 =
· · · = αdn + (1 − α)b′n (bi ∈ R+). Since λ(Aα(G)) = λ(B−1Aα(G)B), then by
Lemma 3.1, we obtain
λ(Aα(G)) = αdℓ + (1− α) b′ℓ
= max
1≤i≤n
αdi +
√
α2d2i +
4
bi
∑
j:j∼i bj(1− α)(αdj + (1− α)b′j)
2

for 1 ≤ ℓ ≤ n.
Taking bi = di in (3.2), and noting that b′i =
1
bi
∑
j:j∼i bj =
1
di
∑
j:j∼i dj = mi, we
obtain the following upper bound for λ(Aα(G)), in terms of vertex degrees and average
vertex 2-degrees.
Corollary 3.3. Let G be a connected graph of order n having vertex degrees di, average
vertex 2-degrees mi (1 ≤ i ≤ n) and 0 < α < 1. Then
λ(Aα(G)) ≤ max
1≤i≤n
αdi +
√
α2d2i +
4(1−α)
di
∑
j:j∼i dj [αdj + (1− α)mj ]
2
 .
Equality holds if and only if αd1+(1−α)m1 = αd2+(1−α)m2 = · · · = αdn+(1−α)mn.
Taking bi =
√
di in (3.2), and noting that b′i =
1
bi
∑
j:j∼i bj =
1√
di
∑
j:j∼i
√
dj = m
′
i
(say), we obtain the following upper bound for λ(Aα(G)), in terms of vertex degrees and
m
′
i.
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 113
Corollary 3.4. Let G be a connected graph of order n having vertex degrees di and let
m
′
i =
1√
di
∑
j:j∼i
√
dj , 1 ≤ i ≤ n and 0 < α < 1. Then
λ(Aα(G)) ≤ max
1≤i≤n
αdi +
√
α2d2i +
4(1−α)√
di
∑
j:j∼i
√
dj [αdj + (1− α)m
′
j ]
2
 .
Equality holds if and only if αd1+(1−α)m
′
1 = αd2+(1−α)m
′
2 = · · · = αdn+(1−α)m
′
n.
Taking bi = 1 in (3.2), and noting that b′i =
1
bi
∑
j:j∼i bj =
∑
j:j∼i 1 = di, we obtain
the following upper bound for λ(Aα(G)), in terms of vertex degrees and average vertex
2-degrees. We note that this upper bound was recently obtained in [15].
Corollary 3.5 ([15]). Let G be a connected graph of order n having vertex degrees di,
average vertex 2-degrees mi (1 ≤ i ≤ n) and 0 < α < 1. Then
λ(Aα(G)) ≤ max
1≤i≤n
{
αdi +
√
α2d2i + 4(1− α)dimi
2
}
.
Equality holds if and only if d1 = d2 = · · · = dn.
Taking bi = mi in (3.2), and noting that b′i =
1
mi
∑
j:j∼i mj = m̄i, we obtain the
following upper bound for λ(Aα(G)), in terms of vertex degrees and the quantity m̄i.
Corollary 3.6. Let G be a connected graph of order n having vertex degrees di, average
vertex 2-degrees mi (1 ≤ i ≤ n) and 0 < α < 1. Then
λ(Aα(G)) ≤ max
1≤i≤n
αdi +
√
α2d2i +
4(1−α)
mi
∑
j:j∼i mj [αdj + (1− α)m̄j ]
2
 ,
where m̄i = 1mi
∑
j:j∼i mj . Equality holds if and only if αd1 + (1 − α)m̄1 = αd2 +
(1− α)m̄2 = · · · = αdn + (1− α)m̄n.
Taking bi = di +mi, bi = di +
√
mi, bi =
√
di +mi, bi =
√
di +
√
mi, bi = 1√di ,
bi =
1√
mi
, bi = 1d2i , bi = d
2
i , etc, and proceeding similarly as above we can obtain some
other new upper bounds for λ(Aα(G)).
4 Relation between ω(G), γ(G) and the generalized adjacency eigen-
values
For a graph G, define ω(G) and γ(G), the clique number and the independence number of
G to be the numbers of vertices of the largest clique and the largest independent set in G,
respectively. In this section, we give bounds for clique number and independence number
of (regular) graph G involving generalized adjacency eigenvalues.
The following lemma, due to Motzkin and Straus [19], links the spectrum of graphs to
its structure.
Lemma 4.1 ([19]). Let F =
{
x = (x1, x2, . . . , xn)
T | xi ≥ 0,
n∑
i=1
xi = 1
}
. Then
1− 1
ω(G)
= max
x∈F
⟨x,Ax⟩.
114 Ars Math. Contemp. 23 (2023) #P1.06 / 97–119
The following result gives a lower bound for ω(G), in terms of the size m, the gen-
eralized adjacency spectral radius λ(Aα(G)), the maximum degree ∆ and the parameter
α.
Theorem 4.2. Let G be a graph of order n, with m edges and maximum degree ∆. Then
ω(G) ≥ 2(1− α)
2m
2(1− α)2m− (λ(Aα(G))− α∆)2
.
Proof. Let x = (x1, x2, . . . , xn)T be the normalized eigenvector corresponding to λ(Aα(G)).
Then
λ(Aα(G)) = α
n∑
i=1
dix
2
i + 2(1− α)
∑
j:j∼i
xixj
≤ α∆
n∑
i=1
x2i + 2(1− α)
∑
j:j∼i
xixj
= α∆+ 2(1− α)
∑
j:j∼i
xixj .
Since λ(Aα(G)) ≥ α(∆ + 1), for α ∈ [0, 12 ], (see [20]), by Cauchy-Schwarz inequality,
we obtain
(λ(Aα(G))− α∆)2 ≤
2(1− α) ∑
j:j∼i
xixj
2 ≤ 2(1− α)2m
2 ∑
j:j∼i
x2ix
2
j
 .
Note that (x21, x
2
2, . . . , x
2
n)
T ≥ 0 and x21 + x22 + · · ·+ x2n = 1. Hence, by Lemma 4.1, we
have
2
∑
j:j∼i
x2ix
2
j ≤ 1−
1
ω(G)
,
then
(λ(Aα(G))− α∆)2
2(1− α)2m
≤ 1− 1
ω(G)
,
that is,
ω(G) ≥ 2(1− α)
2m
2(1− α)2m− (λ(Aα(G))− α∆)2
.
This completes the proof.
Note that Theorem 4.2 extends the Theorem 4.1 proved in [12] for the signless Lapla-
cian spectral radius to generalized adjacency spectral radius.
The following result gives a lower bound for ω(G), when G is a regular graph, in terms
of the order n, the second smallest generalized adjacency eigenvalue λn−1 = λn−1(Aα(G))
and the parameter α
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 115
Theorem 4.3. Let G be a r-regular graph of order n ≥ 3. Then
ω(G) ≥ (1− α)n
2
(1− α) (n2 − nr) + S2(αr − λn−1)
,
where S = minyi ̸=0
1
|yi| and un−1 = (y1, y2, . . . , yn)
T is the normalized eigenvector
corresponding to λn−1, the second smallest eigenvalue of Aα(G).
Proof. Since G is a r-regular graph, we have λ(Aα(G)) = r and the normalized eigenvec-
tor corresponding to λ(Aα(G)) is u1 = e√n , where e = (1, 1, . . . , 1)
T . Let Θ = Sn
and x = en + Θun−1. Then Θyi ≥ −
1
n (i = 1, 2, . . . , n). Since
∑n
i=1 λi(G) =
2αm = αnr and n ≥ 3, we have λ(Aα(G)) ̸= λn−1(G) and ⟨e, un−1⟩ = 0. So,
x ∈
{
(x1, x2, . . . , xn)
T ;xi ≥ 0,
∑n
i=1 xi = 1
}
. By Lemma 4.1, we have
⟨x, Aαx⟩ = α⟨x, Dx⟩+ (1− α)⟨x, Ax⟩
≤ rα⟨x,x⟩+ (1− α)
(
1− 1
ω(G)
)
= αr
(
1
n
+Θ2
)
+ (1− α)
(
1− 1
ω(G)
)
.
On the other hand
⟨x, Aαx⟩ =
〈 e
n
+Θun−1, Aα
( e
n
+Θun−1
)〉
=
〈 e
n
,Aα
e
n
〉
+
〈 e
n
,AαΘun−1
〉
+
〈
Θun−1, Aα
e
n
〉
+ ⟨Θun−1, AαΘun−1⟩
=
nd
n2
+Θ2λn−1.
Then
d
n
+Θ2λn−1 ≤ αr
(
1
n
+Θ2
)
+ (1− α)
(
1− 1
ω(G)
)
,
that is,
ω(G) ≥ 1− α
(1− α)
(
1− rn
)
+Θ2(αr − λn−1)
.
Since Θ = Sn and S = minyi ̸=0
1
|yi| , we have
ω(G) ≥ (1− α)n
2
(1− α) (n2 − nr) + S2(αr − λn−1)
.
This completes the proof.
Note that Theorem 4.3 extends the Theorem 4.4 proved in [12] for the signless Lapla-
cian spectral radius to generalized adjacency spectral radius.
Consider two sequences of real numbers ξ1 ≥ ξ2 ≥ · · · ≥ ξn and η1 ≥ η2 ≥ · · · ≥ ηt
with t < n. The second sequence is said to interlace the first one whenever
ξi ≥ ηi ≥ ξn−t+i,
116 Ars Math. Contemp. 23 (2023) #P1.06 / 97–119
for i = 1, 2, . . . , t. The interlacing is called tight if there exists an integer k ∈ [0, t] such
that ξi = ηi for 1 ≤ i ≤ k and ξn−t+i = ηi for k + 1 ≤ i ≤ t. Suppose rows and columns
of the matrix M are partitioned according to a partitioning of {1, 2, . . . , n}. The partition
is called regular if each block of M has constant row (and column) sum. The following
lemma can be found in [6].
Lemma 4.4 ([6]). Let B be the matrix whose entries are the average row sums of the blocks
of a symmetric partitioned matrix of M. Then
(i) the eigenvalues of B interlace the eigenvalues of M,
(ii) if the interlacing is tight, then the partition is regular.
Next result gives a lower bound for γ(G), in terms of the order n, the sum of first two
largest generalized adjacency eigenvalues, the maximum degree ∆, the minimum degree δ
and the parameter α.
Theorem 4.5. Let G be a simple graph of order n with at least one edge, with minimum
degree δ and maximum degree ∆. Let λ1(G) and λ2(G) are respectively the first and the
second largest eigenvalue of Aα(G). If λ1(G) + λ2(G)− (1 + α)δ ≤ 0, then
γ(G) ≥ λ1(G) + λ2(G)− (1 + α)δ
δ
× n∆
λ1(G) + λ2(G)− 2∆
. (4.1)
Proof. Let G be a simple graph with order n and a partition V (G) = V1 ∪ V2. Let Gi
(i = 1, 2) be the subgraph of G induced by Vi with ni < n vertices and average degree ri
(n1 + n2 = n). Let ti =
∑
v∈Vi
d(v)
ni
for i = 1, 2. Note that
Aα(G) =
(
A11 A12
A21 A22
)
=
(
αD11 + (1− α)A(G1) (1− α)A12
(1− α)A21 αD22 + (1− α)A(G2)
)
,
where D11 = diag(d(v1), . . . , d(vn1)), D22 = diag(d(vn1+1), . . . , d(vn)) and A21 =
AT12. Put M =
(
mij
ni
)
, where mij is the sum of the entries in Aij(G). Hence
M =
(
αt1 + (1− α)r1 (1− α)(t1 − r1)
(1− α)(t2 − r2) αt2 + (1− α)r2
)
and
|ϕI −M | = ϕ2 − (αt1 + (1− α)r1 + αt2 + (1− α)r2)ϕ
− (1− α)2(t1 − r1)(t2 − r2) + (αt1 + (1− α)r1)(αt2 + (1− α)r2).
Then by Lemma 4.4, we have ϕ1(M) ≤ λ1(G) and ϕ2(M) ≤ λ2(G), hence
ϕ1(M) + ϕ2(M) = αt1 + (1− α)r1 + αt2 + (1− α)r2 ≤ λ1(G) + λ2(G).
Note that 2(n2t2−n1t1) = n2(t2+r2)−n1(t1+r1), and hence n2t2−n1t1 = n2r2−n1r1.
Let VG1 be the largest independent set of G, then r1 = 0 and γ(G) = 0, we have
r2 = t2 − n1n2 t1, and
αt1 + αt2 + (1− α)
(
t2 −
n1
n2
t1
)
= αt1 + t2 − (1− α)
n1
n2
t1 ≤ λ1(G) + λ2(G).
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 117
By n = n1 + n2, we get
λ1(G) + λ2(G)− t2 − αt1
t1
n ≥ λ1(G) + λ2(G)− t2 − t1
t1
n1.
Since G has at least one edge, n1 < n. Also we have δ ≤ t1, t2 ≤ ∆, hence
λ1(G) + λ2(G)− (1 + α)δ
δ
n ≥ λ1(G) + λ2(G)− 2∆
∆
n1.
Thus
γ(G) = n1 ≥
λ1(G) + λ2(G)− (1 + α)δ
δ
× n∆
λ1(G) + λ2(G)− 2∆
.
This completes the proof.
Again, we note that Theorem 4.5 extends the Theorem 4.5 proved in [12] for the sign-
less Laplacian spectral radius to generalized adjacency spectral radius.
Remark 4.6. Note that if λ1(G) + λ2(G) − (1 + α)δ > 0, then λ1(G)+λ2(G)−(1+α)δδ ×
n∆
λ1(G)+λ2(G)−2∆ < 0, and the inequality in (4.1) is trivial. Hence, we add the restriction
λ1(G) + λ2(G) − (1 + α)δ ≤ 0, in Theorem 4.5. One can easily see that there exists
graphs with the property that λ1(G) + λ2(G) − (1 + α)δ ≤ 0. For example, we have
specAα(Kn) = {n − 1, αn − 1[n−1]}. Hence, λ1(K3) + λ2(K3) − (1 + α)δ(K3) =
2 + 3α− 1− 2(1 + α) = α− 1 ≤ 0.
If G is an r-regular graph, then λ1(G) = r and ∆ = δ = r. Hence, by Theorem 4.5,
we get the following bound.
Corollary 4.7. Let G be a simple r-regular graph of order n with at least one edge. Then
γ(G) ≥ n(λ2(G)− αr)
λ2(G)− r
,
where λ2(G) is the second largest eigenvalue of Aα(G).
5 Some conclusions
As mentioned in the introduction, for α = 0, the generalized adjacency matrix Aα(G) is
same as the adjacency matrix A(G) and for α = 12 , twice the generalized adjacency ma-
trix Aα(G) is same as the signless Laplacian matrix Q(G). Therefore, if in particular, we
put α = 0 and α = 12 , in all the results obtained in Sections 2, 3 and 4, we obtain the
corresponding bounds for the adjacency spectral radius λ(A(G)) and the signless Lapla-
cian spectral radius λ(Q(G)). We note most of these results we obtained in Section 2, 3
and 4 has been already discussed for the adjacency spectral radius λ(A(G)) or/and for the
signless Laplacian spectral radius λ(Q(G)). Therefore, in this setting our results are the
generalization of these known results.
ORCID iDs
Abdollah Alhevaz https://orcid.org/0000-0001-6167-607X
Maryam Baghipur https://orcid.org/0000-0002-9069-9243
Hilal Ahmad Ganie https://orcid.org/0000-0002-2226-7828
Kinkar Chandra Das https://orcid.org/0000-0003-2576-160X
118 Ars Math. Contemp. 23 (2023) #P1.06 / 97–119
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.07 / 121–127
https://doi.org/10.26493/1855-3974.2144.9e3
(Also available at http://amc-journal.eu)
A note on the neighbour-distinguishing index
of digraphs
Éric Sopena *
Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR5800, F-33400 Talence, France
Mariusz Woźniak
AGH University of Science and Technology,
al. A. Mickiewicza 30, 30-059 Krakow, Poland
Received 9 October 2019, accepted 30 January 2022, published online 20 October 2022
Abstract
In this note, we introduce and study a new version of neighbour-distinguishing arc-
colourings of digraphs. An arc-colouring γ of a digraph D is proper if no two arcs with the
same head or with the same tail are assigned the same colour. For each vertex u of D, we
denote by S−γ (u) and S
+
γ (u) the sets of colours that appear on the incoming arcs and on
the outgoing arcs of u, respectively. An arc colouring γ of D is neighbour-distinguishing
if, for every two adjacent vertices u and v of D, the ordered pairs (S−γ (u), S
+
γ (u)) and
(S−γ (v), S
+
γ (v)) are distinct. The neighbour-distinguishing index of D is then the smallest
number of colours needed for a neighbour-distinguishing arc-colouring of D.
We prove upper bounds on the neighbour-distinguishing index of various classes of
digraphs.
Keywords: Digraph, arc-colouring, neighbour-distinguishing arc-colouring.
Math. Subj. Class. (2020): 05C15, 05C20
1 Introduction
A proper edge-colouring of a graph G is vertex-distinguishing if, for every two vertices u
and v of G, the sets of colours that appear on the edges incident with u and v are distinct.
Vertex-distinguishing proper edge-colourings of graphs were independently introduced by
Burris and Schelp [2], and by Černy, Horňák and Soták [5]. Requiring only adjacent ver-
tices to be distinguished led to the notion of neighbour-distinguishing edge-colourings,
considered in [1, 3, 7].
*Corresponding author.
E-mail addresses: eric.sopena@u-bordeaux.fr (Éric Sopena), mwozniak@agh.edu.pl (Mariusz Woźniak)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
122 Ars Math. Contemp. 23 (2023) #P1.07 / 121–127
Vertex-distinguishing arc-colourings of digraphs have been recently introduced and
studied by Li, Bai, He and Sun [4]. An arc-colouring of a digraph is proper if no two
arcs with the same head or with the same tail are assigned the same colour. Such an arc-
colouring is vertex-distinguishing if, for every two vertices u and v of G,
(i) the sets S−(u) and S−(v) of colours that appear on the incoming arcs of u and v,
respectively, are distinct, and
(ii) the sets S+(u) and S+(v) of colours that appear on the outgoing arcs of u and v,
respectively, are distinct.
In this paper, we introduce and study a neighbour-distinguishing version of arc-colourings
of digraphs, using a slightly different distinction criteria: two neighbours u and v are dis-
tinguished whenever S−(u) ̸= S−(v) or S+(u) ̸= S+(v).
Definitions and notation are introduced in the next section. We prove a general upper
bound on the neighbour-distinguishing index of a digraph in Section 3, and study various
classes of digraphs in Section 4. Concluding remarks are given in Section 5.
2 Definitions and notation
All digraphs we consider are without loops and multiple arcs. For a digraph D, we denote
by V (D) and A(D) its sets of vertices and arcs, respectively. The underlying graph of D,
denoted und(D), is the simple undirected graph obtained from D by replacing each arc uv
(or each pair of arcs uv, vu) by the edge uv.
If uv is an arc of a digraph D, u is the tail and v is the head of uv. For every vertex
u of D, we denote by N+D (u) and N
−
D (u) the sets of out-neighbours and in-neighbours of
u, respectively. Moreover, we denote by d+D(u) = |N
+
D (u)| and d
−
D(u) = |N
−
D (u)| the
outdegree and indegree of u, respectively, and by dD(u) = d+D(u) + d
−
D(u) the degree of
u.
For a digraph D, we denote by δ+(D), δ−(D), ∆+(D) and ∆−(D) the minimum
outdegree, minimum indegree, maximum outdegree and maximum indegree of D, respec-
tively. Moreover, we let
∆∗(D) = max{∆+(D), ∆−(D)}.
A (proper) k-arc-colouring of a digraph D is a mapping γ from V (D) to a set of k
colours (usually {1, . . . , k}) such that, for every vertex u,
(i) any two arcs with head u are assigned distinct colours, and
(ii) any two arcs with tail u are assigned distinct colours.
Note here that two consecutive arcs vu and uw, v and w not necessarily distinct, may be
assigned the same colour. The chromatic index χ′(D) of a digraph D is then the smallest
number k for which D admits a k-arc-colouring.
The following fact is well-known (see e.g. [4, 6, 8]).
Proposition 2.1. For every digraph D, χ′(D) = ∆∗(D).
For every vertex u of a digraph D, and every arc-colouring γ of D, we denote by
S+γ (u) and S
−
γ (u) the sets of colours assigned by γ to the outgoing and incoming arcs
É. Sopena and M. Woźniak: A note on the neighbour-distinguishing index of digraphs 123
of u, respectively. From the definition of an arc-colouring, we get d+D(u) = |S+γ (u)| and
d−D(u) = |S−γ (u)| for every vertex u.
We say that two vertices u and v of a digraph D are distinguished by an arc-colouring
γ of D, if (S+γ (u), S
−
γ (u)) ̸= (S+γ (v), S−γ (v)). Note that we consider here ordered pairs,
so that (A,B) ̸= (B,A) whenever A ̸= B. Note also that if u and v are such that d+D(u) ̸=
d+D(v) or d
−
D(u) ̸= d
−
D(v), which happens in particular if dD(u) ̸= dD(v), then they
are distinguished by every arc-colouring of D. We will write u ≁γ v if u and v are
distinguished by γ and u ∼γ v otherwise.
A k-arc-colouring γ of a digraph D is neighbour-distinguishing if u ≁γ v for ev-
ery arc uv ∈ A(D). Such an arc-colouring will be called an nd-arc-colouring for short.
The neighbour-distinguishing index ndi(D) of a digraph D is then the smallest number of
colours required for an nd-arc-colouring of D.
The following lower bound is easy to establish.
Proposition 2.2. For every digraph D, ndi(D) ≥ χ′(D) = ∆∗(D). Moreover, if there
are two vertices u and v in D with d+D(u) = d
+
D(v) = d
−
D(u) = d
−
D(v) = ∆
∗(D), then
ndi(D) ≥ ∆∗(D) + 1.
Proof. The first statement follows from the definitions. For the second statement, observe
that S+γ (u) = S
+
γ (v) = S
−
γ (u) = S
−
γ (v) = {1, . . . ,∆∗(D)} for any two such vertices u
and v and any ∆∗(D)-arc-colouring γ of D.
3 A general upper bound
If D is an oriented graph, that is, a digraph with no opposite arcs, then every proper edge-
colouring φ of und(D) is an nd-arc-colouring of D since, for every arc uv in D, φ(uv) ∈
S+φ (u) and φ(uv) /∈ S+φ (v), which implies u ≁φ v. Hence, we get the following upper
bound for oriented graphs, thanks to classical Vizing’s bound.
Proposition 3.1. If D is an oriented graph, then
ndi(D) ≤ χ′(und(D)) ≤ ∆(und(D)) + 1 ≤ 2∆∗(D) + 2.
However, a proper edge-colouring of und(D) may produce an arc-colouring of D
which is not neighbour-distinguishing when D contains opposite arcs. Consider for in-
stance the digraph D given by V (D) = {a, b, c, d} and A(D) = {ab, bc, cb, dc}. We then
have und(D) = P4, the path of order 4, and thus χ′(und(D)) = 2. It is then not difficult
to check that for any 2-edge-colouring φ of und(D), S+φ (b) = S
+
φ (c) and S
−
φ (b) = S
−
φ (c).
We will prove that the upper bound given in Proposition 3.1 can be decreased to
2∆∗(D), even when D contains opposite arcs. Recall that a digraph D is k-regular if
d+D(v) = d
−
D(v) = k for every vertex v of D. A k-factor in a digraph D is a spanning
k-regular subdigraph of D. The following result is folklore.
Theorem 3.2. Every k-regular digraph can be decomposed into k arc-disjoint 1-factors.
We first determine the neighbour-distinguishing index of a 1-factor.
Proposition 3.3. If D is a digraph with d+D(u) = d
−
D(u) = 1 for every vertex u of D, then
ndi(D) = 2.
124 Ars Math. Contemp. 23 (2023) #P1.07 / 121–127
Proof. Such a digraph D is a disjoint union of directed cycles and any such cycle needs
at least two colours to be neighbour-distinguished. An nd-arc-colouring of D using two
colours can be obtained as follows. For a directed cycle of even length, use alternately
colours 1 and 2. For a directed cycle of odd length, use the colour 2 on any two consecutive
arcs, and then use alternately colours 1 and 2. The so-obtained 2-arc-colouring is clearly
neighbour-distinguishing, so that ndi(D) = 2.
We are now able to prove the following general upper bound on the neighbour-distin-
guishing index of a digraph.
Theorem 3.4. For every digraph D, ndi(D) ≤ 2∆∗(D).
Proof. Let D′ be any ∆∗(D)-regular digraph containing D as a subdigraph. If D is not
already regular, such a digraph can be obtained from D by adding new arcs, and maybe
new vertices.
By Theorem 3.2, the digraph D′ can be decomposed into ∆∗(D′) = ∆∗(D) arc-
disjoint 1-factors, say F1, . . . , F∆∗(D). By Proposition 3.3, we know that D′ admits an
nd-arc-colouring γ′ using 2∆∗(D′) = 2∆∗(D) colours. We claim that the restriction γ of
γ′ to A(D) is also neighbour-distinguishing.
To see that, let uv be any arc of D, and let t and w be the two vertices such that the
directed walk tuvw belongs to a 1-factor Fi of D′ for some i, 1 ≤ i ≤ ∆∗(D). Note here
that we may have t = w, or w = u and t = v. If γ(uv) ̸= γ′(vw), then γ(uv) ∈ S+γ (u) and
γ(uv) /∈ S+γ (v). Similarly, if γ′(tu) ̸= γ(uv), then γ(uv) ∈ S−γ (v) and γ(uv) /∈ S−γ (u).
Since neither three consecutive arcs nor two opposite arcs in a walk of a 1-factor of D′ are
assigned the same colour by γ′, we get that u ≁γ v for every arc uv of D, as required.
This completes the proof.
4 Neighbour-distinguishing index of some classes of digraphs
We study in this section the neighbour-distinguishing index of several classes of digraphs,
namely complete symmetric digraphs, bipartite digraphs and digraphs whose underlying
graph is k-chromatic, k ≥ 3.
4.1 Complete symmetric digraphs
We denote by K∗n the complete symmetric digraph of order n. Observe first that any proper
edge-colouring ϵ of Kn induces an arc-colouring γ of K∗n defined by γ(uv) = γ(vu) =
ϵ(uv) for every edge uv of Kn. Moreover, since S+γ (u) = S
−
γ (u) = Sϵ(u) for every vertex
u, γ is neighbour-distinguishing whenever ϵ is neighbour-distinguishing. Using a result of
Zhang, Liu and Wang (see Theorem 6 in [7]), we get that ndi(K∗n) = ∆
∗(K∗n) + 1 = n if
n is odd, and ndi(K∗n) ≤ ∆∗(K∗n) + 2 = n+ 1 if n is even.
We prove that the bound in the even case can be decreased by one (we recall the proof
of the odd case to be complete).
Theorem 4.1. For every integer n ≥ 2, ndi(K∗n) = ∆∗(K∗n) + 1 = n.
Proof. Note first that we necessarily have ndi(K∗n) ≥ n for every n ≥ 2 by Proposition 2.2.
Let V (K∗n) = {v0, . . . , vn−1}. If n = 2, we obviously have ndi(K∗2 ) = |A(K∗2 )| = 2 and
the result follows. We can thus assume n ≥ 3. We consider two cases, depending on the
parity of n.
É. Sopena and M. Woźniak: A note on the neighbour-distinguishing index of digraphs 125
Suppose first that n is odd, and consider a partition of the set of edges of Kn into n
disjoint maximal matchings, say M0, . . . ,Mn−1, such that for each i, 0 ≤ i ≤ n − 1, the
matching Mi does not cover the vertex vi. We define an n-arc-colouring γ of K∗n (using
the set of colours {0, . . . , n − 1}) as follows. For every i and j, 0 ≤ i < j ≤ n − 1, we
set γ(vivj) = γ(vjvi) = k if and only if the edge vivj belongs to Mk. Observe now that
for every vertex vi, 0 ≤ i ≤ n − 1, the colour i is the unique colour that does not belong
to S+γ (vi) ∪ S−γ (vi), since vi is not covered by the matching Mi. This implies that γ is an
nd-arc-colouring of K∗n, and thus ndi(K
∗
n) = n, as required.
Suppose now that n is even. Let K ′ be the subgraph of K∗n induced by the set of vertices
{v0, . . . , vn−2} and γ′ be the (n− 1)-arc-colouring of K ′ defined as above. We define an
n-arc-colouring γ of K∗n (using the set of colours {0, . . . , n− 1}) as follows:
1. for every i and j, 0 ≤ i < j ≤ n − 2, j ̸≡ i + 1 (mod n − 1), we set γ(vivj) =
γ′(vivj),
2. for every i, 0 ≤ i ≤ n − 2, we set γ(vivi+1) = n − 1 and γ(vi+1vi) = γ′(vi+1vi)
(subscripts are taken modulo n− 1),
3. for every i, 0 ≤ i ≤ n − 2, we set γ(vn−1vi) = γ′(vi−1vi) and γ(vivn−1) =
γ′(vi+1vi).
Since the colour n−1 belongs to S+γ (vi)∩S−γ (vi) for every i, 0 ≤ i ≤ n−2, and does not
belong to S+γ (vn−1)∪S−γ (vn−1), the vertex vn−1 is distinguished from every other vertex
in K∗n. Moreover, for every vertex vi, 0 ≤ i ≤ n− 2,
S+γ (vi) = S
+
γ′(vi) ∪ {n− 1} and S
−
γ (vi) = S
−
γ′(vi) ∪ {n− 1},
which implies that any two vertices vi and vj , 0 ≤ i < j ≤ n− 2, are distinguished since
γ′ is an nd-arc-colouring of K ′. We thus get that γ is an nd-arc-colouring of K∗n, and thus
ndi(K∗n) ≤ n, as required.
This completes the proof.
4.2 Bipartite digraphs
A digraph D is bipartite if its underlying graph is bipartite. In that case, V (D) = X ∪ Y
with X ∩ Y = ∅ and A(D) ⊆ X × Y ∪ Y ×X . We then have the following result.
Theorem 4.2. If D is a bipartite digraph, then ndi(D) ≤ ∆∗(D) + 2.
Proof. Let V (D) = X ∪ Y be the bipartition of V (D) and γ be any (not necessarily
neighbour-distinguishing) optimal arc-colouring of D using ∆∗(D) colours (such an arc-
colouring exists by Proposition 2.1).
If γ is an nd-arc-colouring we are done. Otherwise, let M1 ⊆ A(D) ∩ (X × Y ) be a
maximal matching from X to Y . We define the arc-colouring γ1 as follows:
γ1(uv) = ∆
∗(D) + 1 if uv ∈ M1, γ1(uv) = γ(uv) otherwise.
Note that if uv is an arc such that u or v is (or both are) covered by M1, then u ≁γ1 v since
the colour ∆∗(D) + 1 appears in exactly one of the sets S+γ1(u) and S
+
γ1(v), or in exactly
one of the sets S−γ1(u) and S
−
γ1(v).
If γ1 is an nd-arc-colouring we are done. Otherwise, let A∼ be the set of arcs uv ∈
A(D) with u ∼γ1 v and M2 ⊆ A∼ ∩ (Y × X) be a maximal matching from Y to X of
A∼. We define the arc-colouring γ2 as follows:
126 Ars Math. Contemp. 23 (2023) #P1.07 / 121–127
γ2(uv) = ∆
∗(D) + 2 if uv ∈ M2, γ2(uv) = γ1(uv) otherwise.
Again, note that if uv is an arc such that u or v is (or both are) covered by M2, then u ≁γ2 v.
Moreover, since M2 is a matching of A∼, pairs of vertices that were distinguished by γ1
are still distinguished by γ2.
Hence, every arc uv such that u and v were not distinguished by γ1 are now distin-
guished by γ2 which is thus an nd-arc-colouring of D using ∆∗(D) + 2 colours. This
concludes the proof.
The upper bound given in Theorem 4.2 can be decreased when the underlying graph of
D is a tree.
Theorem 4.3. If D is a digraph whose underlying graph is a tree, then ndi(D) ≤ ∆∗(D)+
1.
Proof. The proof is by induction on the order n of D. The result clearly holds if n ≤ 2. Let
now D be a digraph of order n ≥ 3, such that the underlying graph und(D) of D is a tree,
and P = v1 . . . vk, k ≤ n, be a path in und(D) with maximal length. By the induction
hypothesis, there exists an nd-arc-colouring γ of D − vk using at most ∆∗(D − vk) + 1
colours. We will extend γ to an nd-arc-colouring of D using at most ∆∗(D) + 1 colours.
If ∆∗(D) = ∆∗(D− vk) + 1, we assign the new colour ∆∗(D) + 1 to the at most two
arcs incident with vk so that the so-obtained arc-colouring is clearly neighbour-distinguishing.
Suppose now that ∆∗(D) = ∆∗(D − vk). If all neighbours of vk−1 are leaves, the
underlying graph of D is a star. In that case, there is at most one arc linking vk−1 and vk,
and colouring this arc with any admissible colour produces an nd-arc-colouring of D. If
the underlying graph of D is not a star, then, by the maximality of P , we get that vk−1
has exactly one neighbour which is not a leaf, namely vk−2. This implies that the only
conflict that might appear when colouring the arcs linking vk and vk−1 is between vk−2
and vk−1 (recall that two neighbours with distinct indegree or outdegree are necessarily
distinguished).
Since d+D(vk−2) ≤ ∆∗(D) and d
−
D(vk−2) ≤ ∆∗(D), there necessarily exist a colour a
such that S+γ (vk−2) ̸= S+γ (vk−1)∪{a}, and a colour b such that S−γ (vk−2) ̸= S−γ (vk−1)∪
{b}. Therefore, the at most two arcs incident with vk can be coloured, using a and/or b, in
such a way that the so-obtained arc-colouring is neighbour-distinguishing.
This completes the proof.
4.3 Digraphs whose underlying graph is k-chromatic
Since the set of edges of every k-colourable graph can be partitionned in ⌈log k⌉ parts each
inducing a bipartite graph (see e.g. Lemma 4.1 in [1]), Theorem 4.2 leads to the following
general upper bound:
Corollary 4.4. If D is a digraph whose underlying graph has chromatic number k ≥ 3,
then ndi(D) ≤ ∆∗(D) + 2⌈log k⌉.
Proof. Starting from an optimal arc-colouring of D with ∆∗(D) colours, it suffices to use
two new colours for each of the ⌈log k⌉ bipartite parts (obtained from any optimal vertex-
colouring of the underlying graph of D), as shown in the proof of Theorem 4.2, in order to
get an nd-arc-colouring of D.
É. Sopena and M. Woźniak: A note on the neighbour-distinguishing index of digraphs 127
5 Discussion
In this note, we have introduced and studied a new version of neighbour-distinguishing arc-
colourings of digraphs. Pursuing this line of research, we propose the following questions.
1. Is there any general upper bound on the neighbour-distinguishing index of symmetric
digraphs?
2. Is there any general upper bound on the neighbour-distinguishing index of not nec-
essarily symmetric complete digraphs?
3. Is there any general upper bound on the neighbour-distinguishing index of directed
acyclic graphs?
4. The general bound given in Corollary 4.4 is certainly not optimal. In particular, is it
possible to improve this bound for digraphs whose underlying graph is 3-colourable?
We finally propose the following conjecture.
Conjecture 5.1. For every digraph D, ndi(D) ≤ ∆∗(D) + 1.
ORCID iDs
Eric Sopena https://orcid.org/0000-0002-9570-1840
Mariusz Woźniak https://orcid.org/0000-0003-4769-0056
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.08 / 129–144
https://doi.org/10.26493/1855-3974.2530.e7c
(Also available at http://amc-journal.eu)
Complexity of circulant graphs with non-fixed
jumps, its arithmetic properties and
asymptotics*
Alexander Mednykh , Ilya Mednykh †
Sobolev Institute of Mathematics, 630090, Novosibirsk, Russia
Novosibirsk State University, 630090, Novosibirsk, Russia
Received 11 January 2021, accepted 23 March 2022, published online 21 October 2022
Abstract
In the present paper, we investigate a family of circulant graphs with non-fixed jumps
Gn = Cβn(s1, . . . , sk, α1n, . . . , αℓn),
1 ≤ s1 < . . . < sk < [
βn
2
], 1 ≤ α1 < . . . < αℓ ≤ [
β
2
].
Here n is an arbitrary large natural number and integers s1, . . . , sk, α1, . . . , αℓ, β are sup-
posed to be fixed.
First, we present an explicit formula for the number of spanning trees in the graph Gn.
This formula is a product of βsk−1 factors, each given by the n-th Chebyshev polynomial
of the first kind evaluated at the roots of some prescribed polynomial of degree sk. Next, we
provide some arithmetic properties of the complexity function. We show that the number
of spanning trees in Gn can be represented in the form τ(n) = p n a(n)2, where a(n)
is an integer sequence and p is a given natural number depending on parity of β and n.
Finally, we find an asymptotic formula for τ(n) through the Mahler measure of the Laurent
polynomials differing by a constant from 2k −
∑k
i=1(z
si + z−si).
Keywords: Spanning tree, circulant graph, Laplacian matrix, Chebyshev polynomial, Mahler mea-
sure.
Math. Subj. Class. (2020): 05C30, 05A18
*The authors are grateful to anonymous referees for helpful remarks and suggestions. The work was supported
by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1613 with the Ministry of Science
and Higher Education of the Russian Federation.
†Corresponding author.
E-mail addresses: smedn@mail.ru (Alexander Mednykh), ilyamednykh@mail.ru (Ilya Mednykh)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
130 Ars Math. Contemp. 23 (2023) #P1.08 / 129–144
1 Introduction
The complexity of a finite connected graph G, denoted by τ(G), is the number of span-
ning trees of G. The famous Kirchhoff’s Matrix Tree Theorem [13] states that τ(G) can
be expressed as the product of non-zero Laplacian eigenvalues of G divided by the number
of its vertices. Since then, a lot of papers devoted to the complexity of various classes of
graphs were published. In particular, explicit formulae were derived for complete multipar-
tite graphs [16], wheels [2], fans [10], prisms [1], anti-prisms [32], ladders [26], Möbius
ladders [27], lattices [28] and other families. The complexity of circulant graphs has been
the subject of study by many authors [4, 5, 17, 34, 35, 36, 37, 38].
Starting with Boesch and Prodinger [2] the idea to calculate the complexity of graphs by
making use of Chebyshev polynomials was implemented. This idea provided a way to find
complexity of circulant graphs and their natural generalisations in [4, 14, 19, 25, 36, 38].
Recently, asymptotical behavior of complexity for some families of graphs was investi-
gated from the point of view of so called Malher measure [9, 29, 30]. For general properties
of the Mahler measure see, for example [31] and [7]. It worth mentioning that the Mahler
measure is related to the growth of groups, values of some hypergeometric functions and
volumes of hyperbolic manifolds [3].
For a sequence of graphs Gn, one can consider the number of vertices v(Gn) and the
number of spanning trees τ(Gn) as functions of n. Assuming that limn→∞
log τ(Gn)
v(Gn)
exists,
it is called the thermodynamic limit of the family Gn [20]. This number plays an important
role in statistical physics and was investigated by many authors [12, 28, 29, 30, 33].
The purpose of this paper is to present new formulas for the number of spanning trees
in circulant graphs with non-fixed jumps and investigate their arithmetical properties and
asymptotics. We mention that the number of spanning trees for such graphs was found
earlier in [5, 8, 17, 19, 37, 38]. Our results are different from those obtained in the cited
papers. Moreover, by the authors opinion, the obtained formulas are more convenient for
analytical investigation.
The content of the paper is lined up as follows. Basic definitions and preliminary results
are given in Sections 2 and 3. Then, in the Section 4, we present an explicit formula for the
number of spanning trees in the undirected circulant graph
Cβn(s1, s2, . . . , sk, α1n, α2n, . . . , αℓn),
1 ≤ s1 < . . . < sk < [
βn
2
], 1 ≤ α1 < . . . < αℓ ≤ [
β
2
].
This formula is a product of βsk−1 factors, each given by the n-th Chebyshev polynomial
of the first kind evaluated at the roots of a prescribed polynomial of degree sk. Through the
paper, we will assume that β > 1 and ℓ > 0. The case β = 1 and ℓ = 0 of the circulant
graphs with bounded jumps has been investigated in our previous papers [22, 23].
Next, in the Section 5, we provide some arithmetic properties of the complexity func-
tion. More precisely, we show that the number of spanning trees of the circulant graph can
be represented in the form τ(n) = β pn a(n)2, where a(n) is an integer sequence and p is a
prescribed natural number depending only on parity of n and β. Later, in the Section 6, we
use explicit formulas for the number of spanning trees to produce its asymptotics through
the Mahler measures of the finite set of Laurent polynomials
Pu(z) = 2k −
k∑
i=1
(zsi + z−si) + 4
ℓ∑
m=1
sin2(
π uαm
β
), u = 0, 1, . . . , β − 1.
A. Mednykh and I. Mednykh: Complexity of circulant graphs with non-fixed jumps, . . . 131
As a consequence (Corollary 6.2), we prove that the thermodynamic limit of sequence
Cβn(s1, s2, . . . , sk, α1n, α2n, . . . , αℓn) as n → ∞ is the arithmetic mean of small Mahler
measures of Laurent polynomials Pu(z), u = 0, 1, . . . , β−1. In the Section 7, we illustrate
the obtained results by a series of examples.
2 Basic definitions and preliminary facts
Consider a connected finite graph G, allowed to have multiple edges but without loops. We
denote the vertex and edge set of G by V (G) and E(G), respectively. Given u, v ∈ V (G),
we set auv to be equal to the number of edges between vertices u and v. The matrix A =
A(G) = {auv}u,v∈V (G) is called the adjacency matrix of the graph G. The degree d(v) of
a vertex v ∈ V (G) is defined by d(v) =
∑
u∈V (G) auv. Let D = D(G) be the diagonal
matrix indexed by the elements of V (G) with dvv = d(v). The matrix L = L(G) =
D(G)−A(G) is called the Laplacian matrix, or simply Laplacian, of the graph G.
In what follows, by In we denote the identity matrix of order n.
Let s1, s2, . . . , sk be integers such that 1 ≤ s1, s2, . . . , sk ≤ n2 . The graph
G = Cn(s1, s2, . . . , sk) with n vertices 0, 1, 2, . . . , n− 1 is called circulant graph if the
vertex i, 0 ≤ i ≤ n − 1 is adjacent to the vertices i ± s1, i ± s2, . . . , i ± sk (mod n). All
vertices of the graph G have even degree 2k. If there is i such that si = n2 then graph G
has multiple edges.
We call an n×n matrix circulant, and denote it by circ(a0, a1, . . . , an−1) if it is of the
form
circ(a0, a1, . . . , an−1) =

a0 a1 a2 . . . an−1
an−1 a0 a1 . . . an−2
...
. . .
...
a1 a2 a3 . . . a0
 .
It easy to chose enumeration of vertices such that adjacency and Laplacian matrices for
the circulant graph are circulant matrices. The converse is also true. If the Laplacian matrix
of a graph is circulant then the graph is also circulant.
In this paper, we consider a particular class of circulant graphs, namely circulant graphs
with non-fixed jumps. They are defined as before, with special restrictions on the number
of vertices and structure of jumps.
More precisely, we will deal with circulant graphs
Gn = Cβn(s1, s2, . . . , sk, α1n, α2n, . . . , αℓn)
on β n vertices and jumps s1, s2, . . . , sk, α1n, α2n, . . . , αℓn satisfying the inequalities
1 ≤ s1 < . . . < sk < [βn2 ], 1 ≤ α1 < . . . < αℓ ≤ [
β
2 ]. Mostly, we are interest-
ing in investigation of such graphs for sufficiently large n. In what follows, the numbers
s1, s2, . . . , sk, α1, α2, . . . , αℓ, β are supposed to be fixed positive integers.
In particular, graph Gn has no multiple edges if αℓ < β2 . If αℓ =
β
2 , it has exactly
two edges between vertices vi and vi+ β n2 , where indices are taken mod β n. In the latter
case, β is certainly an even positive integer. A typical example is graph C2n(1, n) which,
under the above agreement, represents a Moebius ladder graph on 2n vertices with double
steps. Circulant graphs with non-fixed jumps have been the subject of investigation in many
papers [8, 17, 24, 38].
132 Ars Math. Contemp. 23 (2023) #P1.08 / 129–144
Warning. In series of papers [5, 22, 23, 37] devoted to circulant graphs with odd
degree of vertices the notation C2n(1, n) stands for the Moebius ladder with ordinary steps.
The degree of vertices of such graph is three. These families of graphs are outside of
consideration in the present paper.
Recall [6] that the eigenvalues of matrix C = circ(a0, a1, . . . , an−1) are given by the
following simple formulas λj = L(ζjn), j = 0, 1, . . . , n − 1, where L(x) = a0 + a1x +
. . . + an−1x
n−1 and ζn is a primitive n-th root of unity. Moreover, the circulant matrix
C = L(T ), where T = circ(0, 1, 0, . . . , 0) is the matrix representation of the shift operator
T : (x0, x1, . . . , xn−2, xn−1) → (x1, x2, . . . , xn−1, x0).
Let P (z) = a0 + a1z+ . . .+ adzd = ad
∏d
k=1(z−αk) be a non-constant polynomial
with complex coefficients. Then, following Mahler [21] its Mahler measure is defined to
be
M(P ) := exp(
∫ 1
0
log |P (e2πit)|dt), (2.1)
the geometric mean of |P (z)| for z on the unit circle. However, M(P ) had appeared earlier
in a paper by Lehmer [15], in an alternative form
M(P ) = |ad|
∏
|αk|>1
|αk|. (2.2)
The equivalence of the two definitions follows immediately from Jensen’s formula [11]∫ 1
0
log |e2πit − α|dt = log+ |α|,
where log+ x denotes max(0, log x). We will also deal with the small Mahler measure
which is defined as
m(P ) := logM(P ) =
∫ 1
0
log |P (e2πit)|dt.
The concept of Mahler measure can be naturally extended to the class of Laurent polyno-
mials P (z) = a0zp+a1zp+1+ . . .+ad−1zp+d−1+adzp+d = adzp
∏d
k=1(z−αk), where
a0, ad ̸= 0 and p is an arbitrary and not necessarily positive integer.
3 Associated polynomials and their properties
The aim of this section is to introduce a few polynomials naturally associated with the
circulant graph
Gn = Cβn(s1, . . . , sk, α1n, . . . , αℓn),
1 ≤ s1 < . . . < sk < [
βn
2
], 1 ≤ α1 < . . . < αℓ ≤ [
β
2
].
We start with the Laurent polynomial
L(z) = 2(k + ℓ)−
k∑
i=1
(zsi + z−si)−
ℓ∑
m=1
(zαmn + z−αmn)
A. Mednykh and I. Mednykh: Complexity of circulant graphs with non-fixed jumps, . . . 133
responsible for the structure of Laplacian of graph Gn. More precisely, the Laplacian of
Gn is given by the matrix
L = L(T ) = 2(k + ℓ)Iβn −
k∑
i=1
(T si + T−si)−
ℓ∑
m=1
(Tαmn + T−αmn),
where T the circulant matrix circ(0, 1, . . . , 0︸ ︷︷ ︸
βn
). We decompose L(z) into the sum of two
polynomials L(z) = P (z) + p(zn), where P (z) = 2k −
∑k
i=1(z
si + z−si) and
p(z) = 2ℓ −
∑ℓ
m=1(z
αm + z−αm). Now, we have to introduce a family of Laurent poly-
nomials differing by a constant from P (z). They are Pu(z) = P (z) + p(e
2π i u
β ), u =
0, 1, . . . , β−1. One can check that Pu(z) = 2k−
∑k
i=1(z
si+z−si)+4
∑ℓ
m=1 sin
2(π uαmβ ).
In particular, P0(z) = P (z).
We note that all the above Laurent polynomials are palindromic, that is they are invari-
ant under replacement z by 1/z. Any non-trivial palindromic Laurent polynomial can be
represented in the form P(z) = asz−s+as−1z−(s−1)+ . . .+a0+ . . .+as−1zs−1+aszs,
where as ̸= 0. We will refer to 2s as a degree of the polynomial P(z). Since P(z) = P( 1z ),
the following polynomial of degree s is well defined
Q(w) = P(w +
√
w2 − 1).
We will call it a Chebyshev trasform of P(z). Since Tk(w) = (w+
√
w2−1)k+(w+
√
w2−1)−k
2
is the k-th Chebyshev polynomial of the first kind, one can easy deduce that
Q(w) = a0 + 2a1T1(w) + . . .+ 2as−1Ts−1(w) + 2asTs(w).
Also, we have P(z) = Q( 12 (z +
1
z )).
Throughout the paper, we will use the following observation. If z1, 1/z1, . . . , zs, 1/zs
is the list of all the roots of P(z), then wk = 12 (zk +
1
zk
), k = 1, 2, . . . , s are all the roots
of the polynomial Q(w).
By direct calculation, we obtain that the Chebyshev transform of polynomial Pu(z) is
Qu(w) = 2k − 2
k∑
i=1
Tsi(w) + 4
ℓ∑
m=1
sin2(
π uαm
β
).
In particular, if zs(u), 1/zs(u), s = 1, 2, . . . , sk are the roots of Pu(z), then
ws(u) =
1
2 (zs(u) + zs(u)
−1), s = 1, 2, . . . , sk are all roots of the algebraic equation∑k
i=1 Tsi(w) = k + 2
∑ℓ
m=1 sin
2(π uαmβ ). We also need the following lemma.
Lemma 3.1. Let gcd(α1, α2, . . . , αℓ, β) = 1. Suppose that Pu(z) = 0, where 0 < u < β.
Then |z| ≠ 1.
Proof. Recall that Pu(z) = P (z)+ p(e
2π i u
β ), where P (z) = 2k−
∑k
i=1(z
si + z−si) and
p(z) = 2ℓ−
∑ℓ
m=1(z
αm + z−αm). We show that p(e
2π i u
β ) = 4
∑ℓ
m=1 sin
2(π uαmβ ) > 0.
Indeed, suppose that p(e
2π i u
β ) = 0. Then there are integers mj such that uαj = mjβ, j =
1, 2, . . . , ℓ. Hence
B = gcd(uα1, . . . , u αℓ, u β) = u gcd(α1, . . . , αℓ, β) = u < β.
134 Ars Math. Contemp. 23 (2023) #P1.08 / 129–144
From the other side
B = gcd(m1β, . . . ,mℓβ, u β) = β gcd(m1, . . . ,mℓ, u) ≥ β.
Contradiction. Now, let |z| = 1. Then z = eiφ, for some φ ∈ R. We have
Pu(e
iφ) = P (eiφ) + p(e
2π i u
β ) = 2k −
k∑
j=1
(eisjφ + e−isjφ) + 4
ℓ∑
m=1
sin2(
π uαm
β
)
= 2
k∑
j=1
(1− cos(sjφ)) + 4
ℓ∑
m=1
sin2(
π uαm
β
) > 0.
Hence, Pu(z) > 0 and lemma is proved.
4 Complexity of circulant graphs with non-fixed jumps
The aim of this section is to find new formulas for the numbers of spanning trees of circu-
lant graph Cβn(s1, s2, . . . , sk, α1n, α2n, . . . , αℓn) in terms of Chebyshev polynomials. It
should be noted that nearby results were obtained earlier by different methods in the papers
[5, 8, 17, 19, 37, 38].
Theorem 4.1. The number of spanning trees in the circulant graph with non-fixed jumps
Cβn(s1, . . . , sk, α1n, . . . , αℓn), 1 ≤ s1 < . . . < sk < [
βn
2
], 1 ≤ α1 < . . . < αℓ ≤ [
β
2
]
is given by the formula
τ(n) =
n
β q
β−1∏
u=0
sk∏
j=1,
wj(0)̸=1
|2Tn(wj(u))− 2 cos(
2πu
β
)|,
where for each u = 0, 1, . . . , β − 1 the numbers wj(u), j = 1, 2, . . . , sk, are all the
roots of the equation
∑k
i=1 Tsi(w) = k + 2
∑ℓ
m=1 sin
2(π uαmβ ), Ts(w) is the Chebyshev
polynomial of the first kind and q = s21 + s
2
2 + . . .+ s
2
k.
Proof. Let G = Cβn(s1, s2, . . . , sk, α1n, α2n, . . . , αℓn). By the celebrated Kirchhoff the-
orem, the number of spanning trees τ(n) in Gn is equal to the product of non-zero eigen-
values of the Laplacian of the graph Gn divided by the number of its vertices βn. To
investigate the spectrum of Laplacian matrix, we denote by T the βn×βn circulant matrix
circ(0, 1, . . . , 0). Consider the Laurent polynomial
L(z) = 2(k + ℓ)−
k∑
i=1
(zsi + z−si)−
ℓ∑
m=1
(zαmn + z−αmn).
Then the Laplacian of Gn is given by the matrix
L = L(T ) = 2(k + ℓ)Iβn −
k∑
i=1
(T si + T−si)−
ℓ∑
m=1
(Tαmn + T−αmn).
A. Mednykh and I. Mednykh: Complexity of circulant graphs with non-fixed jumps, . . . 135
The eigenvalues of the circulant matrix T are ζjβn, j = 0, 1, . . . , βn − 1, where ζℓ =
e
2πi
ℓ . Since all of them are distinct, the matrix T is similar to the diagonal matrix T =
diag(1, ζβn, . . . , ζ
βn−1
βn ). To find spectrum of L, without loss of generality, one can assume
that T = T. Then L is a diagonal matrix. This essentially simplifies the problem of finding
eigenvalues of L. Indeed, let λ be an eigenvalue of L and x be the respective eigenvector.
Then we have the following system of linear equations
((2(k + ℓ)− λ)Iβn −
k∑
i=1
(T si + T−si)−
ℓ∑
m=1
(Tαmn + T−αmn))x = 0.
Let ej = (0, . . . , 1︸︷︷︸
j−th
, . . . , 0), j = 1, . . . , βn. The (j, j)-th entry of T is equal to ζj−1βn .
Then, for j = 0, . . . , βn− 1, the matrix L has an eigenvalue
λj = L(ζ
j
βn) = 2(k + ℓ)−
k∑
i=1
(ζjsiβn + ζ
−jsi
βn )−
ℓ∑
m=1
(ζjαmβ + ζ
−jαm
β ), (4.1)
with eigenvector ej+1. Since all graphs under consideration are supposed to be connected,
we have λ0 = 0 and λj > 0, j = 1, 2, . . . , βn− 1. Hence
τ(n) =
1
βn
βn−1∏
j=1
L(ζjβn). (4.2)
By setting j = βt+ u, where 0 ≤ t ≤ n− 1, 0 ≤ u ≤ β − 1, we rewrite the formula
(4.2) in the form
τ(n) = (
1
n
n−1∏
t=1
L(ζβtβn))(
1
β
β−1∏
u=1
n−1∏
t=0
L(ζtβ+uβn )). (4.3)
It is easy to see that τ(n) is the product of two numbers τ1(n) = 1n
∏n−1
t=1 L(ζ
βt
βn) and
τ2(n) =
1
β
∏β−1
u=1
∏n−1
t=0 L(ζ
tβ+u
βn ).
We note that
L(ζβtβn) = 2k −
k∑
i=1
(ζβtsiβn + ζ
−βtsi
βn ) = 2k −
k∑
i=1
(ζtsin + ζ
−tsi
n ) = P (ζ
t
n), 1 ≤ t ≤ n− 1.
The numbers µt = P (ζtn), 1 ≤ t ≤ n−1 run through all non-zero Laplacian eigenvalues of
circulant graph Cn(s1, s2, . . . , sk) with fixed jumps s1, s2, . . . , sk and n vertices. So τ1(n)
coincide with the number of spanning trees in Cn(s1, s2, . . . , sk). By ([23], Corollary 1)
we get
τ1(n) =
n
q
sk∏
j=1,
wj(0)̸=1
|2Tn(wj(0))− 2|, (4.4)
where wj(0), j = 1, 2, . . . , sk, are all the roots of the equation
∑k
i=1 Tsi(w) = k.
136 Ars Math. Contemp. 23 (2023) #P1.08 / 129–144
In order to continue the calculation of τ(n) we have to find the product
τ2(n) =
1
β
β−1∏
u=1
n−1∏
t=0
L(ζtβ+uβn ).
Recall that L(z) = P (z) + p(zn). Since (ζβt+uβn )
n = ζβt+uβ = ζ
u
β , we obtain
L(ζβt+uβn ) = P (ζ
βt+u
βn ) + p(ζ
βt+u
β ) = P (ζ
βt+u
βn ) + p(ζ
u
β ) = Pu(ζ
βt+u
βn ),
where Pu(z) = P (z) + p(ζuβ ). By Section 3, we already know that
Pu(z) = −
sk∏
j=1
(z − zj(u))(z − zj(u)−1),
where wj(u) = 12 (zj(u) + zj(u)
−1), j = 1, 2, . . . , sk are all roots of the equation∑k
i=1 Tsi(w) = k + 2
∑ℓ
d=1 sin
2(π uαdβ ).
We note that ζtβ+uβn = e
i(2πt+ωu)
n , where ωu = 2πuβ . Then
∏n−1
t=0 L(ζ
tβ+u
βn ) =∏n−1
t=0 Pu(e
i(2πt+ωu)
n ). To evaluate the latter product, we need following lemma.
Lemma 4.2. Let H(z) =
∏m
s=1(z − zs)(z − z−1s ) and ω be a real number. Then
n−1∏
t=0
H(e
i(2πt+ω)
n ) = (−eiω)m
m∏
s=1
(2Tn(ws)− 2 cos(ω)),
where ws = 12 (zs + z
−1
s ), s = 1, . . . ,m and Tn(w) is the n-th Chebyshev polynomial of
the first kind.
Proof of Lemma 4.2. We note that 12 (z
n + z−n) = Tn(
1
2 (z + z
−1)). By the substitution
z = ei φ, this follows from the evident identity cos(nφ) = Tn(cosφ). Then we have
n−1∏
t=0
H(e
i(2πt+ω)
n ) =
n−1∏
t=0
m∏
s=1
(e
i(2πt+ω)
n − zs)(e
i(2πt+ω)
n − z−1s )
=
m∏
s=1
n−1∏
t=0
(−e
i(2πt+ω)
n z−1s )(zs − e
i(2πt+ω)
n )(zs − e−
i(2πt+ω)
n )
=
m∏
s=1
(−eiωzs−n)
n−1∏
t=0
(zs − e
i(2πt+ω)
n )(zs − e−
i(2πt+ω)
n )
=
m∏
s=1
(−eiωzs−n)(z2ns − 2 cos(ω)zns + 1)
=
m∏
s=1
(−eiω)(2 z
n
s + z
−n
s
2
− 2 cos(ω))
= (−eiω)m
m∏
s=1
(2Tn(ws)− 2 cos(ω)).
A. Mednykh and I. Mednykh: Complexity of circulant graphs with non-fixed jumps, . . . 137
Since Pu(z) = −Hu(z), where Hu(z) =
∏sk
j=1(z−zj(u))(z−zj(u)−1), by Lemma 4.2
we get
n−1∏
t=0
Pu(e
i(2πt+ωu)
n ) = (−1)n(−e
2π i u
β )sk
sk∏
j=1
(2Tn(wj(u))− 2 cos(
2π u
β
)).
Then,
τ2(n) =
1
β
β−1∏
u=1
n−1∏
t=0
L(ζβt+uβn ) =
1
β
β−1∏
u=1
n−1∏
t=0
Pu(e
i(2πj+ωu)
n )
=
(−1)n(β−1)
β
β−1∏
u=1
(−e
2π i u
β )sk
sk∏
j=1
(2Tn(wj(u))− 2 cos(
2π u
β
)) (4.5)
=
(−1)n(β−1)
β
β−1∏
u=1
sk∏
j=1
(2Tn(wj(u))− 2 cos(
2π u
β
)).
Since the number τ2(n) is a product of positive eigenvalues of Gn divided by β, from (4.5)
we have
τ2(n) =
1
β
β−1∏
u=1
sk∏
j=1
|2Tn(wj(u))− 2 cos(
2π u
β
)|. (4.6)
Combining Equations (4.4) and (4.6) we finish the proof of the theorem.
As the first consequence from Theorem 4.1 we have the following result obtained earlier
by Justine Louis [19] in a slightly different form.
Corollary 4.3. The number of spanning trees in the circulant graphs with non-fixed jumps
Cβn(1, α1n, α2n, . . . , αℓn), where 1 ≤ α1 < α2 < . . . < αℓ ≤ [β2 ] is given by the
formula
τ(n) =
n 2β−1
β
β−1∏
u=1
(Tn(1 + 2
ℓ∑
m=1
sin2(
π uαm
β
))− cos(2π u
β
)),
where Tn(w) is the Chebyshev polynomial of the first kind.
Proof. Follows directly from the theorem.
The next corollary is new.
Corollary 4.4. The number of spanning trees in the circulant graphs with non-fixed jumps
Cβn(1, 2, α1n, α2n, . . . , αℓn), where 1 ≤ α1 < α2 < . . . < αℓ ≤ [β2 ] is given by the
formula
τ(n) =
nF 2n
β
β−1∏
u=1
2∏
j=1
|2Tn(wj(u))− 2 cos(
2π u
β
)|,
where Fn is the n-th Fibonacci number, Tn(w) is the Chebyshev polynomial of the first
kind and w1,2(u) =
(
−1±
√
25 + 16
∑ℓ
m=1 sin
2(π uαmβ )
)
/4.
138 Ars Math. Contemp. 23 (2023) #P1.08 / 129–144
We note that nF 2n is the number of spanning trees in the graph Cn(1, 2).
Proof. In this case, k = 2, s1 = 1, s2 = 2 and q = s21 + s
2
2 = 5. Given u we find
wj(u), j = 1, 2 as the roots of the algebraic equation
T1(w) + T2(w) = 2 + 2
ℓ∑
m=1
sin2(
π uαm
β
),
where T1(w) = w and T2(w) = 2w2 − 1. For u = 0 the roots are w1(0) = 1 and w2(0) =
−3/2. Hence, by (4.4), τ1(n) = n5 |2Tn(−
3
2 )−2| =
n
5 |(
−3+
√
5
2 )
n+(−3−
√
5
2 )
n−2| = nF 2n
gives the well-known formula for the number of spanning trees in the graph Cn(1, 2). (See,
for example, [2], Theorem 4). For u > 0 the numbers w1(u) and w2(u) are roots of the
quadratic equation
2w2 + w − 3− 2
ℓ∑
m=1
sin2(
π uαm
β
) = 0.
By (4.6) we get τ2(n) = 1β
∏β−1
u=1
∏2
j=1 |2Tn(wj(u)) − 2 cos(
2π u
β )|. Since τ(n) =
τ1(n)τ2(n), the result follows.
5 Arithmetic properties of the complexity for circulant graphs
It was noted in the series of paper [14, 22, 23, 25] that in many important cases the complex-
ity of graphs is given by the formula τ(n) = p n a(n)2, where a(n) is an integer sequence
and p is a prescribed constant depending only on parity of n.
The aim of the next theorem is to explain this phenomena for circulant graphs with
non-fixed jumps. Recall that any positive integer p can be uniquely represented in the form
p = q r2, where p and q are positive integers and q is square-free. We will call q the
square-free part of p.
Theorem 5.1. Let τ(n) be the number of spanning trees of the circulant graph
Gn = Cβn(s1, s2, . . . , sk, α1n, α2n, . . . , αℓn),
where 1 ≤ s1 < s2 < . . . < sk < [βn2 ], 1 ≤ α1 < α2 < . . . , αℓ ≤ [
β
2 ].
Denote by p and q the number of odd elements in the sequences s1, s2, . . . , sk and
α1, α2, . . . , αℓ, respectively. Let r be the square-free part of p and s be the square-free
part of p+ q. Then there exists an integer sequence a(n) such that
10 τ(n) = β n a(n)2, if n and β are odd;
20 τ(n) = β r n a(n)2, if n is even;
30 τ(n) = β sn a(n)2, if n is odd and β is even.
Proof. The number of odd elements in the sequences s1, s2, . . . , sk and α1, α2, . . . , αℓ,
respectively is counted by the formulas p =
∑k
i=1
1−(−1)si
2 and q =
∑ℓ
i=1
1−(−1)αi
2 . We
already know that all non-zero Laplacian eigenvalues of the graph Gn are given by the
formulas λj = L(ζ
j
βn), j = 1, . . . , βn− 1, where ζβn = e
2πi
βn and
L(z) = 2(k + l)−
k∑
i=1
(zsi + z−si)−
ℓ∑
m=1
(znαm + z−nαm).
A. Mednykh and I. Mednykh: Complexity of circulant graphs with non-fixed jumps, . . . 139
We note that λβn−j = L(ζ
βn−j
βn ) = L(ζ
j
βn) = λj .
By the Kirchhoff theorem we have βn τ(n) =
∏βn−1
j=1 λj . Since λβn−j = λj , we
obtain βn τ(n) = (
∏ βn−1
2
j=1 λj)
2 if βn is odd and βn τ(n) = λ βn
2
(
∏ βn
2 −1
j=1 λj)
2 if βn is
even. We note that each algebraic number λj comes into the above products together with
all its Galois conjugate [18]. So, the numbers c(n) =
∏ βn−1
2
j=1 λj and d(n) =
∏ βn
2 −1
j=1 λj
are integers. Also, for even n we have
λ βn
2
= L(−1) = 2(k + l)−
k∑
i=1
((−1)si + (−1)−si)−
ℓ∑
m=1
((−1)nαm + (−1)−nαm)
= 2k −
k∑
i=1
((−1)si + (−1)−si) = 4
k∑
i=1
1− (−1)si
2
= 4p.
If n is odd and β is even, the number βn2 is integer again. Then we obtain
λ βn
2
= L(−1) = 2(k + l)−
k∑
i=1
((−1)si + (−1)−si)−
ℓ∑
m=1
((−1)αm + (−1)−αm)
= 4
k∑
i=1
1− (−1)si
2
+ 4
ℓ∑
m=1
1− (−1)αm
2
= 4p+ 4q.
Therefore, β n τ(n) = c(n)2 if β and n are odd, β n τ(n) = 4p d(n)2 if n is even and
β n τ(n) = 4(p+ q) d(n)2 if n is odd and β is even. Let r be the square-free part of p and
s be the square-free part of p + q. Then there are integers u and v such that p = ru2 and
s = (p+ q)v2. Hence,
1◦
τ(n)
β n
=
(
c(n)
β n
)2
if n and β are odd,
2◦
τ(n)
β n
= r
(
2u d(n)
β n
)2
if n is even and
3◦
τ(n)
β n
= s
(
2 v d(n)
β n
)2
if n is odd and β is even.
Consider an automorphism group Zβn = ⟨g⟩ of the graph Gn generated by the element
g circularly permuting vertices v0, v1, . . . , vβ n−1 by the rule vi → vi+1 and the addition
in the indices is done modulo βn. The action of such a group is uniquely defined on the set
of all edges of Gn, except for those that connect diametrically opposite vertices. Consider
separately two cases αℓ = β/2 and αℓ < β/2.
In the first case, we have two parallel edges between the diametrically opposite vertices
vi and vi+ β n2 , where the indices are taken modβ n. To distinguish them, we orient one of
this edges by the arrow from vi and vi+ β n2 and the other one by the arrow from vi+ β n2 to
vi. As a result, we get exactly β n oriented edges. Denote the edge oriented from vi+ β n2 to
vi by ei and define the action of g on such edges by the rule ei → ei+1, where i is taken
mod β n.
140 Ars Math. Contemp. 23 (2023) #P1.08 / 129–144
In the second case, we have αℓ < β2 and sk <
β n
2 . Therefore, all jumps α1n, . . . , αℓn
and s1, . . . , sk of the graph Gn are strictly less then β n2 and Gn has no edges between the
diametrically opposite edges. That is, the action of group Zβn is well defined on its edges.
So, one can conclude that group Zβn acts fixed point free on the set vertices and on the
set of edges of Gn.
We are aimed to show that it also acts freely on the set of the spanning trees in the
graph. Indeed, suppose that some non-trivial element γ of Zβn leaves a spanning tree A
in the graph Gn invariant. Then γ fixes the center of A. The center of a tree is a vertex
or an edge. The first case is impossible, since γ acts freely on the set of vertices. In the
second case, γ permutes the endpoints of an edge connecting the opposite vertices of Gn.
This means that β n is even, and γ is the unique involution in the group Zβn. This is also
impossible, since the group is acting without fixed edges.
So, the cyclic group Zβn acts on the set of spanning trees of the graph Gn fixed point
free. Therefore τ(n) is a multiple of β n and their quotient τ(n)β n is an integer.
Setting a(n) = c(n)β n in the case 1
◦, a(n) = 2u d(n)β n in the case 2
◦ and a(n) = 2 v d(n)β n
in the case 3◦ we conclude that number a(n) is always integer and the statement of the
theorem follows.
6 Asymptotic for the number of spanning trees
In this section, we give asymptotic formulas for the number of spanning trees for circulant
graphs. It is interesting to compare these results with those in papers [5, 8, 17, 19, 37],
where the similar results were obtained by different methods.
Theorem 6.1. Let gcd(s1, s2, . . . , sk) = d and gcd(α1, α2, . . . , αℓ, β) = 1. Then the
number of spanning trees in the circulant graph
Cβn(s1, s2, . . . , sk, α1n, α2n, . . . , αℓn),
1 ≤ s1 < s2 < . . . < sk < [
βn
2
], 1 ≤ α1 < α2 < . . . < αℓ ≤ [
β
2
],
has the following asymptotic
τ(n) ∼ nd
2
β q
An, as n → ∞ and (n, d) = 1,
where q = s21+s
2
2+. . .+s
2
k, A =
∏β−1
u=0 M(Pu) and M(Pu) = exp(
∫ 1
0
log |Pu(e2πit)|dt)
is the Mahler measure of Laurent polynomial
Pu(z) = 2k −
k∑
i=1
(zsi + z−si) + 4
ℓ∑
m=1
sin2(
πuαm
β
).
Proof. By Theorem 4.1, τ(n) = τ1(n)τ2(n), where τ1(n) is the number of spanning trees
in Cn(s1, s2, . . . , sk) and τ2(n) = 1β
∏β−1
u=1
∏sk
j=1 |2Tn(wj(u)) − 2 cos(
2π u
β )|. By ([23],
Theorem 5) we already know that
τ1(n) ∼
nd2
q
An0 , as n → ∞ and (n, d) = 1,
A. Mednykh and I. Mednykh: Complexity of circulant graphs with non-fixed jumps, . . . 141
where A0 is the Mahler measure of Laurent polynomial P0(z). So, we have to find asymp-
totics for τ2(n) only.
By Lemma 3.1, for any integer u, 0 < u < β we obtain Tn(wj(u)) = 12 (zj(u)
n +
zj(u)
−n), where the zj(u) and 1/zj(u) are roots of the polynomial Pu(z) satisfying the
inequality |zj(u)| ≠ 1, j = 1, 2, . . . , sk. Replacing zj(u) by 1/zj(u), if necessary, we can
assume that |zj(u)| > 1 for all j = 1, 2, . . . , sk. Then Tn(wj(u)) ∼ 12zj(u)
n, as n tends
to ∞. So |2Tn(wj(u))− 2 cos( 2π uβ )| ∼ |zj(u)|
n, n → ∞. Hence
sk∏
j=1
|2Tn(wj(u))− 2 cos(
2π u
β
)| ∼
sk∏
s=1
|zj(u)|n =
∏
Pu(z)=0, |z|>1
|z|n = Anu,
where Au =
∏
Pu(z)=0, |z|>1|z| coincides with the Mahler measure of Pu(z). As a result,
τ2(n) =
1
β
β−1∏
u=1
sk∏
j=1
|2Tn(wj(u))− 2 cos(
2π u
β
)| ∼ 1
β
β−1∏
u=1
Anu.
Finally, τ(n) = τ1(n)τ2(n) ∼ nd
2
β q
∏β−1
u=0 A
n
u, as n → ∞ and (n, d) = 1. Since Au =
M(Pu), the result follows.
As an immediate consequence of above theorem we have the following result obtained
earlier in ([8], Theorem 3) by completely different methods.
Corollary 6.2. The thermodynamic limit of the sequence Cβn(s1, s2, . . . , sk,
α1n, α2n, . . . , αℓn) of circulant graphs is equal to the arithmetic mean of small Mahler
measures of Laurent polynomials Pu(z), u = 0, 1, . . . , β − 1. More precisely,
lim
n→∞
log τ(Cβn(s1, s2, . . . , sk, α1n, α2n, . . . , αℓn))
β n
=
1
β
β−1∑
u=0
m(Pu),
where m(Pu) =
∫ 1
0
log |Pu(e2πit)|dt and Pu(z) = 2k −
∑k
i=1(z
si + z−si) +
4
∑ℓ
m=1 sin
2(π uαmβ ).
7 Examples
1. Graph C2n(1, n). (Möbius ladder with double steps). By Theorem 4.1, we have
τ(n) = τ(C2n(1, n)) = n (Tn(3) + 1). Compare this result with ([38], Theorem 4).
Recall [2] that the number of spanninig trees in the Möbius ladder with single steps
is given by the formula n (Tn(2) + 1).
2. Graph C2n(1, 2, n). We have τ(n) = 2nF 2n |Tn(−1−
√
41
4 ) − 1||Tn(
−1+
√
41
4 ) − 1|.
By Theorem 5.1, one can find an integer sequence a(n) such that τ(n) = 2na(n)2
if n is even and τ(n) = na(n)2 if n is odd.
3. Graph C2n(1, 2, 3, n). Here τ(n) = 8n7 (Tn(θ1)−1)(Tn(θ2)−1)
∏3
p=1(Tn(ωp)+1),
where θ1 = −3+
√
−7
4 , θ2 =
−3−
√
−7
4 and ωp, p = 1, 2, 3 are roots of the cubic
equation 2w3 + w2 − w − 3 = 0. We have τ(n) = 6na(n)2 is n is odd and τ(n) =
4na(n)2 is n is even. Also, τ(n) ∼ n28A
n, n → ∞, where A ≈ 42.4038.
142 Ars Math. Contemp. 23 (2023) #P1.08 / 129–144
4. Graph C3n(1, n). We have
τ(n) =
n
3
(2Tn(
5
2
) + 1)2 =
n
3
((
5 +
√
21
2
)n + (
5−
√
21
2
)n + 1)2.
See also ([38], Theorem 5). We note that τ(n) = 3na(n)2, where a(n) satisfies
the recursive relation a(n) = 6a(n − 1) − 6a(n − 2) + a(n − 3) with initial data
a(1) = 2, a(2) = 8, a(3) = 37.
5. Graph C3n(1, 2, n). By Theorem 4.1, we obtain
τ(n) =
n
3
F 2n(2Tn(ω1) + 1)
2(2Tn(ω2) + 1)
2,
where ω1 = −1+
√
37
4 and ω2 =
−1−
√
37
4 . By Theorem 5.1, τ(n) = 3na(n)
2 for
some integer sequence a(n).
6. Graph C6n(1, n, 3n). Now, we get
τ(n) =
n
3
(2Tn(
5
2
) + 1)2(2Tn(
7
2
)− 1)2(Tn(5) + 1).
For a suitable integer sequence a(n), one has τ(n) = 6na(n)2 if n is even and
τ(n) = 18na(n)2 if n is odd.
7. Graph C12n(1, 3n, 4n). In this case
τ(n) =
2n
3
Tn(2)
2(2Tn(
5
2
) + 1)2(Tn(3) + 1)(4Tn(
7
2
)2 − 3)2(2Tn(
9
2
)− 1)2.
By Theorem 5.1, one can conclude that τ(n) = 3na(n)2 if n is even and τ(n) =
6na(n)2 if n is odd, for some sequence a(n) of even numbers.
ORCID iDs
Alexander Mednykh https://orcid.org/0000-0003-3084-1225
Ilya Mednykh https://orcid.org/0000-0001-7682-3917
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.09 / 145–162
https://doi.org/10.26493/1855-3974.2550.d96
(Also available at http://amc-journal.eu)
On generalized Minkowski arrangements*
Máté Kadlicskó
Department of Geometry, Budapest University of Technology,
Egry József utca 1., Budapest, Hungary, 1111
Zsolt Lángi †
MTA-BME Morphodynamics Research Group and Department of Geometry,
Budapest University of Technology,
Egry József utca 1., Budapest, Hungary, 1111
Received 9 Februar 2021, accepted 14 December 2021, published online 28 October 2022
Abstract
The concept of a Minkowski arrangement was introduced by Fejes Tóth in 1965 as a
family of centrally symmetric convex bodies with the property that no member of the family
contains the center of any other member in its interior. This notion was generalized by Fejes
Tóth in 1967, who called a family of centrally symmetric convex bodies a generalized
Minkowski arrangement of order µ for some 0 < µ < 1 if no member K of the family
overlaps the homothetic copy of any other member K ′ with ratio µ and with the same
center as K ′. In this note we prove a sharp upper bound on the total area of the elements
of a generalized Minkowski arrangement of order µ of finitely many circular disks in the
Euclidean plane. This result is a common generalization of a similar result of Fejes Tóth
for Minkowski arrangements of circular disks, and a result of Böröczky and Szabó about
the maximum density of a generalized Minkowski arrangement of circular disks in the
plane. In addition, we give a sharp upper bound on the density of a generalized Minkowski
arrangement of homothetic copies of a centrally symmetric convex body.
Keywords: Arrangement, Minkowski arrangement, density, homothetic copy.
Math. Subj. Class. (2020): 52C15, 52C26, 52A10
*The authors express their gratitude to K. Bezdek for directing their attention to this interesting problem, and
to two anonymous referees for many helpful suggestions. The second named author is supported by the National
Research, Development and Innovation Office, NKFI, K-119670, the János Bolyai Research Scholarship of the
Hungarian Academy of Sciences, and the BME IE-VIZ TKP2020 and ÚNKP-20-5 New National Excellence
Programs by the Ministry of Innovation and Technology.
†Corresponsing author.
E-mail addresses: kadlicsko.mate@gmail.com (Máté Kadlicskó), zlangi@math.bme.hu (Zsolt Lángi)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
146 Ars Math. Contemp. 23 (2023) #P1.09 / 145–162
1 Introduction
The notion of a Minkowski arrangement of convex bodies was introduced by L. Fejes
Tóth in [6], who defined it as a family F of centrally symmetric convex bodies in the
d-dimensional Euclidean space Rd, with the property that no member of F contains the
center of any other member of F in its interior. He used this concept to show, in particular,
that the density of a Minkowski arrangement of homothets of any given plane convex body
with positive homogeneity is at most four. Here an arrangement is meant to have positive
homogeneity if the set of the homothety ratios is bounded from both directions by positive
constants. It is worth mentioning that the above result is a generalization of the planar case
of the famous Minkowski Theorem from lattice geometry [11]. Furthermore, Fejes Tóth
proved in [6] that the density of a Minkowski arrangement of circular disks in R2 with
positive homogeneity is maximal for a Minkowski arrangement of congruent circular disks
whose centers are the points of a hexagonal lattice and each disk contains the centers of six
other members on its boundary.
In [7], extending the investigation to finite Minkowski arrangements, Fejes Tóth gave a
sharp upper bound on the total area of the members of a Minkowski arrangement of finitely
many circular disks, and showed that this result immediately implies the density estimate
in [6] for infinite Minkowski circle-arrangements. Following a different direction, in [8]
for any 0 < µ < 1 Fejes Tóth defined a generalized Minkowski arrangements of order µ as
a family F of centrally symmetric convex bodies with the property that for any two distinct
members K,K ′ of F , K does not overlap the µ-core of K ′, defined as the homothetic
copy of K ′ of ratio µ and concentric with K ′. In this paper he made the conjecture that
for any 0 < µ ≤
√
3 − 1, the density of a generalized Minkowski arrangement of circular
disks with positive homogeneity is maximal for a generalized Minkowski arrangement of
congruent disks whose centers are the points of a hexagonal lattice and each disk touches
the µ-core of six other members of the family. According to [8], this conjecture was verified
by Böröczky and Szabó in a seminar talk in 1965, though the first written proof seems to be
published only in [4] in 2002. It was observed both in [8] and [4] that if
√
3− 1 < µ < 1,
then, since the above hexagonal arrangement does not cover the plane, that arrangement
has no maximal density.
In this paper we prove a sharp estimate on the total area of a generalized Minkowski
arrangement of finitely many circular disks, with a characterization of the equality case.
Our result includes the result in [7] as a special case, and immediately implies the one in
[4]. The proof of our statement relies on tools from both [4, 7], but uses also some new
ideas. In addition, we also generalize a result from Fejes Tóth [6] to find a sharp upper
bound on the density of a generalized Minkowski arrangement of homothetic copies of a
centrally symmetric convex body.
For completeness, we mention that similar statements for (generalized) Minkowski ar-
rangements in other geometries and in higher dimensional spaces were examined, e.g. in
[5, 9, 13]. Minkowski arrangements consisting of congruent convex bodies were consid-
ered in [3]. Estimates for the maximum cardinality of mutually intersecting members in a
(generalized) Minkowski arrangement can be found in [10, 14, 15, 17]. The problem inves-
tigated in this paper is similar in nature to those dealing with the volume of the convex hull
of a family of convex bodies, which has a rich literature. This includes a result of Oler [16]
(see also [2]), which is also of lattice geometric origin [20], and the notion of parametric
density of Betke, Henk and Wills [1]. In particular, our problem is closely related to the
notion of density with respect to outer parallel domains defined in [2]. Applications of
M. Kadlicskó and Z. Lángi: On generalized Minkowski arrangements 147
(generalized) Minkowski arrangements in other branches of mathematics can be found in
[18, 19].
As a preliminary observation, we start with the following generalization of Remark 2
of [6], stating the same property for (not generalized) Minkowski arrangements of plane
convex bodies. In Proposition 1.1, by vold(·) we denote d-dimensional volume, and by Bd
we denote the closed Euclidean unit ball centered at the origin.
Proposition 1.1. Let 0 < µ < 1, let K ⊂ Rd be an origin-symmetric convex body and
let F = {x1 + λ1K,x2 + λ2K, . . . } be a generalized Minkowski arrangement of order µ,
where xi ∈ Rd, λi > 0 for each i = 1, 2, . . . . Assume that F is of positive homogeneity,
that is, there are constants 0 < C1 < C2 satisfying C1 ≤ λi ≤ C2 for all values of i, and
define the (upper) density δ(F) of F in the usual way as
δ(F) = lim sup
R→∞
∑
xi∈RBd vold(xi + λiK)
vold(RBd)
,
if it exists. Then
δ(F) ≤ 2
d
(1 + µ)d
, (1.1)
where equality is attained, e.g. if {x1, x2, . . .} is a lattice with K as its fundamental region,
and λi = 2/(1 + µ) for all values of i.
Proof. Note that the equality part of Proposition 1.1 clearly holds, and thus, we prove only
the inequality in (1.1). Let || · ||K : Rd → [0,∞) denote the norm with K as its unit ball.
Then, by the definition of a generalized Minkowski arrangement, we have
||xi − xj ||K ≥ max{λi + µλj , λj + µλi}
≥ 1
2
((λi + µλj) + (λj + µλi))
=
1 + µ
2
(λi + λj),
implying that the homothets xi+(λi/2) ·(1 + µ)K are pairwise non-overlapping. In other
words, the family F ′ = {xi + (λi/2) · (1 + µ)K : i = 1, 2, . . .} is a packing. Thus, the
density of F ′ is at most one, from which (1.1) readily follows. Furthermore, if K is the
fundamental region of a lattice formed by the xi’s and λi = 2/(1 + µ) for all values of i,
then F ′ is a tiling, implying the equality case.
Following the terminology of Fejes Tóth in [7] and to permit a simpler formulation
of our main result, in the remaining part of the paper we consider generalized Minkowski
arrangements of open circular disks, where we note that generalized Minkowski arrange-
ments can be defined for families of open circular disks in the same way as for families of
closed circular disks.
To state our main result, we need some preparation, where we denote the boundary of
a set by bd(·). Consider some generalized Minkowski arrangement F = {Bi = xi +
ρi int(B
2) : i = 1, 2, . . . , n} of open circular disks in R2 of order µ, where 0 < µ < 1. Set
U(F) =
⋃n
i=1 Bi =
⋃
F . Then each circular arc Γ in bd(U(F)) corresponds to a circular
sector, which can be obtained as the union of the segments connecting a point of Γ to the
center of the disk in F whose boundary contains Γ. We call the union of these circular
148 Ars Math. Contemp. 23 (2023) #P1.09 / 145–162
sectors the outer shell of F . Now consider a point p ∈ bd(U(F)) belonging to at least
two members of F , say Bi and Bj , such that xi, xj and p are not collinear. Assume that
the convex angular region bounded by the two closed half lines starting at p and passing
through xi and xj , respectively, do not contain the center of another element of F in its
interior which contains p on its boundary. We call the union of the triangles conv{p, xi, xj}
satisfying these conditions the inner shell of F . We denote the inner and the outer shell of F
by I(F) and O(F), respectively. Finally, we call the set C(F) = U(F)\(I(F)∪O(F)) the
core of F (cf. Figure 1). Clearly, the outer shell of any generalized Minkowski arrangement
of open circular disks is nonempty, but there are arrangements for which I(F) = ∅ or
C(F) = ∅.
Figure 1: The outer and inner shell, and the core of an arrangement, shown in white, light
grey and dark grey, respectively.
If the intersection of two members of F is nonempty, then we call this intersection a
digon. If a digon touches the µ-cores of both disks defining it, we call the digon thick.
A digon which is not contained in a third member of F is called a free digon. Our main
theorem is as follows, where area(X) denotes the area of the set X .
Theorem 1.2. Let 0 < µ ≤
√
3−1, and let F = {Bi = xi+ρi int(B2) : i = 1, 2, . . . , n}
be a generalized Minkowski arrangement of finitely many open circular disks of order µ.
Then
T = π
n∑
i=1
ρ2i ≤
2π√
3(1 + µ)2
area(C(F))+
+
4 · arccos( 1+µ2 )
(1 + µ) ·
√
(3 + µ)(1− µ)
area(I(F)) + area(O(F)),
M. Kadlicskó and Z. Lángi: On generalized Minkowski arrangements 149
where T is the total area of the circles, with equality if and only if each free digon in F is
thick.
In the paper, for any points x, y, z ∈ R2, we denote by [x, y] the closed segment with
endpoints x, y, by [x, y, z] the triangle conv{x, y, z}, by |x| the Euclidean norm of x, and if
x and z are distinct from y, by ∠xyz we denote the measure of the angle between the closed
half lines starting at y and passing through x and z. Note that according to our definition,
∠xyz is at most π for any x, z ̸= y. For brevity we call an open circular disk a disk, and
a generalized Minkowski arrangement of disks of order µ a µ-arrangement. Throughout
Sections 2 and 3 we assume that 0 < µ ≤
√
3− 1.
In Section 2, we prove some preliminary lemmas. In Section 3, we prove Theorem 1.2.
Finally, in Section 4, we collect additional remarks and questions.
2 Preliminaries
For any Bi, Bj ∈ F , if Bi ∩Bj ̸= ∅, we call the two intersection points of bd(Bi) and
bd(Bj) the vertices of the digon Bi ∩Bj .
First, we recall the following lemma of Fejes Tóth [7, Lemma 2]. To prove it, we ob-
serve that for any µ > 0, a generalized Minkowski arrangement of order µ is a Minkowski
arrangement as well.
Lemma 2.1. Let Bi, Bj , Bk ∈ F such that the digon Bi ∩ Bj is contained in Bk. Then
the digon Bi ∩Bk is free (with respect to F).
From now on, we call the maximal subfamilies F ′ of F (with respect to containment)
with the property that
⋃
Bi∈F ′ Bi is connected the connected components of F . Our next
lemma has been proved by Fejes Tóth in [7] for Minkowski arrangements of order µ = 0.
His argument can be applied to prove Lemma 2.2 for an arbitrary value of µ. Here we
include this proof for completeness.
Lemma 2.2. If F ′ is a connected component of F in which each free digon is thick, then
the elements of F ′ are congruent.
Proof. We need to show that for any Bi, Bj ∈ F ′, Bi and Bj are congruent. Observe that
by connectedness, we may assume that Bi ∩ Bj is a digon. If Bi ∩ Bj is free, then it is
thick, which implies that Bi and Bj are congruent. If Bi ∩ Bj is not free, then there is a
disk Bk ∈ F ′ containing it. By Lemma 2.1, the digons Bi ∩ Bk and Bj ∩ Bk are free.
Thus Bk is congruent to both Bi and Bj .
In the remaining part of Section 2, we examine densities of some circular sectors in
certain triangles. The computations in the proofs of these lemmas were carried out by a
Maple 18.00 software.
Lemma 2.3. Let 0 < γ < π and A,B > 0 be arbitrary. Let T = [x, y, z] be a triangle such
that ∠xzy = γ, and |x− z| = A and |y − z| = B. Let ∆ = ∆(γ,A,B), α = α(γ,A,B)
and β = β(γ,A,B) denote the functions with variables γ,A,B whose values are the area
and the angles of T at x and y, respectively, and set fA,B(γ) =
(
αA2 + βB2
)
/∆. Then,
for any A,B > 0, the function fA,B(γ) is strictly decreasing on the interval γ ∈ (0, π).
150 Ars Math. Contemp. 23 (2023) #P1.09 / 145–162
B
A
γ
z
β
y
α
x
Figure 2: Notation in Lemma 2.3.
Proof. Without loss of generality, assume that A ≤ B, and let g = αA2 + βB2. Then, by
an elementary computation, we have that
g = A2 arccot
A−B cos γ
B sin γ
+B2 arccot
B −A cos γ
A sin γ
, and ∆ =
1
2
AB sin γ.
We regard g and ∆ as functions of γ. We intend to show that g′∆− g∆′ is negative on the
interval (0, π) for all A,B > 0. Let h = g′ · ∆/∆′ − g, and note that this expression is
continuous on (0, π/2) and (π/2, π) for all A,B > 0. By differentiating and simplifying,
we obtain
h′ =
−2
(
A2(1 + cos2(γ)) +B2(1 + cos2(γ))− 4AB cos(γ)
)
A2B2 sin2(γ)
cos2 (γ)(A2 +B2 − 2AB cos(γ))2
,
which is negative on its domain. This implies that g′∆ − g∆′ is strictly decreasing on
(0, π/2) and strictly increasing on (π/2, π). On the other hand, we have limγ→0+(g′∆ −
g∆′) = −A3Bπ, and limγ→π− (g′∆− g∆′) = 0. This yields the assertion.
Lemma 2.4. Consider two disks Bi, Bj ∈ F such that |xi − xj | < ρi + ρj , and let v be a
vertex of the digon Bi ∩ Bj . Let T = [xi, xj , v], ∆ = area(T ), and let αi = ∠vxixj and
αj = ∠vxjxi. Then
1
2
αiρ
2
i +
1
2
αjρ
2
j ≤
4 arccos 1+µ2
(1 + µ)
√
(1− µ)(3 + µ)
∆, (2.1)
with equality if and only if ρi = ρj and |xi − xj | = ρi(1 + µ).
Proof. First, an elementary computation shows that if ρi = ρj and |xi − xj | = ρi(1 + µ),
then there is equality in (2.1).
Without loss of generality, let ρi = 1, and 0 < ρj = ρ ≤ 1. By Lemma 2.3, we
may assume that |xi − xj | = 1 + µρ. Thus, the side lengths of T are 1, ρ, 1 + µρ.
M. Kadlicskó and Z. Lángi: On generalized Minkowski arrangements 151
v
αi
xi
αj
xj
ρi ρj
Figure 3: Notation in Lemma 2.4.
Applying the Law of Cosines and Heron’s formula to T we obtain that
1
2αiρ
2
i +
1
2αjρ
2
j
∆
=
f(ρ, µ)
g(ρ, µ)
,
where
f(ρ, µ) =
1
2
arccos
1 + (1 + µρ)2 − r2
2(1 + µρ)
+
1
2
ρ2 arccos
ρ2 + (1 + µρ)2 − 1
2ρ(1 + µρ)
,
and
g(ρ, µ) = ρ
√
2 + ρ+ µρ)(2− ρ+ µρ)(1− µ2).
In the remaining part we show that
f(ρ, µ)
g(ρ, µ)
<
4 arccos 1+µ2
(1 + µ)
√
(1− µ)(3 + µ)
if 0 < ρ < 1 and 0 ≤ µ ≤
√
3− 1. To do it we distinguish two separate cases.
Case 1: 0 < ρ ≤ 1/5. In this case we estimate f(ρ, µ)/g(ρ, µ) as follows. Let the
part of [xi, xj ] covered by both disks Bi and Bj be denoted by S. Then S is a segment of
length (1−µ)ρ. On the other hand, if Ai denotes the convex circular sector of Bi bounded
by the radii [xi, v] and [xi, xj ] ∩ Bi, and we define Aj analogously, then the sets Ai ∩ Aj
and (Ai ∪ Aj) \ T are covered by the rectangle with S as a side which contains v on the
side parallel to S. The area of this rectangle is twice the area of the triangle conv(S∪{v}),
implying that
f(ρ, µ)
g(ρ, µ)
≤ 1 + 2(1− µ)ρ
1 + µρ
.
We show that if 0 < ρ ≤ 1/5, then the right-hand side quantity in this inequality is strictly
less than the right-hand side quantity in (2.1). By differentiating with respect to ρ, we
see that as a function of ρ, 1 + (2(1− µ)ρ) /(1 + µρ) is strictly increasing on its domain
152 Ars Math. Contemp. 23 (2023) #P1.09 / 145–162
and attains its maximum at ρ = 1/5. Thus, using the fact that this maximum is equal to
(7− µ)/(5 + µ), we need to show that
4 arccos 1+µ2
(1 + µ)
√
(1− µ)(3 + µ)
− 7− µ
5 + µ
> 0.
Clearly, the function
µ 7→
arccos 1+µ2
1+µ
2
is strictly decreasing on the interval [0,
√
3 − 1]. By differentiation one can easily check
that the function
µ 7→ 7− µ
5 + µ
√
(1− µ)(3 + µ)
is also strictly increasing on the same interval. Thus, we obtain that the above expression
is minimal if µ =
√
3− 1, implying that it is at least 0.11570 . . ..
Case 2: 1/5 < ρ ≤ 1.
We show that in this case the partial derivative ∂ρ (f(ρ, µ)/g(ρ, µ)), or equivalently, the
quantity h(ρ, µ) = f ′ρ(ρ, µ)g(ρ, µ) − g′ρ(ρ, µ)f(ρ, µ) is strictly positive. By plotting the
latter quantity on the rectangle 0 ≤ µ ≤
√
3 − 1, 1/5 ≤ ρ ≤ 1, its minimum seems to be
approximately 0.00146046085. To use this fact, we upper bound the two partial derivatives
of this function, and compute its values on a grid. In particular, using the monotonicity
properties of the functions f, g, we obtain that under our conditions |f(ρ, µ)| < 1.25 and
|g(ρ, µ)| ≤ 0.5. Furthermore, using the inequalities 0 ≤ µ ≤
√
3 − 1, 1/5 ≤ ρ ≤ 1 and
also the triangle inequality to estimate the derivatives of f and g, we obtain that
|f ′ρ(ρ, µ)| < 1.95, |f ′µ(ρ, µ)| < 2.8, |f ′′ρρ(ρ, µ)| < 2.95, |f ′′ρµ(ρ, µ)| < 9.8,
and
|g′ρ(ρ, µ)| < 0.93, |g′µ(ρ, µ)| < 1.08, |g′′ρρ(ρ, µ)| < 2.64, |g′′ρµ(ρ, µ)| < 15.1.
These inequalities imply that |h′ρ(ρ, µ)| < 4.78 and |h′µ(ρ, µ)| < 28.49, and hence, for any
∆ρ and ∆µ, we have h(ρ + ∆ρ, µ + ∆µ) > h(ρ, µ) − 4.78|∆ρ| − 28.49|∆µ|. Thus, we
divided the rectangle [0.2, 1] × [0,
√
3 − 1] into a 8691 × 8691 grid, and by numerically
computing the value of h(ρ, µ) at the gridpoints, we showed that at any such point the value
of h (up to 12 digits) is at least 0.00144. According to our estimates above, this implies
that h(ρ, µ) ≥ 0.00002 for all values of ρ and µ.
Before our next lemma, recall that B2 denotes the closed unit disk centered at the
origin.
Lemma 2.5. For some 0 < ν < 1, let x, y, z ∈ R2 be non-collinear points, and let
{Bu = u + ρuB2 : u ∈ {x, y, z}} be a ν-arrangement of disks; that is, assume that for
any {u, v} ⊂ {x, y, z}, we have |u − v| ≥ max{ρu, ρv} + νmin{ρu, ρv}. Assume that
for any {u, v} ⊂ {x, y, z}, Bu ∩ Bv ̸= ∅, and that the union of the three disks covers the
triangle [x, y, z]. Then ν ≤
√
3− 1.
M. Kadlicskó and Z. Lángi: On generalized Minkowski arrangements 153
x y
z
ρj
ρx
ρy
ρz
T
Figure 4: Notation in Lemma 2.5. The circles drawn with dotted lines represent the µ-cores
of the disks.
Proof. Without loss of generality, assume that 0 < ρz ≤ ρy ≤ ρx. Since the disks
are compact sets, by the Knaster-Kuratowski-Mazurkiewicz lemma [12], there is a point
q of T belonging to all the disks, or in other words, there is some point q ∈ T such
that |q − u| ≤ ρu for any u ∈ {x, y, z}. Recalling the notation T = [x, y, z] from the
introduction, let T ′ = [x′, y′, z′] be a triangle with edge lengths |y′ − x′| = ρx + νρy ,
|z′ − x′| = ρx + νρz and |z′ − y′| = ρy + νρz , and note that these lengths satisfy the
triangle inequality.
We show that the disks x′+ρxB2, y′+ρyB2 and z′+ρzB2 and T ′ satisfy the conditions
in the lemma. To do this, we show the following, more general statement, which, together
with the trivial observation that any edge of T ′ is covered by the two disks centered at
its endpoints, clearly implies what we want: For any triangles T = [x, y, z] and T ′ =
[x′, y′, z′] satisfying |u′ − v′| ≤ |u − v| for any u, v ∈ {x, y, z}, and for any point q ∈ T
there is a point q′ ∈ T ′ such that |q′ − u′| ≤ |q − u| for any u ∈ {x, y, z}. The main
tool in the proof of this statement is the following straightforward consequence of the Law
of Cosines, stating that if the side lengths of a triangle are A,B,C, and the angle of the
triangle opposite of the side of length C is γ, then for any fixed values of A and B, C is a
strictly increasing function of γ on the interval (0, π).
To apply it, observe that if we fix x, y and q, and rotate [x, z] around x towards [x, q],
we strictly decrease |z − y| and |z − q| and do not change |y − x|, |z − x|, |x − q| and
|y−q|. Thus, we may replace z by a point z∗ satisfying |z∗−y| = |z′−y′|, or the property
that z∗, q, x are collinear. Repeating this transformation by x or y playing the role of z
we obtain either a triangle congruent to T ′ in which q satisfies the required conditions, or
a triangle in which q is a boundary point. In other words, without loss of generality we
may assume that q ∈ bd(T ). If q ∈ {x, y, z}, then the statement is trivial, and so we
assume that q is a relative interior point of, say, [x, y]. In this case, if |z − x| > |z′ − x′| or
|z − y| > |z′ − y′|, then we may rotate [y, z] or [x, z] around y or x, respectively. Finally,
if |y − x| > |y′ − x′|, then one of the angles ∠yxz or ∠xyz, say ∠xyz, is acute, and then
we may rotate [z, y] around z towards [z, q]. This implies the statement.
154 Ars Math. Contemp. 23 (2023) #P1.09 / 145–162
By our argument, it is sufficient to prove Lemma 2.5 under the assumption that |y−x| =
ρx+νρy , |z−x| = ρx+νρz and |z−y| = ρy+νρz . Consider the case that ρx > ρy . Let q be
a point of T belonging to each disk, implying that |q−u| ≤ ρu for all u ∈ {x, y, z}. Clearly,
from our conditions it follows that |x−q| > ρx−ρy . Let us define a 1-parameter family of
configurations, with the parameter t ∈ [0, ρx−ρy], by setting x(t) = x−tw, where w is the
unit vector in the direction of x− q, ρx(t) = ρx − t, and keeping q, y, z, ρy, ρz fixed. Note
that in this family q ∈ Bx(t) = x(t)+ρx(t)B2, which implies that |x(t)−u| ≤ ρx(t)+ρu
for u ∈ {y, z}. Thus, for any {u, v} ⊂ {x(t), y, z}, there is a point of [u, v] belonging to
both Bu and Bv . This, together with the property that q belongs to all three disks and using
the convexity of the disks, yields that the triangle [x(t), y, z] is covered by Bx(t)∪By∪Bz .
Let the angle between u − x(t) and w be denoted by φ. Then, using the linearity of
directional derivatives, we have that for f(t) = |x(t) − u|, f ′(t) = − cosφ ≥ −1 for
u ∈ {y, z}, implying |x(t) − u| ≥ |x − u| − t = ρx(t) + νρu for u ∈ {y, z}, and also
that the configuration is a ν-arrangement for all values of t. Hence, all configurations in
this family, and in particular, the configuration with t = ρx − ρy satisfies the conditions
in the lemma. Thus, repeating again the argument in the first part of the proof, we may
assume that ρx = ρy ≥ ρz , |y − x| = (1 + µ)ρx and |z − x| = |z − y| = ρx + νρz .
Finally, if ρx = ρy > ρz , then we may assume that q lies on the symmetry axis of T
and satisfies |x − q| = |y − q| > ρx − ρz . In this case we apply a similar argument
by moving x and y towards q at unit speed and decreasing ρx = ρy simultaneously till
they reach ρz , and, again repeating the argument in the first part of the proof, obtain that
the family {ū + ρzB2 : ū ∈ {x̄, ȳ, z̄}}, where T̄ = [x̄, ȳ, z̄] is a regular triangle of side
lengths (1 + ν)ρz , covers T̄ . Thus, the inequality ν ≤
√
3 − 1 follows by an elementary
computation.
In our next lemma, for any disk Bi ∈ F we denote by B̄i the closure xi + ρiB2 of Bi.
Lemma 2.6. Let Bi, Bj , Bk ∈ F such that B̄u ∩ B̄v ̸⊆ Bw for any {u, v, w} = {i, j, k}.
Let T = [xi, xj , xk], ∆ = area(T ), and αu = ∠xvxuxw. If T ⊂ B̄i ∪ B̄j ∪ B̄k, then
1
2
∑
u∈{i,j,k}
αuρ
2
u ≤
2π√
3(1 + µ)2
∆, (2.2)
with equality if and only if ρi = ρj = ρk, and T is a regular triangle of side length
(1 + µ)ρi.
Proof. In the proof we call
δ =
∑
u∈{i,j,k} αuρ
2
u
2∆
the density of the configuration.
Consider the 1-parameter families of disks Bu(ν) = xu+(1 + µ) / (1 + ν) ρu int(B2),
where u ∈ {i, j, k} and ν ∈ [µ, 1]. Observe that the three disks Bu(ν), where u ∈ {i, j, k},
form a ν-arrangement for any ν ≥ µ. Indeed, in this case for any {u, v} ⊂ {i, j, k}, if
ρu ≤ ρv , we have
1 + µ
1 + ν
ρv + ν
(
1 + µ
1 + ν
ρu
)
= ρv + µρu −
ν − µ
1 + ν
(ρv − ρu) ≤ ρv + µρu ≤ |xu − xv|.
M. Kadlicskó and Z. Lángi: On generalized Minkowski arrangements 155
Furthermore, for any ν ≥ µ, we have
(1 + µ)2
∑
u∈{i,j,k}
αuρ
2
u = (1 + ν)
2
∑
u∈{i,j,k}
αu
(
1 + µ
1 + ν
)2
ρ2u.
Thus, it is sufficient to prove the assertion for the maximal value ν̄ of ν such that the
conditions T ⊂ B̄i(ν)∪B̄j(ν)∪B̄k(ν) and B̄u∩B̄v ̸⊆ Bw are satisfied for any {u, v, w} =
{i, j, k}. Since the relation B̄u ∩ B̄v ̸⊆ Bw implies, in particular, that B̄u ∩ B̄v ̸= ∅, in this
case the conditions of Lemma 2.5 are satisfied, yielding ν̄ ≤
√
3 − 1. Hence, with a little
abuse of notation, we may assume that ν̄ = µ. Then one of the following holds:
(i) The intersection of the disks B̄u is a single point.
(ii) For some {u, v, w} = {i, j, k}, B̄u ∩ B̄v ⊂ B̄w and B̄u ∩ B̄v ̸⊂ Bw.
Before investigating (i) and (ii), we remark that during this process, which we refer to
as µ-increasing process, even though there might be non-maximal values of ν for which the
modified configuration satisfies the conditions of the lemma and also (i) or (ii), we always
choose the maximal value. This value is determined by the centers of the original disks and
the ratios of their radii.
First, consider (i). Then, clearly, the unique intersection point q of the disks lies in T ,
and note that either q lies in the boundary of all three disks, or two disks touch at q. We
describe the proof only in the first case, as in the second one we may apply a straightforward
modification of our argument. Thus, in this case we may decompose T into three triangles
[xi, xj , q], [xi, xk, q] and [xj , xk, q] satisfying the conditions in Lemma 2.4, and obtain
1
2
∑
u∈{i,j,k}
αuρ
2
u ≤
4 arccos 1+µ2
(1 + µ)
√
(1− µ)(3 + µ)
∆ ≤ 2π√
3(1 + µ)2
∆,
where the second inequality follows from the fact that the two expressions are equal if
µ =
√
3− 1, and (
2 arccos
1 + µ
2
−
π
√
(1− µ)(3 + µ)√
3(1 + µ)
)′
> 0
if µ ∈ [0,
√
3 − 1]. Here, by Lemma 2.4, equality holds only if ρi = ρj = ρk, and T is a
regular triangle of side length (1+µ)ρi. On the other hand, under these conditions in (2.2)
we have equality. This implies Lemma 2.6 for (i).
In the remaining part of the proof, we show that if (ii) is satisfied, the density of the
configuration is strictly less than 2π/
(√
3(1 + µ)2
)
. Let q be a common point of bd(B̄w)
and, say, B̄u. If q is a relative interior point of an arc in bd(B̄u ∩ B̄v), then one of the disks
is contained in another one, which contradicts the fact that the disks Bu, Bv, Bw form a
µ-arrangement. Thus, we have that either B̄u ∩ B̄v = {q}, or that q is a vertex of the digon
Bu ∩ Bv . If B̄u ∩ B̄v = {q}, then the conditions of (i) are satisfied, and thus, we assume
that q is a vertex of the digon Bu ∩ Bv . By choosing a suitable coordinate system and
rescaling and relabeling, if necessary, we may assume that Bu = int(B2), xv lies on the
positive half of the x-axis, and xw is written in the form xw = (ζw, ηw), where ηw > 0,
156 Ars Math. Contemp. 23 (2023) #P1.09 / 145–162
B̄u
B̄w
B̄v
Figure 5: An illustration for the proof of Lemma 2.6.
and the radical line of Bu and Bv separates xv and xw (cf. Figure 5). Set ρ = ρw. We
show that ηw > (1 + µ)ρ/2.
Case 1: if ρ ≥ 1. Then we have |xw| ≥ ρ+ µ.
Let the radical line of Bu and Bv be the line {x = t} for some 0 < t ≤ 1. Then, as
this line separates xv and xw, we have ζw ≤ t, and by (ii) we have q = (t,−
√
1− t2).
This implies that |xw − q| ≤ |xw − xu|, |xw − xv|, from which we have 0 ≤ ζw. Let
S denote the half-infinite strip S = {(ζ, η) ∈ R2 : 0 ≤ ζ ≤ t, η ≥ 0}, and set s =
(t,−
√
1− t2+ρ). Note that by our considerations, xw ∈ S and |xw− q| = ρ, which yield
ηw ≤ −
√
1− t2 + ρ. From this it follows that ρ + µ ≤ |xw| ≤ |s|, or in other words, we
have t2 + (ρ −
√
1− t2)2 ≥ (ρ + µ)2. By solving this inequality for t with parameters ρ
and µ, we obtain that t ≥ t0, 1 ≤ ρ ≤
(
1− µ2
)
/ (2µ) and 0 ≤ µ ≤
√
2− 1, where
t0 =
√
1−
(
1− 2µρ− µ2
2ρ
)2
.
Let p = (ζp, ηp) be the unique point in S with |p| = ρ + µ and |p − q| = ρ, and
observe that ηw ≥ ηp. Now we find the minimal value of ηp if t is permitted to change
and ρ is fixed. Set p′ = (ζp,−
√
1− ζ2p). Since the bisector of [p′, q] separates p′ and
p, it follows that |p − p′| ≥ |p − q| = ρ with equality only if p′ = q and p = s, or in
other words, if t = t0. This yields that ζp is maximal if t = t0. On the other hand, since
|p| = ρ + µ and p lies in the first quadrant, ηp is minimal if ζp is maximal. Thus, for a
fixed value of ρ, ηp is minimal if t = t0 and p = s = (t0,−
√
1− t20 + ρ), implying that
ηw ≥ −
√
1− t20 + ρ =
(
2ρ2 + µ2 + 2µρ− 1
)
/(2ρ). Now, ρ ≥ 1 and µ < 1 yields that
2ρ2 + µ2 + 2µρ− 1
2ρ
− (1 + µ)ρ
2
=
ρ2 − µρ2 + 2µρ− 1
2ρ
≥ µ
2ρ
> 0,
implying the statement.
M. Kadlicskó and Z. Lángi: On generalized Minkowski arrangements 157
Case 2: if 0 < ρ ≤ 1. In this case the inequality ηw > (1 + µ)ρ/2 follows by a similar
consideration.
In the remaining part of the proof, let
σ(µ) =
2π√
3(1 + µ)2
.
Now we prove the lemma for (ii). Suppose for contradiction that for some configuration
{Bu, Bv, Bw} satisfying (ii) the density is at least σ(µ); here we label the disks as in the
previous part of the proof. Let B′w = x
′
w + ρw int(B
2) denote the reflection of Bw to
the line through [xu, xv]. By the inequality ηw > (1 + µ)ρ/2 proved in the two previous
cases, we have that {Bu, Bv, Bw, B′w} is a µ-arrangement, where we observe that by the
strict inequality, Bw and B′w do not touch each others cores. Furthermore, each triangle
[xu, xw, x
′
w] and [xv, xw, x
′
w] is covered by the three disks from this family centered at
the vertices of the triangle, and the intersection of no two disks from one of these triples
is contained in the third one. Thus, the conditions of Lemma 2.6 are satisfied for both
{Bu, Bw, B′w} and {Bv, Bw, B′w}. Observe that as by our assumption the density in T is
σ(µ), it follows that the density in at least one of the triangles [xu, xw, x′w] and [xv, xw, x
′
w],
say in T ′ = [xu, xw, x′w], is at least σ(µ). In other words, under our condition there
is an axially symmetric arrangement with density at least σ(µ). Now we apply the µ-
increasing process as in the first part of the proof and obtain a µ′-arrangement {B̂u =
xu + (1 + µ)/(1 + µ
′)ρu int(B
2), B̂w = xw + (1 + µ)/(1 + µ
′)ρw int(B
2), B̂′w = x
′
w +
(1 + µ)/(1 + µ′)ρw int(B
2)} with density σ(µ′) and µ′ ≥ µ that satisfies either (i) or (ii).
If it satisfies (i), we have that the density of this configuration is at most σ(µ′) with equality
if only if T ′ is a regular triangle of side length (1 + µ′)ρ, where ρ is the common radius of
the three disks. On the other hand, this implies that in case of equality, the disks centered at
xw and x′w touch each others’ cores which, by the properties of the µ-increasing process,
contradicts the fact that Bw and B′w do not touch each others’ µ-cores. Thus, we have that
the configuration satisfies (ii).
From Lemma 2.1 it follows that B̂w ∩ B̂′w ⊂ B̂u. Thus, applying the previous con-
sideration with B̂u playing the role of Bw, we obtain that the distance of xu from the
line through [xw, x′w] is greater than (1 + µ
′)/2ρu. Thus, defining B̂′u = x
′
u + (1 + µ)/
(1 + µ′)ρu int(B
2) as the reflection of B′u about the line through [xw, x
′
w], we have that
{B̂u, B̂w, B̂′w, B̂′u} is a µ′-arrangement such that {B̂u, B̂′u, B̂w} and {B̂u, B̂′u, B̂′w} satisfy
the conditions of Lemma 2.6. Without loss of generality, we may assume that the density
of {B̂u, B̂′u, B̂w} is at least σ(µ′). Again applying the µ-increasing procedure described
in the beginning of the proof, we obtain a µ′′-arrangement of three disks, with µ′′ ≥ µ′,
concentric with the original ones that satisfy the conditions of the lemma and also (i) or
(ii). Like in the previous paragraph, (i) leads to a contradiction, and we have that it satisfies
(ii). Now, again repeating the argument we obtain a µ′′′ -arrangement{
y +
1 + µ
1 + µ′′′
ρu int(B
2), xw +
1 + µ
1 + µ′′′′
ρw int(B
2), x′w +
1 + µ
1 + µ′′′
ρw int(B
2)
}
,
with density at least σ(µ′′′) and µ′′′ ≥ µ′′, that satisfies the conditions of the lemma,
where either y = xu or y = x′u. On the other hand, since in the µ-increasing process we
choose the maximal value of the parameter satisfying the required conditions, this yields
that µ′ = µ′′ = µ′′′. But in this case the property that {B̂u, B̂′u, B̂w} satisfies (ii) yields
that {B̂u, B̂′u, B̂w} does not; a contradiction.
158 Ars Math. Contemp. 23 (2023) #P1.09 / 145–162
3 Proof of Theorem 1.2
The idea of the proof follows that in [7] with suitable modifications. In the proof we de-
compose U(F) =
⋃n
i=1 Bi, by associating a polygon to each vertex of certain free digons
formed by two disks. Before doing it, we first prove some properties of µ-arrangements.
Let q be a vertex of a free digon, say, D = B1 ∩ B2. We show that the convex
angular region R bounded by the closed half lines starting at q and passing through x1
and x2, respectively, does not contain the center of any element of F different from B1
and B2 containing q on its boundary. Indeed, suppose for contradiction that there is a disk
B3 = x3+ρ3 int(B
2) ∈ F with q ∈ bd(B3) and x3 ∈ R. Since [q, x1, x2]\{q} ⊂ B1∪B2,
from this and the fact that F is a Minkowski-arrangement, it follows that the line through
[x1, x2] strictly separates x3 from q. As this line is the bisector of the segment [q, q′], where
q′ is the vertex of D different from q, from this it also follows that |x3 − q| > |x3 − q′|.
Thus, q′ ∈ B3.
Observe that in a Minkowski arrangement any disk intersects the boundary of another
one in an arc shorter than a semicircle. This implies, in particular, that B3 ∩ bd(B1) and
B3 ∩ bd(B2) are arcs shorter than a semicircle. On the other hand, from this the fact that
q, q′ ∈ B3 yields that bd(D) ⊂ B3, implying, by the properties of convexity, that D ⊂ B3,
which contradicts our assumption that D is a free digon.
Note that, in particular, we have shown that if a member of F contains both vertices of
a digon, then it contains the digon.
B1 B2
B(t)
Figure 6: The 1-parameter family of disks inscribed in B1 ∩B2.
Observe that the disks inscribed in D can be written as a 1-parameter family of disks
B(t) continuous with respect to Hausdorff distance, where t ∈ (0, 1) and B(t) tends to
{q} as t → 0+ (cf. Figure 6); here the term ‘inscribed’ means that the disk is contained
in Bi ∩ Bj and touches both disks from inside. We show that if some member Bk of F ,
different from B1 and B2, contains B(t) for some value of t, then Bk contains exactly
one vertex of D. Indeed, assume that some Bk contains some B(t) but it does not contain
any vertex of D. Then for i ∈ {1, 2}, Bk ∩ bd(Bi) is a circular arc Γi in bd(D). Let
Li be the half line starting at the midpoint of Γi, and pointing in the direction of the outer
normal vector of Bi at this point. Note that as D is a plane convex body, L1 ∩ L2 = ∅. On
the other hand, since B1, B2, Bk are a Minkowski arrangement, from this it follows that
xk ∈ L1∩L2; a contradiction. The property that no Bk contains both vertices of D follows
from the fact that D is a free digon. Thus, if q ∈ Bk for an element Bk ∈ F , then there is
M. Kadlicskó and Z. Lángi: On generalized Minkowski arrangements 159
some value t0 ∈ (0, 1) such that B(t) ⊆ Bk if and only if t ∈ (0, t0].
In the proof, we call the disks Bi, Bj adjacent, if Bi ∩ Bj is a digon, and there is a
member of the family B(t) defined in the previous paragraph that is not contained in any
element of F different from Bi and Bj . Here, we remark that any two adjacent disks define
a free digon, and if a vertex of a free digon is a boundary point of U(F), then the digon is
defined by a pair of adjacent disks.
Consider a pair of adjacent disks, say B1 and B2, and let q be a vertex of D = B1∩B2.
If q is a boundary point of the union U(F), then we call the triangle [x1, x2, q] a shell
triangle, and observe that by the consideration in the previous paragraph, the union of shell
triangles coincides with the inner shell of F .
If q is not a boundary point of U(F), then there is a maximal value t0 ∈ (0, 1) such
that B(t0) = x+ ρB2 is contained in an element Bi of F satisfying q ∈ Bi. Then, clearly,
B(t0) touches any such Bi from inside, and since B1 and B2 are adjacent, there is no
element of F containing B(t0) and the vertex of D different from q. Without loss of gen-
erality, assume that the elements of F touched by B(t0) from inside are B1, B2, . . . , Bk.
Since B1 and B2 are adjacent and there is no element of F containing both B(t0) and the
vertex of D different from q, we have that the tangent points of B1 and B2 on bd(B(t0)) are
consecutive points among the tangent points of all the disks Bi, where 1 ≤ i ≤ k. Thus,
we may assume that the tangent points of B1, B2, . . . , Bk on B(t0) are in this counter-
clockwise order on bd(B(t0)). Let x denote the center of B(t0). Since F is a Minkowski
arrangement, for any 1 ≤ i < j ≤ k, the triangle [x, xi, xj ] contains the center of no ele-
ment of F apart from Bi and Bj , which yields that the points x1, x2, . . . , xk are in convex
position, and their convex hull Pq contains x in its interior but it does not contain the center
of any element of F different from x1, x2, . . . , xk (cf. also [7]). We call Pq a core polygon.
We remark that since F is a µ-arrangement, the longest side of the triangle [x, xi, xi+1],
for i = 1, 2 . . . , k, is [xi, xi+1]. This implies that ∠xixxi+1 > π/3, and also that k < 6.
Furthermore, it is easy to see that for any i = 1, 2, . . . , k, the disks Bi and Bi+1 are
adjacent. Thus, any edge of a core polygon is an edge of another core polygon or a shell
triangle. This property, combined with the observation that no core polygon or shell triangle
contains any center of an element of F other than their vertices, implies that core polygons
cover the core of F without interstices and overlap (see also [7]).
Let us decompose all core polygons of F into triangles, which we call core triangles, by
drawing all diagonals in the polygon starting at a fixed vertex, and note that the conditions
in Lemma 2.6 are satisfied for all core triangles. Now, the inequality part of Theorem 1.2
follows from Lemmas 2.4 and 2.6, with equality if and only if each core triangle is a regular
triangle [xi, xj , xk] of side length (1 + µ)ρ, where ρ = ρi = ρj = ρk, and each shell
triangle [xi, xj , q], where q is a vertex of the digon Bi ∩ Bj is an isosceles triangle whose
base is of length (1 + µ)ρ, and ρ = ρi = ρj . Furthermore, since to decompose a core
polygon into core triangles we can draw diagonals starting at any vertex of the polygon, we
have that in case of equality in the inequality in Theorem 1.2, all sides and all diagonals of
any core polygon are of equal length. From this we have that all core polygons are regular
triangles, implying that all free digons in F are thick.
On the other hand, assume that all free digons in F are thick. Then, from Lemma 2.2
it follows that any connected component of F contains congruent disks. Since an adjacent
pair of disks defines a free digon, from this we have that, in a component consisting of disks
of radius ρ > 0, the distance between the centers of two disks defining a shell triangle, and
the edge-lengths of any core polygon, are equal to (1 + µ)ρ. Furthermore, since all disks
160 Ars Math. Contemp. 23 (2023) #P1.09 / 145–162
centered at the vertices of a core polygon are touched by the same disk from inside, we also
have that all core polygons in the component are regular k-gons of edge-length (1 + µ)ρ,
where 3 ≤ k ≤ 5. This and the fact that any edge of a core polygon connects the vertices
of an adjacent pair of disks yield that if the intersection of any two disks centered at two
different vertices of a core polygon is more than one point, then it is a free digon. Thus,
any diagonal of a core polygon in this component is of length (1 + µ)ρ, implying that any
core polygon is a regular triangle, from which the equality in Theorem 1.2 readily follows.
4 Remarks and open questions
Remark 4.1. If
√
3 − 1 < µ < 1, then by Lemma 2.5, C(F) = ∅ for any µ-arrangement
F of order µ.
Remark 4.2. Observe that the proof of Theorem 1.2 can be extended to some value µ >√
3 − 1 if and only if Lemma 2.4 can be extended to this value µ. Nevertheless, from
the continuity of the functions in the proof of Lemma 2.4, it follows that there is some
µ0 >
√
3−1 such that the lemma holds for any µ ∈ (
√
3−1, µ0]. Nevertheless, we cannot
extend the proof for all µ < 1 due to numeric problems.
Remark 4.2 readily implies Remark 4.3.
Remark 4.3. There is some µ0 >
√
3 − 1 such that if µ ∈ (
√
3 − 1, µ0], and F is a
µ-arrangment of finitely many disks, then the total area of the disks is
T ≤
4 · arccos( 1+µ2 )
(1 + µ) ·
√
(3 + µ)(1− µ)
area(I(F)) + area(O(F)),
with equality if and only if every free digon in F is thick.
Conjecture 4.4. The statement in Remark 4.3 holds for any µ-arrangement of finitely many
disks with
√
3− 1 < µ < 1.
Let 0 < µ < 1 and let F = {Ki : i = 1, 2, . . .} be a generalized Minkowski arrange-
ment of order µ of homothets of an origin-symmetric convex body in Rd with positive
homogeneity. Then we define the (upper) density of F with respect to U(F) as
δU (F) = lim sup
R→∞
∑
Bi⊂RB2 area (Bi)
area
(⋃
Bi⊂RB2 Bi
) .
Clearly, we have δ(F) ≤ δU (F) for any arrangement F .
Our next statement is an immediate consequence of Theorem 1.2 and Remark 4.3.
Corollary 4.5. There is some value
√
3− 1 < µ0 < 1 such that for any µ-arrangement F
of Euclidean disks in R2, we have
δU (F) ≤

2π√
3(1+µ)2
, if 0 ≤ µ ≤
√
3− 1, and
4·arccos( 1+µ2 )
(1+µ)·
√
(3+µ)(1−µ)
, if
√
3− 1 < µ ≤ µ0.
For any 0 ≤ µ < 1, let u, v ∈ R2 be two unit vectors whose angle is π3 , and let
Fhex(µ) denote the family of disks of radius (1 + µ) whose set of centers is the lattice
M. Kadlicskó and Z. Lángi: On generalized Minkowski arrangements 161
{ku + mv : k,m ∈ Z}. Then Fhex(µ) is a µ-arrangement, and by Corollary 4.5, for
any µ ∈ [0,
√
3 − 1], it has maximal density on the family of µ-arrangements of positive
homogeneity. Nevertheless, as Fejes Tóth observed in [8] (see also [4] or Section 1), the
same does not hold if µ >
√
3 − 1. Indeed, an elementary computation shows that in
this case Fhex(µ) does not cover the plane, and thus, by adding disks to it that lie in the
uncovered part of the plane we can obtain a µ-arrangement with greater density.
Fejes Tóth suggested the following construction to obtain µ-arrangements with large
densities. Let τ > 0 be sufficiently small, and, with a little abuse of notation, let τFhex(µ)
denote the family of the homothetic copies of the disks in Fhex(µ) of homothety ratio τ
and the origin as the center of homothety. Let F1hex(µ) denote the µ-arrangement obtained
by adding those elements of τFhex(µ) to Fhex(µ) that do not overlap any element of it.
Iteratively, if for some positive integer k, Fkhex(µ) is defined, then let F
k+1
hex (µ) denote the
union of Fkhex(µ) and the subfamily of those elements of τk+1Fhex(µ) that do not overlap
any element of it. Then, as was observed also in [8], choosing suitable values for τ and
k, the value of δU (Fhex(µ)) can be approximated arbitrarily well by δ(Fkhex(µ)). We note
that the same idea immediately leads to the following observation.
Remark 4.6. The supremums of δ(F) and δU (F) coincide on the family of the µ-arrange-
ments F in R2 of positive homogeneity.
We finish the paper with the following conjecture.
Conjecture 4.7. For any µ ∈ (
√
3 − 1, 1) and any µ-arrangement F in R2, we have
δ(F) ≤ δU (Fhex(µ)).
ORCID iDs
Máté Kadlicskó https://orcid.org/0000-0003-2948-8823
Zsolt Lángi https://orcid.org/0000-0002-5999-5343
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.10 / 163–189
https://doi.org/10.26493/1855-3974.2730.6ac
(Also available at http://amc-journal.eu)
Braid representatives minimizing the number of
simple walks*
Hans U. Boden † , Matthew Shimoda ‡
Mathematics & Statistics, McMaster University, Hamilton, Ontario, Canada
Received 15 November 2021, accepted 19 July 2022, published online 21 November 2022
Abstract
Given a knot, we develop methods for finding a braid representative that minimizes
the number of simple walks. Such braids lead to an efficient method for computing the
colored Jones polynomial of the knot, following an approach developed by Armond and
implemented by Hajij and Levitt. We use this method to compute the colored Jones poly-
nomial in closed form for the knots 52, 61, and 72. The set of simple walks can change
under reflection, rotation, and cyclic permutation of the braid, and we prove an invariance
property which relates the simple walks of a braid to those of its reflection under cyclic
permutation. We study the growth rate of the number of simple walks for families of torus
knots. Finally, we present a table of braid words that minimize the number of simple walks
for knots up to 13 crossings.
Keywords: Knots, braids, simple walk, colored Jones polynomial.
Math. Subj. Class. (2020): 57K10, 57K14
1 Introduction
The Jones polynomial VLptq is an invariant of knots and links defined using quantum rep-
resentations of braids. It can be uniquely characterized as the polynomial-valued invariant
of oriented links with V⃝ptq “ 1 for ⃝ the unknot and satisfying the skein relation
t´1VL` ptq ´ tVL´ ptq “ pt1{2 ´ t´1{2qVL0ptq,
*The authors are especially grateful to Alexander Stoimenow for providing crucial input. They would also like
to thank Homayun Karimi, Robert Osburn, Andrew Nicas, Will Rushworth, and Cornelia Van Cott for valuable
feedback.
†Corresponding author. The author would like to acknowledge funding from the Natural Sciences and Engi-
neering Research Council of Canada.
‡The author acknowledges funding from a USRA award and a Stewart award from McMaster University.
E-mail addresses: boden@mcmaster.ca (Hans U. Boden), mattshimoda@hotmail.com (Matthew Shimoda)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
164 Ars Math. Contemp. 23 (2023) #P1.10 / 163–189
where L`, L´, L0 are identical outside a neighborhood, where they are as pictured
L` L´ L0.
The Jones polynomial admits a combinatorial state sum formula that can be used to com-
pute it, but the complexity of the computation grows exponentially with the crossing num-
ber. So this method is impractical for computations that involve links with a large number
of crossings.
The colored Jones polynomial is a powerful knot invariant packaged as a sequence of
Laurent polynomials JN,Kpqq for N ě 2, with N “ 2 giving the usual Jones polynomial.
It encodes subtle geometric information about the knot complement and appears in several
famous open problems in quantum topology, including (i) the Volume Conjecture [18, 23];
(ii) the Slope Conjecture [12, 17]; and (iii) the AJ Conjecture [11]. The first relates the limit
of JN,Kpqq for q “ e2πi{N as N Ñ 8 to the hyperbolic volume of the knot complement;
the second posits that every Jones slope of a knot is the slope of an incompressible surface
in the knot complement; and the third asserts that the recurrence relation for the N -th
colored Jones polynomials is given by the A-polynomial of [9], a plane curve associated to
the character variety of SLp2,Cq representations of the knot group.
For further background information on the colored Jones polynomial and its relation to
the geometry of 3-manifolds, we refer the reader to the books [24] and [10] and their ex-
tensive bibliographies. The colored Jones polynomial also has intriguing number theoretic
interpretations that will not be discussed in this paper; for more details about these aspects,
we refer the reader to the recent papers [3, 4, 20, 21] and their bibliographies.
Our goal in this paper is to study a probabilistic method for computing the colored
Jones polynomial. This approach was first developed by Huynh and Lê in [15], and it
was later described in terms of walks along braids by Armond [2]. Armond identified
the special role played by simple walks, resulting in an extremely efficient algorithm for
computing JN,Kpqq, which has been implemented by Hajij and Levitt [14]. The algorithm
is exponential in the number of simple walks on the braid, so it is natural to try minimize
the number of simple walks before executing the program of [14]. However, as we shall
see, this number is highly dependent on the braid representative chosen.
We study how the number of simple walks changes under taking reflection, rotation,
and cyclic permutation of a given braid. We also examine the growth rate of the number
of simple walks for two families of torus knots. For instance, for the family of p2, nq torus
knots, the simple walks satisfy a Fibonacci recurrence and grow exponentially in n. For
the family of p3, nq torus knots, the simple walks satisfy a tribonacci recurrence and also
grow exponentially in n. We further prove that the total number of simple walks on a braid
and its reflection is invariant under cyclic permutation. This fact is used to facilitate finding
braid representatives with the least number of simple walks. For knots up to 13 crossings,
we developed a program that finds minimal braid representatives. When these braids are
used in conjunction with the program of Hajij and Levitt [14], this provides an efficient
method for computing the colored Jones polynomial for these knots.
We close this section with a brief synopsis of the rest of this paper. In Section 2, we re-
view the method from [2, 15] for computing the colored Jones polynomial. In Section 3, we
use it to compute JN,Kpqq in closed form for the knots 52, 61, and 72. These computations
were originally performed by Masbaum using skein theory, see [22]. In Section 4, we recall
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 165
the basic results about braid representatives for knots and study the effect of the Markov
moves on the set of simple walks. In Section 5, we introduce the set of semi-simple walks,
and we show that it is invariant under cyclic permutation of the braid word. In Sections 6
and 7, we study the growth rate of the number of simple walks for two families of torus
knots. In Section 8, we present the output of a program for finding braid representatives
that minimize the number of simple walks.
2 The colored Jones polynomial and walks along braids
Given a knot K and integer N ě 2, the colored Jones polynomial JN,Kpqq is a Laurent
polynomial in the variable q1{2. It is normalized so that JN,⃝pqq “ 1, where ⃝ is the un-
knot. When N “ 2, the colored Jones polynomial agrees with the usual Jones polynomial.
In general, the N -th colored Jones polynomial of a knot K can be expressed in terms of
the usual Jones polynomial of the pN ´ 1q strand cable of K. However, since the crossing
number of the pN ´ 1q strand cable of a knot is pN ´ 1q2 times the crossing number of the
knot, this does not lead to a practical method for computing the colored Jones polynomial.
One approach for computing the colored Jones polynomial is presented by Huynh and
Lê [15]. Starting with a braid β whose closure is the given knot, Huynh and Lê use meth-
ods from quantum algebra to express the colored Jones polynomial as the inverse of the
quantum determinant of an almost quantum matrix. The matrix is constructed through the
product of Burau matrices, which we obtain from the crossing and orientation properties of
β.
A second approach is presented by Armond [2]. It is based on a probabilistic inter-
pretation of the colored Jones polynomial and involves counting walks along braids. This
method is closely related to the previous one, and in fact it provides a visual representation
of the quantum algebra approach. The idea is to view walks along the braid as traversing
the strands of the braid from the bottom to the top and to record information about the
crossings and their orientations as a product of operators. The end result is the same as
that obtained by taking the quantum determinant of the deformation of Burau matrices, but
Armond’s approach is more accessible and requires less background material on operator
theory. One interesting aspect is that the complexity of the computation is sensitive to the
choice of braid word, and this will be explored further in Section 4. For now, we focus on
describing Armond’s approach and the special role played by the simple walks.
We begin by introducing a little terminology from braid theory.
Definition 2.1. A braid is a set of m strands running from top to bottom with no reversals
in vertical direction. The strands may cross each other, but only two strands can participate
at each crossing.
Given a braid β, a braid word is an expression of the form
β “ σε1i1 σ
ε2
i2
. . . σεℓiℓ ,
where εi “ ˘1 and σi is a symbol. Braid words are read from left to right, and braids are
drawn from top to bottom. For 1 ď i ď m ´ 1, σi represents the braid with one crossing
where the pi`1q-st strand crosses over the i-th strand. The inverse σ´1i represents the braid
where the i-th strand crosses over the pi ` 1q-st strand.
166 Ars Math. Contemp. 23 (2023) #P1.10 / 163–189
The braid word β “ σε1i1 σ
ε2
i2
. . . σεℓiℓ has ℓ crossings, and we say it has braid length ℓ.
If β is a braid on m strands, we say it has braid width m. Note that braid words are not
uniquely determined by the braid. Applying a braid relation (see below) will alter the word
without changing the braid. The writhe of a braid is defined to be the sum of the signs on
all its crossings. For example, the braid word above has writhe wpβq “
řℓ
i“1 εi.
The braid group on m strands is denoted Bm. Abstractly, it is the group with generators
σ1, . . . , σm´1 and relations
(i) σiσj “ σjσi for 1 ď i, j ď m ´ 1 with |i ´ j| ą 1 and
(ii) σiσi`1σi “ σi`1σiσi`1 for 1 ď i ď m ´ 2.
Relation (i) is called far commutativity and (ii) is called the Yang-Baxter relation (or braid
relation). The group operation is given by concatenation of words or, equivalently, by
stacking geometric braids, one on top of the other.
Next, we introduce the notions of paths and walks along braids. A path starts at the bot-
tom of the braid and traverses arcs of the braid, sometimes jumping down, until it reaches
the top of the braid. If the path starts at strand i on the bottom and ends at strand j at the
top, we say it is a path from i to j. Whenever the path encounters a crossing, if it is on the
overstrand, it is allowed to jump down to the undercrossing arc. If it is on the understrand,
then it must stay on that strand. At each crossing the path encounters, a weight from the
set tai,εi , bi,εi , ci,εi | i “ 1, . . . , ℓu is assigned. The weight will depend on the crossing,
its sign, and the arcs traversed by the path at that crossing. For example, if the path jumps
down at the i-th crossing, it is assigned the weight ai,εi . Otherwise, it is assigned the weight
bi,εi if the path traverses the understrand and ci,εi if it traverses the overstrand. Note that,
at a given crossing, the path will follow the braid unless it jumps down there. The total
weight of the path is the product of the weights of the crossings.
A walk W along β consists of a set J Ď t1, . . . ,mu, a permutation π of J , and a
collection of paths with exactly one path in the collection from j to πpjq for each j P J .
The weight of a walk is p´1qp´qq|J|`invpπq times the product of the weights of the paths,
where |J | is the cardinality of J and invpπq is the number of inversions in π, i.e., the
number of pairs of elements in J with i ă j and πpiq ą πpjq.
The paths in a walk are ordered from left to right, using their starting strand at the
bottom of the braid. This induces an ordering on the weights. The order of the weights is
important, and that is because the operators associated to the weights are non-commuting.
Operators based at different crossings do commute, but operators at the same crossing do
not. Thus, the effect of non-commutativity can be computed locally at each crossing of
the braid. Given a walk W and a crossing i of the braid, we use Wpiq to denote the local
weight of paths of W through crossing i in the given order. If no path of W passes through
crossing i, then we set the local weight Wpiq “ 1.
A stack of walks is any ordered collection W1 ¨ ¨ ¨Wk of walks. Visually, this can be
viewed as stacking the walks on top of one another with W1 at the top and Wk on the
bottom. The weight of the stack is the product of the weights of the paths in the appropriate
order. If two paths belong to different walks, then the path in the higher walk is multiplied
to the left of the path in the lower walk. If two paths belong to the same walk, then the path
which begins to the left of the other path is said to be above and is multiplied to the left
of the other path. Given a stack S “ W1 ¨ ¨ ¨Wk and a crossing i of the braid, we let Spiq
be the local weight of the stack at i; it is equal to the product pW1qpiq ¨ ¨ ¨ pWkqpiq of local
weights of the walks of the stack at i in the given order.
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 167
With walks along braids established, we now show how they can be used to com-
pute the colored Jones polynomial. Let R “ Zrq, q´1s. Let x̂ and τx be operators on
the ring Rrx˘1, y˘1, u˘1s given by x̂fpx, y, . . . q “ xfpx, y, . . . q and τxfpx, y, . . . q “
fpqx, y, . . . q. The operators ŷ, û, τy , τu are defined similarly. We associate operators to
each of the crossing weights using the formulas:
a` “ pû ´ ŷτx´1qτy´1, b` “ û2, c` “ x̂τy´2τu´1,
a´ “ pτy ´ x̂´1qτx´1τu, b´ “ û2, c´ “ ŷ´1τy´1τu.
By taking the summation of all walks on the braid and writing their weights in terms of
the above operators, we obtain the operator P . Letting P act on the constant 1 and making
the substitutions x “ z, y “ z and u “ 1, we obtain a polynomial EpP q. Let EN pP q
denote the polynomial obtained by making the substitution z “ qN´1 to EpP q.
The next result shows how to compute the colored Jones polynomial of a knot from its
braid representative. It was proved by Huynh and Lê in [15] and appears as Theorem 2.3
in [2].
Theorem 2.2. Let K be a knot obtained as the closure of a braid β P Bm. Its colored
Jones polynomial is given by
JN,Kpqq “ qpN´1qpwpβq´m`1q{2
8
ÿ
n“0
EN pPnq, (1)
where the operator P is the sum of the weights of the walks on β with J Ď t2, . . . ,mu.
Stacks of walks are produced when we take the product of the weights of walks. This
occurs in the step when we expand the operator P to the power of n.
Huynh and Lê also gave a useful method for evaluating the terms EN pPnq which avoids
operator theory. The key result is the following lemma from [15] which computes EN pPnq
directly from the weights once they have been put in a preferred order. In the following,
we suppress the dependence of the weights on the crossing and write a˘, b˘, c˘ instead of
ai,˘, bi,˘, ci,˘.
Lemma 2.3.
EN pbs`cr`ad`q “ qrpN´1´dq
d´1
ź
i“0
p1 ´ qN´1´r´iq
EN pbs´cr´ad´q “ q´rpN´1q
d´1
ź
i“0
p1 ´ qr`i`1´N q
We will apply this lemma to the local weights Spiq of each stack at each crossing.
However, before we can apply Lemma 2.3, we must first put the local weights at a crossing
into the preferred order. This can always be achieved using the following relations:
a`b` “ b`a`, a`c` “ qc`a`, b`c` “ q2c`b`
a´b´ “ q2b´a´, a´c´ “ q´1c´a´, b´c´ “ q´2c´b´
168 Ars Math. Contemp. 23 (2023) #P1.10 / 163–189
Once the local weights have been put into the preferred order at each crossing, The-
orem 2.2 and Lemma 2.3 can be applied locally at each crossing to compute the colored
Jones polynomial.
The computation is simplified by the observation that only simple walks contribute to
the colored Jones polynomial [2, Lemma 2.5(a)]. Here, a walk is said to be simple if no two
paths intersect in the walk, otherwise it is non-simple. It turns out that non-simple walks
occur in cancelling pairs, so for the purpose of computing JN,Kpqq, it is enough to consider
only simple walks.
The computation is further simplified by the fact that, for any stack of walks, the evalu-
ation of its weights will vanish if the walks in the stack traverse the same arc on N or more
different levels [2, Lemma 2.5(b)]. This is extremely useful because it reduces the com-
plexity of the computation and guarantees that only finitely many terms of
ř8
n“0 EN pPnq
contribute to the N -th colored Jones polynomial. In particular, for a fixed N , there will
always be an upper bound to the integers n which need to be considered in the infinite sum
of Equation (1). In practice, this bound can be determined by comparing the arcs traversed
by the set of all simple walks for a given braid word.
3 The colored Jones polynomial in closed form
In this section, we apply Theorem 2.2 and Lemma 2.3 to compute JN,Kpqq, the colored
Jones polynomial, for the knots 52, 61 and 72. This is achieved by choosing favorable braid
representatives, namely those with only a few simple walks.
In [22], Masbaum uses skein theory to compute the colored Jones polynomial for all
twist knots, a class which includes 52, 61, and 72. More general calculations of the colored
Jones polynomial for the double twist knots can be found in [20, 21].
  
Figure 1: The simple walks A and B shown as arcs with zebra stripes for the braid
σ´12 σ1σ
3
2σ1 with closure the knot 52.
Example 3.1. The braid word σ´12 σ1σ32σ1 represents the knot 52 and has two simple walks
with J Ď t2, 3u. They are A “ qa1,´c3,`a4,`b5,` and B “ q3b1,´c1,´c2,`c3,`a4,`
b5,`b6,` (see Figure 1). Notice that A has J “ t3u and B has J “ t2, 3u. Since the walks
A and B both traverse the third strand at top and bottom, stacks only need to be considered
up to level N ´ 1.
Using Theorem 2.2, we can write the colored Jones polynomial as the following:
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 169
JN,Kpqq “ qp1´Nq
N´1
ÿ
n“0
EN ppA ` Bqnq,
“ qp1´Nq
N´1
ÿ
n“0
EN ppqa1,´c3,`a4,`b5,` ` q3b1,´c1,´c2,`c3,`a4,`b5,`b6,`qnq.
We will expand the above expression using the q-binomial theorem. For that purpose,
we introduce the Gaussian binomial coefficients (or q-binomial coefficients), which are
defined by
ˆ
n
k
˙
q
“
k´1
ź
i“0
ˆ
1 ´ qn´i
1 ´ qi`1
˙
.
The expansion of the above expression includes a sum of products of A’s and B’s, which
can be interpreted as stacks.
To apply Lemma 2.3, the weights at each crossing must be placed into the order bscrad,
and it is preferable to expand pA`Bqn as a sum of terms of the form BkAn´k. Since the lo-
cal weights at different crossings commute, the only potential issue with non-commutativity
of A and B is at the first crossing. Since a1,´b1,´c1,´ “ qb1,´c1,´a1,´, we have AB “
qBA, so we can adjust for inversions using the q-binomial coefficient:
pA ` Bqn “
n
ÿ
k“0
ˆ
n
k
˙
q
BkAn´k.
Next we use Lemma 2.3 to apply EN p¨q to evaluate each stack. First, these walks
have weights bi,` indexed alone at the fifth and sixth crossings, which evaluates to 1.
Additionally, the ci,˘ weights at the first and second crossings in walk B always cancel
out since EN pc1,´q “ q´pN´1q and EN pc2,`q “ qN´1. We still have c3,` in each walk A
and walk B. Therefore, for each walk in a stack, the term A or B, qN´1 is contributed.
Similarly, the weight a4,` appears in both walks, so a stack consisting of n walks will
contribute
śn
i“1p1 ´ qN´iq. Meanwhile, the weight a1,´ is only in walk A, so the stack
BkAn´k will contribute
śn´k
i“1 p1 ´ qn`i´N q. Additionally, we need to adjust for the
correct order of b1,´ and c1,´ in the weights in products containing Bk. The number of
times the relation is applied increases quadratically with the exponent of B. That is, for a
weight containing Bk, its contribution is qk
2´k. Finally, the remaining powers of q arise
from the existing variables in the simple walks.
Applying Theorem 2.2, the colored Jones polynomial of 52 can be written in closed
form as
JN,Kpqq “ qN´1
N´1
ÿ
n“0
n
ÿ
k“0
ˆ
n
k
˙
q
qnN`kpk`1q
n
ź
i“1
`
1 ´ qN´i
˘
n´k
ź
i“1
`
1 ´ qn`i´N
˘
.
Example 3.2. The braid word σ1σ2σ1´1σ3´1σ2σ3´1σ1 represents the knot 61 and has
three simple walks with J Ď t2, 3, 4u. They are A “ qa4,´a6,´, B “ q3c2,`a3,´a4,´b5,`
b6,´c6,´ and C “ q5c1,`c2,`b3,´c3,´a4,´b5,`b6,´c6,´b7,` (see Figure 2). Notice that A
has J “ t4u, B has J “ t3, 4u, and C has J “ t2, 3, 4u. Since the walks A,B and C all
traverse the fourth strand at top and bottom, stacks only need to be considered up to level
N ´ 1.
170 Ars Math. Contemp. 23 (2023) #P1.10 / 163–189
   
Figure 2: The simple walks A,B and C shown as arcs with zebra stripes for the braid
σ1σ2σ1
´1σ3
´1σ2σ3
´1σ1 with closure the knot 61.
Using Theorem 2.2, the colored Jones polynomial for 61 can be written as
JN,Kpqq “ qp1´Nq
N´1
ÿ
n“0
EN ppA ` B ` Cqnq ,
“ qp1´Nq
N´1
ÿ
n“0
EN
`
pqa4,´a6,´ ` q3c2,`a3,´a4,´b5,`b6,´c6,´
` q5c1,`c2,`b3,´c3,´a4,´b5,`b6,´c6,´b7,`qn
˘
.
We use the q-multinomial theorem to expand the terms pA ` B ` Cqn, and to that end
we recall the definition of the q-multinomial coefficients.
   
Figure 3: The simple walks A,B and C as arcs with zebra stripes on the braid
σ1σ2σ
3
3σ
´1
1 σ2σ1σ
´1
3 with closure the knot 72.
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 171
Given an integer r ě 1 and sequence of nonnegative integers m1,m2, . . . ,mr such
that n “ m1 ` ¨ ¨ ¨ ` mr, let
ˆ
n
m1,m2, . . . ,mr
˙
q
“ rnsq!rm1sq!rm2sq! . . . rmrsq!
, (2)
where rnsq “ qn´1 ` ¨ ¨ ¨ ` q ` 1 and rnsq! “ r1sq ¨ ¨ ¨ rnsq . The term
`
n
m1,m2,...,mr
˘
q
is
called the Gaussian multinomial coefficient or q-multinomial coefficient.
With the weights indexed at the third and sixth crossings, the natural order of the walks
is C,B,A, as this will allow use of Lemma 2.3 with only a few other adjustments. The use
of the q-multinomial coefficient is suitable for this order because inversions with respect to
the alphabet C,B,A are reversed with multiplication by q. That is, AB “ qBA, AC “
qCA and BC “ qCB. The expansion of the trinomial takes the form
pA ` B ` Cqn “
n
ÿ
m“0
m
ÿ
k“0
ˆ
n
n ´ m,m ´ k, k
˙
q
Cn´mBm´kAk, (3)
where
`
n
n´m,m´k,k
˘
q
denotes the q-trinomial coefficient of Equation (2).
We can now expand the trinomial for any power n and use Lemma 2.3 to apply EN p¨q
and evaluate each stack. The b˘ and c˘ weights are evaluated the same as in Example 3.1.
The weight a4,´ appears in each of A,B,C, so for every term Cn´mBm´kAk, it will
contribute
śn
i“1p1 ´ qi´N q. Meanwhile, the weight a3,´ only appears in the walk B,
so for the term Cn´mBm´kAk, it contributes
śm´k
i“1 p1 ´ qn´m`i´N q. Similarly, the
weight a6,´ only appears in the walk A, so for the term Cn´mBm´kAk, it contributes
śm´k
i“1 p1 ´ qn´k`i´N q.
Finally, we need to adjust for the correct order of the terms b6,´ and c6,´ in the products
containing Cn´mBm´k. The number of times the relation is applied increases quadrati-
cally with the sum of the exponents of B and C, which is pn´mq`pm´kq “ n´k. That
is, for the term Cn´mBm´kAk, applying the relation introduces a factor of qpn´kq
2´pn´kq.
We follow the same logic for products Cn´m to adjust for the order of the weights b3,´
and c3,´ at the third crossing.
Applying Theorem 2.2, the colored Jones polynomial of 61 can be written in closed
form as
JK,N pqq “ q1´N
N´1
ÿ
n“0
n
ÿ
m“0
m
ÿ
k“0
ˆ
n
n ´ m,m ´ k, k
˙
q
q3n´k´m`pn´kq
2`pn´mq2
ˆ
n
ź
i“1
p1 ´ qi´N q
m´k
ź
i“1
p1 ´ qn´m`i´N q
k
ź
i“1
p1 ´ qn´k`i´N q.
Example 3.3. The braid word σ1σ2σ33σ
´1
1 σ2σ1σ
´1
3 represents the knot 72 and has three
simple walks with J Ď t2, 3, 4u. They are A “ qc3,`a4,`b5,`a9,´, which has has
J “ t4u; B “ q3c2,`c3,´a4,`b5,`a6,´b7,`b9,´c9,´, which has has J “ t3, 4u; and
C “ q5c2,`c3,`a4,`b5,`b6,´c6,´b7,`b8,´b9,´c9,´, which has J “ t2, 3, 4u (see Fig-
ure 3). Since the walks A,B and C all traverse the fourth strand at top and bottom, stacks
only need to be considered up to N ´ 1.
172 Ars Math. Contemp. 23 (2023) #P1.10 / 163–189
We use Theorem 2.2 to write the colored Jones polynomial for 72 as:
JN,Kpqq “ qp1´Nq
N´1
ÿ
n“0
EN ppA ` B ` Cqnq ,
“ qp1´Nq
N´1
ÿ
n“0
EN
`
pqc3,`a4,`b5,`a9,´ ` q3c2,`c3,´a4,`b5,`a6,´b7,`b9,´c9,´
` q5c2,`c3,`a4,`b5,`b6,´c6,´b7,`b8,´b9,´c9,´qn
˘
.
With the weights indexed at the sixth and ninth crossings, the most natural order of the
walks is C,B,A. Note that AB “ qBA, AC “ qCA and BC “ qCB. Therefore, the
expansion of the trinomial is as given in equation (3) above.
We expand the trinomial for all powers of n and use Lemma 2.3 to apply EN p¨q and
evaluate each stack. The b˘ and c˘ weights are evaluated the same as in Examples 3.1
and 3.2. The weight a4,` appears in each of A,B,C, so for every term Cn´mBm´kAk,
it will contribute
śn
i“1p1 ´ qN´iq. Meanwhile, the weight a6,´ only appears in the walk
B, so for the term Cn´mBm´kAk, it contributes
śm´k
i“1 p1 ´ qn´m`i´N q. Similarly, the
weight a9,´ only appears in the walk A, so for the term Cn´mBm´kAk, it contributes
śk
i“1p1 ´ qn´k`i´N q.
Finally, we need to adjust for the correct order of the terms b9,´ and c9,´ in the products
containing Cn´mBm´k. The number of times the relation is applied increases quadrati-
cally with pn ´ mq ` pm ´ kq “ n ´ k. That is, for the term Cn´mBm´kAk, applying
the relation introduces a factor of qpn´kq
2´pn´kq. We follow the same logic for products
Cn´m to adjust for the weights at the sixth crossing.
Applying Theorem 2.2, the colored Jones polynomial of 72 can be written in closed
form as
JK,N pqq “ qN´1
N´1
ÿ
n“0
n
ÿ
m“0
m
ÿ
k“0
ˆ
n
n ´ m,m ´ k, k
˙
q
qnN`2n´m´k`pn´kq
2`pn´mq2
ˆ
n
ź
i“1
p1 ´ qN´iq
m´k
ź
i“1
p1 ´ qn´m`i´N q
k
ź
i“1
p1 ´ qn´k`i´N q.
For hyperbolic knots, the Volume Conjecture asserts that
lim
NÑ8
log |JN,Kpe2πi{N q|
N
“ VolpS
3 ∖Kq
2π
.
For more information about this important open problem, see the book [24]. The knots in
Examples 3.1, 3.2, and 3.3 are all hyperbolic, and the Volume Conjecture has been verified
for each of them. For 52, this was proved by Ohtsuki [25]; for 61, it was proved by Ohtsuki
and Yokota [27]; and for 72, it follows from Ohtsuki’s work on 7-crossing hyperbolic knots
[26]. It would be interesting to use the formulas given here to independently verify the
Volume Conjecture for these knots.
4 Representing knots as braids
One of the objectives in this paper is to find braid representations of knots that minimize
the number of simple walks. To do that, we will apply several different operations that
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 173
alter the braid word without changing its representative knot. These operations include
reflection, rotation, and cyclic permutation of the braid word, and each of them can be used
to reduce the number of simple walks. We begin with a review of some standard material
on representing knots as braids (see also [5]).
· · ·
β
· · ·
· · ·
α
β
α−1
· · ·
Conjugation
· · ·
β
· · ·
· · ·
β
· · ·
· · ·
β
· · ·
Stabilization
Figure 4: The Markov moves.
Given a braid β, its closure is denoted pβ and is the knot or link obtained by connect-
ing the strands on top with the corresponding strands on bottom without introducing any
additional crossings.
The next result is called Alexander’s theorem and was first proved in 1923, see [1].
Theorem 4.1. Every oriented knot or link is equivalent to the closure pβ for some braid
β P Bm.
Definition 4.2. The Markov moves include conjugation and stabilization (see Figure 4).
Given a braid β P Bm, conjugation involves replacing it with αβα´1 for some α P Bm.
Stabilization involves replacing β with either βσm or βσ´1m . Note that conjugation pre-
serves the braid width and stabilization increases it by one.
The next result is attributed to Markov. For a proof, see [5, Theorem 2.3].
Theorem 4.3. Two braids have equivalent link closures if and only if they are related by a
sequence of Markov moves.
Let β P Bm, and let
SWβ “ tW | W is a simple walk on β with J Ď t2, . . . ,muu (4)
be the set of simple walks on the braid β.
Given a knot, we are interested in finding the braid representative that minimizes the
number of simple walks. Of course, the set of simple walks depends on the braid represen-
tative chosen. In fact, it depends on the braid word since it is not preserved under insertion
(or deletion) of σiσ´1i or σ
´1
i σi into the braid word. This is the analogue, for braids, of the
Reidemeister II move. We explain this important point next.
174 Ars Math. Contemp. 23 (2023) #P1.10 / 163–189
Given a walk on a braid, we say that an arc of the braid is active if it is traversed by a
path of the walk. Similarly, we say that a crossing is active if the walk jumps down from
overcrossing arc to undercrossing arc at that crossing. Thus, the active crossings are the
ones with the local weight ai,˘. In Figures 1, 2, and 3, the active arcs are depicted with
zebra stripes.
Now consider two braid words: γ “ αβ and γ1 “ ασiσ´1i β. (A similar argument
applies to γ1 “ ασ´1i σiβ.) We will show that SWγ Ď SWγ1 .
Suppose W is a simple walk on γ. If both strands i, i ` 1 are active or if they are both
not active, then W extends in a unique way to a simple walk on γ1. If one of the strands
i, i ` 1 is active and the other is not, then W extends to a simple walk on γ1, but possibly
in more than one way. This proves the claim, and in particular, we see that the number of
simple walks is non-decreasing under an elementary insertion.
Recall that a braid word is said to be reduced if it does not contain an occurrence of
σiσ
´1
i or σ
´1
i σi. By the above considerations, for any given knot, we can always assume
that its braid representative is given by a reduced word.
In a similar way, one can show that the set of simple walks is invariant under far com-
mutativity and the Yang-Baxter relation. For far commutativity, this is straightforward, and
we leave the details to the reader. For the Yang-Baxter relation, consider the braid words
γ “ ασiσi`1σiβ and γ1 “ ασi`1σiσi`1β and assume the relevant crossings are j, j ` 1,
and j ` 2.
We claim that any simple walk on γ extends in a unique way to a simple walk on γ1.
There are several cases, depending on which of the three crossings j, j`1, j`2 are active.
If none of the crossings are active, then it extends to a simple walk on γ1. If one of the
crossings is active, and if we make the corresponding crossing on γ1 active, and then it
extends to a simple walk on γ1. If two of the crossings of γ are active, then they must be j
and j`2, and it extends to a simple walk on γ1 again with j and j`2 active crossings. Note
that it is not possible for all three crossings to be active. This shows that |SWγ | “ |SWγ1 |
under the Yang-Baxter relation.
One can also apply the Markov moves to a braid and consider their effect on the set of
simple walks. For instance, under conjugation, one would expect that the resulting braid
word will have a larger set of simple walks. A special case is cyclic permutation, which
involves replacing β “ σε1i1 σ
ε2
i2
¨ ¨ ¨σεℓiℓ with β
1 “ σε2i2 ¨ ¨ ¨σ
εℓ
iℓ
σε1i1 . We will study the effect
of cyclic permutation on the set of simple walks in the next section.
Under stabilization, we will show that the set of simple walks is non-decreasing. Let
β P Bm, β1 “ βσ˘1m P Bm`1, and suppose W is a simple walk on β. If m R J, then
W extends uniquely to a simple walk on β1. If m P J and β1 “ βσm, then we can
extend W to a simple walk on β1 by either making the extra crossing active or by setting
J 1 “ J Y tm ` 1u. If m P J and β1 “ βσ´1m , then we can extend W to a simple walk on
β1 with J 1 “ J Y tm`1u. Notice that for β1 “ βσ´1m , we have one additional simple walk
that does not come from β, namely the one with J “ tm` 1u and the extra crossing made
active. In particular, it follows that |SWβ | ď |SWβ1 |.
Let K be a knot and suppose β P Bm`1 is a braid representative for K with m ě 1.
If β is conjugate to a braid of the form γσ˘m for some braid γ P Bm, then β is said to be
reducible. The braid β is said to be irreducible if it is not reducible.
The next result summarizes our discussion.
Proposition 4.4. If K is a knot, then any braid representative for K that minimizes the
number of simple walks can be assumed to be given by a reduced and irreducible braid word.
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 175
In addition, one can apply symmetry operations to alter the braid word without chang-
ing its knot or link closure. We will use these operations to find braid representatives that
minimize the number of simple walks. The three operations we will consider are called
reflection, rotation, and reversal, and we introduce them next.
For a given braid β, its reflection is denoted β˚ and is obtained by switching all the
crossings of β. If β represents the knot K, then β˚ represents its mirror image K˚. If
β “ σε1i1 ¨ ¨ ¨σ
εℓ
iℓ
, then its reflection is the braid word given by β˚ “ σ´ε1i1 ¨ ¨ ¨σ
´εℓ
iℓ
(see
Figure 5).
For the purposes of computing the colored Jones polynomial, one can use either β or
β˚, since the invariants are related by the simple formula JN,Kpqq “ JN,K˚ pq´1q. The
two braids will have completely different sets of simple walks. In fact, as we shall see
in the next section, the simple walks on β and β˚ are disjoint and complementary to one
another. There is an obvious computational advantage to working with the braid having
fewer simple walks.
In fact, there are other symmetries that can be applied to get a new braid representative
for a knot (or its mirror image). For example, given a braid word β representing a knot, if
one rotates it 1800 in the plane, one obtains a new braid word representing the same knot.
Specifically, if β “ σε1i1 ¨ ¨ ¨σ
εℓ
iℓ
, then the rotated braid word is denoted β: and is given by
β: “ σεℓm´iℓ ¨ ¨ ¨σ
ε1
m´i1 (see Figure 5).
Another example is braid reversal, which is given by reversing the order of the braid
word. Again, the new braid represents the same knot. If β “ σε1i1 ¨ ¨ ¨σ
εℓ
iℓ
, then its reversal
is denoted βr and is given by βr “ σεℓiℓ ¨ ¨ ¨σ
ε1
i1
(see Figure 5). Notice that βr is the braid
obtained from β by rotating it 1800 around a horizontal line in the plane.
There is a one-to-one correspondence between the sets of simple walks on β and βr.
Under the correspondence, the walks have the same set of active crossings, and the weights
for the over- and undercrossings pbi,˘, ci,˘q are switched. In fact, as we shall see, a simple
walk is completely determined by its set of active crossings, and it follows that |SWβ | “
|SWβr |.
β β∗ β† βr
Figure 5: A braid β representing the knot 52 and its reflection β˚, rotation β:, and reversal
βr.
Given a braid word for a knot, applying reflection, rotation, or cyclic permutation will
alter its set of simple walks. Since the computation of the colored Jones polynomial is
exponential in the number of simple walks, it is advantageous to choose the braid represen-
tative that minimizes the number of simple walks.
176 Ars Math. Contemp. 23 (2023) #P1.10 / 163–189
5 Semi-simple walks and cyclic permutation
The main result in this section is an invariance property which asserts that under the cyclic
permutation, the total number of simple walks on a braid β and its reflection β˚ does not
change. To see this, we introduce the notion of semi-simple walks and study their behavior
under reflection.
Recall the definition of SWβ in Equation (4). Previously, we identified walks W with
their weights, given by the ordered product of operators tai,˘, bi,˘, ci,˘u for each crossing
traversed. However, it will be more convenient to record W using only the operators ai,˘,
and we can do so with no loss of information. In the following, we write ai instead of ai,˘;
it is notationally more compact and the sign ˘ can be recovered from the braid word. Thus,
there is a one-to-one correspondence between simple walks on β and (certain) monomials
in ta1, . . . , aℓu, as we shall now explain.
Given a simple walk W , recall that the active crossings are where the walk jumps
down from the overcrossing arc to the undercrossing arc. If the i-th crossing is active, we
record this with ai. As usual, the crossings are labeled 1, 2, . . . , ℓ from top to bottom of
the braid. The collection of active crossings of W determines a monomial in ta1, . . . , aℓu,
and thus we see that a simple walk determines a monomial. Conversely, the monomial in
ta1, . . . , aℓu uniquely determines the simple walk W . We will explain this below, but be-
fore we do, notice that not every monomial corresponds to a simple walk. For example, the
trefoil braid σ31 has three crossings and so there are 2
3 “ 8 possible monomials. However,
it has only one simple walk corresponding to the monomial a2.
Suppose then that ai1 ¨ ¨ ¨ aik is a monomial, indicating that crossings i1, . . . , ik are
active. We perform an oriented smoothing at each active crossing. Since the walk jumps
down there, the crossing type determines which of the arcs are active and which are not.
Specifically, if the crossing is positive, the active arc is the one on the left, and if the
crossing is negative, the active arc is the one on the right (see Figure 6). At each active
crossing, we mark the active arc using some marking scheme. (In all figures, the active
arcs have zebra stripes.) We then extend the marking along the arc through any inactive
crossings and around the back of the braid closure, continuing again through the braid and
around the back as many times as necessary, until reaching another active crossing.
σi σ−1i
Figure 6: An active crossing and its oriented smoothing for σi and σ´1i . The active arc has
zebra stripes and the inactive arc is solid.
One of two things will happen at this active crossing. Either the extended marking is
on an inactive arc, in which case the monomial does not correspond to a valid simple walk,
or it is on the active arc. If the second case holds for all extended arcs, then this is a valid
simple walk. Notice that, in that case, every arc of the partial smoothing of the braid closure
is connected to either an active or an inactive arc. To see that we argue by contradiction. If
the partial smoothing of pβ contains an arc that is not connected to an active or an inactive
arc, then we can follow it along pβ and it will only pass through inactive crossings before
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 177
returning to itself. Therefore, it determines a sublink of the closure of β, which contradicts
the assumption that the closure of β is a knot.
Note that, in the second case, the simple walk is determined by the active or marked
arcs, and once those are specified we can read the full operator by recording all the crossings
it passes through. This explains why the simple walk is determined by its active crossings.
Note that, in order for the marked arcs to be a simple walk, the first strand of the braid at
the top and bottom must be inactive.
This process can alternatively be understood in terms of taking the partial smoothing of
a knot K at a subset of crossings. Given a subset of crossings, we can perform the partial
oriented smoothing of K at the selected crossings. In general, this will produce a link. Our
assumption is that the resulting link can be partitioned into two sublinks, one containing
the active (or marked) arcs the other containing the inactive (or unmarked) arcs.
So, for a given monomial to correspond to a simple walk, it must be the case that the
active and inactive arcs of the braid are contained in different components of the partial
smoothing. Further, it must be the case that the first strand of the braid (at bottom) is
inactive.
We can apply this to understand the behavior of simple walks under taking reflection.
In the mirror image, all the crossings are switched. So, using same monomials to record
simple walk on β and β˚, it follows that taking reflection is equivalent to switching the
active and inactive arcs. That is because each positive crossing of β is negative in β˚ and
vice versa. This is illustrated by switching from left to right or vice versa in Figure 6.
The walks on β and β˚ with the same monomial are dual. We explain this from the
point of view of the full set of weights. Let W be a walk on β with monomial ai1 ¨ ¨ ¨ aik ,
and let W˚ be the corresponding walk on β˚ with monomial ai1 ¨ ¨ ¨ aiℓ . Then W and W˚
trace out disjoint arcs of the braid obtained from β by smoothing the crossings i1, . . . , ik.
In terms of the weights, the walks W and W˚ have the same set of active crossings, but
at the inactive crossings, their local weights are opposite. Specifically, if i is an inactive
crossing and the local weight Wpiq “ bi, then W˚piq “ ci. If instead Wpiq “ ci, then
W˚piq “ bi. Likewise, if Wpiq “ 1, then W
˚
piq “ bici, and if Wpiq “ bici then W
˚
piq “ 1.
Lemma 5.1. The simple walks with J Ď t2, . . . ,mu on a braid β and its reflection β˚ are
disjoint. In other words, SWβ X SWβ˚ “ ∅.
Proof. Suppose W is a simple walk on β with monomial ai1 ¨ ¨ ¨ aik . Then the partial
smoothing of β at the crossings i1, . . . , ik can be partitioned into active and inactive arcs.
Here, we mark the active arcs, with the first strand of the braid at the top inactive. For the
same monomial on the mirror image β˚, the active and inactive arcs will be switched. In
particular, the first strand on β˚ will be active at the top and marked as such. Therefore,
the monomial ai1 ¨ ¨ ¨ aik will not correspond to a valid simple walk on β˚. It follows that
SWβ X SWβ˚ “ ∅, and this completes the proof.
Definition 5.2. Given a braid word β, we say that a walk W on β is semi-simple if it
is a simple walk on β or on β˚ with J Ď t2, . . . ,mu. We use Sβ to denote the set of
semi-simple walks on β. Therefore, Sβ “ SWβ Y SWβ˚ .
Since SWβ and SWβ˚ are disjoint, it follows that |Sβ | “ |SWβ | ` |SWβ˚ |, where
|S| denotes the cardinality of the finite set S.
We leave it as an exercise to show that every monomial ai for 1 ď i ď ℓ corresponds
to a simple walk on either β or β˚. Thus |Sβ | ě n.
178 Ars Math. Contemp. 23 (2023) #P1.10 / 163–189
Theorem 5.3. The set of semi-simple walks Sβ is invariant under cyclic permutation of
the braid word.
The theorem is a direct consequence of the next two lemmas. The first lemma implies
that cyclic permutation of β does not alter the set of simple walks unless β starts with σ1
or σ´11 .
Lemma 5.4. Suppose β “ σε1i1 σ
ε2
i2
¨ ¨ ¨σεℓiℓ is a braid word with i1 ‰ 1. Let β
1 “
σε2i2 ¨ ¨ ¨σ
εℓ
iℓ
σε1i1 be the braid obtained by cyclic permutation. Then SWβ “ SWβ1 .
Proof. Suppose W P SWβ is a simple walk on β with J Ď t2, . . . ,mu. Let W 1 be the
corresponding simple walk on β1, with underlying set J 1. There are three possible cases,
depending on whether i1 and i1 ` 1 lie in J . First, if neither i1 nor i1 ` 1 lie in J , then
cyclic permutation has no effect and W 1 is a simple walk on β1 with J 1 “ J . Second, if
exactly one of i1, i1 ` 1 lies in J , then J 1 ‰ J , but W 1 is nevertheless a simple walk on β1
with J 1 Ď t2, . . . ,mu. Third, if both i1 and i1 ` 1 are in J , then J 1 “ J and W is a valid
simple walk on β1. This completes the proof of the lemma.
The second lemma studies the effect of cyclic permutation for braids that start with σ1
or σ´11 . We will show that cyclic permutation of a braid β has the potential to exchange
simple walks between SWβ and SWβ˚ , but it does not alter the set of semi-simple walks.
Lemma 5.5. Suppose β “ σε1i1 σ
ε2
i2
¨ ¨ ¨σεℓiℓ is a braid word with i1 “ 1. Let β
1 “
σε2i2 ¨ ¨ ¨σ
εℓ
iℓ
σε1i1 be the braid obtained by cyclic permutation. Then Sβ “ Sβ1 .
Proof. Suppose W P SWβ is a simple walk on β with J Ď t2, . . . ,mu. Let W 1 be the
corresponding simple walk on β1, with underlying set J 1. There are two possible cases,
depending on whether or not J contains 2. If 2 R J, then J 1 “ J and so W 1 P SWβ1 .
Similarly, if 2 P J and the monomial for W contains a1 (in which case β necessarily begins
with σ´11 ), then again J
1 “ J and W 1 P SWβ1 . However, if 2 P J and the monomial for
W does not contain a1, then 1 P J 1 and so W 1 R SWβ1 . However, the dual walk pW 1q˚
is simple walk on pβ1q˚ with pJ 1q˚ Ď t2, . . . ,mu, and hence pW 1q˚ P SWpbe1q˚ . In
particular, it follows that the set Sβ “ SWβ Y SWβ˚ of semi-simple walks is unchanged
by cyclic permutation. This completes the proof of the lemma.
The next result states that, up to reordering the crossings, the set of semi-simple walks
on a braid word and its rotation are equal.
Proposition 5.6. Let β “ σε1i1 ¨ ¨ ¨σ
εℓ
iℓ
be a braid word on m strands, and let β: “
σε1m´i1 ¨ ¨ ¨σ
εℓ
m´iℓ be its rotation. Then the sets of semi-simple walks of β and β
: are equal,
namely Sβ “ Sβ: .
Proof. The braid rotation β: is obtained from β by rotating it 1800 in the plane. In order to
relate the semi-simple walks on β and β:, we index the crossings of β: from top to bottom
using n, . . . , 1, and we identify semi-simple walks on β and β: with the subsets of active
crossings. In this way, every semi-simple walk on β and β: corresponds to a monomial
ai1 . . . aik indicating that i1, . . . , ik are the active crossings.
Given a monomial ai1 . . . aik , the semi-simple walk on β is obtained by taking the
smoothing of β at each crossing i1, . . . , ik and locally marking the active and inactive arcs
using a marking scheme that differentiates them. (We mark the active arcs using zebra
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 179
stripes.) Then extend the markings around the braid closure. Since ai1 . . . aik corresponds
to a semi-simple walk, the markings on the active and inactive arcs will not coincide.
The same will be true for β:, provided one follows the same procedures at the corre-
sponding crossings. Since β: is obtained by a 1800 rotation which interchanges the first
and last strands of the braid, this will not preserve SWβ since the new walk may not satisfy
J Ď t2, . . . ,mu. Nevertheless, the semi-simple walks of β and β: are preserved. This
completes the proof.
6 Simple walks on p2, nq torus braids
In this section, we show that the number of simple walks on the braid βn with closure
the p2, nq torus link is given by the n-th term in the Fibonacci sequence 1, 2, 3, 5, 8, 13,
21, 34, 55, 89, 144, . . . We will find a closed form solution for the number of simple walks
and use it to show they grow exponentially in n.
For n ě 1 let βn “ σ´n1 . The closure of βn is the p2, nq torus knot if n is odd and the
p2, nq torus link if n is even. For that reason, we refer to βn as the (negative) p2, nq torus
braid.
Proposition 6.1. Let fpnq be the number of simple walks on the p2, nq torus braid βn.
Then
fpnq “
ˆ
5 `
?
5
10
˙ ˆ
1 `
?
5
2
˙n
`
ˆ
5 ´
?
5
10
˙ ˆ
1 ´
?
5
2
˙n
.
Proof. The first step is to show that the simple walks on βn satisfy the Fibonacci recurrence
relation
fpnq “ fpn ´ 1q ` fpn ´ 2q. (5)
To do this, we establish a bijective correspondence between the set of simple walks on
βn and the union of the sets of simple walks on βn´1 and βn´2. This is accomplished by
extending simple walks on βn´1 and βn´2 to simple walks on βn.
In the following, we identify simple walks with their weights, which we write as mono-
mials in tai, bi, ci | i “ 1, . . . , nu. Notice that all simple walks under consideration will
have J “ t2u.
Given a simple walk w1 on βn´1, set w “ w1an. Then w is a simple walk on βn. See
Figure 7 (left). Since J “ t2u, this is actually the only way to extend w1 to a simple walk
on βn .
Similarly, given a simple walk w1 on βn´2, set w “ w1bn´1cn. Then w is again a
simple walk on βn. See Figure 7 (right). This is the only way to extend w1 to a simple walk
on βn which avoids simple walks extended from βn´1.
 
Figure 7: Extending simple walks from βn´1 and βn´2 to βn.
The two sets of simple walks are disjoint. This can be verified by noting that they
traverse different strands between the pn ´ 1q-st and n-th crossings. Equivalently, one can
180 Ars Math. Contemp. 23 (2023) #P1.10 / 163–189
see this by comparing their weights at the n-th crossing. The simple walks extended from
βn´1 have weight an, whereas those extended from βn´2 have weight cn.
Every simple walk on βn is an extension of one on βn´1 or βn´2. To that end, let w be
a simple walk on βn. Since J “ t2u, at the n-th crossing, either the walk jumps down and
has weight an, or it stays on the overstrand and has weight cn. In the first case, w “ w1an
for a simple walk w1 on βn´1. In the second, w “ w1bn´1cn for some simple walk w1 on
βn´2. This establishes the bijective correspondence, and Equation (5) follows directly.
The second step is to solve the recurrence Relation (5). It is a homogeneous linear
recurrence relation with constant coefficients and characteristic polynomial
pptq “ tn ´ tn´1 ´ tn´2 “ tn´2pt2 ´ t ´ 1q.
This polynomial has two non-zero roots:
t “ 1 ˘
?
5
2
.
Therefore, its general solution is given by fpnq “ c1p 1`
?
5
2 q
n ` c2p 1´
?
5
2 q
n. Using the
values fp1q “ 1 an fp2q “ 2, it follows that
1 “ c1
ˆ
1 `
?
5
2
˙
` c2
ˆ
1 ´
?
5
2
˙
,
2 “ c1
ˆ
1 `
?
5
2
˙2
` c2
ˆ
1 ´
?
5
2
˙2
.
Solving for c1, c2, we find that
c1 “
5 `
?
5
10
and c2 “
5 ´
?
5
10
.
The formula for fpnq follows, and this completes the proof.
7 Simple walks on p3, nq torus braids
In this section, we show that the number of simple walks on the braid γn with closure the
p3, nq torus link is given by the n-th term of the sequence 0, 1, 4, 5, 10, 19, 34, 63, 116, 213,
392, 721, 1326, . . . We will find a closed form solution for the number of simple walks and
use it to show they grow exponentially in n.
For n ě 1, let γn “ pσ´11 σ´12 qn. The closure of γn is the p3, nq torus knot if
n ı 0 pmod 3q and the p3, nq torus link if n ” 0 pmod 3q. For that reason, we refer
to γn as the (negative) p3, nq torus braid.
Proposition 7.1. Let gpnq be the number of simple walks on the p3, nq torus braid γn.
Then
gpnq “ c1αn ` c2βn ` c3γn,
where α, β, γ are the roots of t3 ´ t2 ´ t´ 1 (see Equation (7) for explicit formulas for the
roots) and where
c1 “
1 ` 3α´1
´α2 ` 4α ´ 1 , c2 “
1 ` 3β´1
´β2 ` 4β ´ 1 , c3 “
1 ` 3γ´1
´γ2 ` 4γ ´ 1 .
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 181
Proof. We claim that gpnq satisfies the tribonacci recurrence relation:
gpnq “ gpn ´ 1q ` gpn ´ 2q ` gpn ´ 3q. (6)
The proof of the claim is long, so we first show how to solve the recurrence relation to get
the formula for gpnq.
The tribonacci numbers T pnq are the sequence 0, 1, 1, 2, 4, 7, 13, . . . for n ě 0, and
they also satisfy (6). We will use the closed form solution for T pnq to find a closed form
solution for gpnq. The recurrence Relation (6) has characteristic polynomial pptq “ tn ´
tn´1´tn´2´tn´3 “ tn´3pt3´t2´t´1q. It has one nonzero real root α and two complex
roots β and γ given by
α “ 13 p1 `
3
b
19 ` 3
?
33 ` 3
b
19 ´ 3
?
33q,
β “ 12 p1 ´ α `
a
´3α2 ` 2α ` 5q,
γ “ 12 p1 ´ α ´
a
´3α2 ` 2α ` 5q.
(7)
The closed form solution for T pnq is a linear combination of powers of the roots of the
characteristic polynomial:
T pnq “ α
n
´α2 ` 4α ´ 1 `
βn
´β2 ` 4β ´ 1 `
γn
´γ2 ` 4γ ´ 1 .
The sequence gpnq is related to the tribonacci sequence by the equation
gpnq “ T pnq ` 3T pn ´ 1q.
From this we can write gpnq “ c1αn`c2βn`c3γn and use the values gp1q “ 0, gp2q “
1, gp3q “ 4 to solve for the coefficients c1, c2, c3:
c1 “
1 ` 3α´1
´α2 ` 4α ´ 1 , c2 “
1 ` 3β´1
´β2 ` 4β ´ 1 , c3 “
1 ` 3γ´1
´γ2 ` 4γ ´ 1 .
It remains to prove the claim, namely to show that the simple walks on γn satisfy the
recurrence Relation (6) for all n ě 4. To do this, we establish a bijective correspondence
between the simple walks on γn and the union of the sets of simple walks on γn´1, γn´2
and γn´3. In general, for braids on three strands, the simple walks will have J “ t2u,
J “ t3u or J “ t2, 3u. For the pn, 3q torus braids, all three occur.
We claim that every simple walk on γn´1, γn´2 and γn´3 can be extended to a simple
walk along γn. We prove this by considering the three cases separately. As before, we will
identify simple walks with their weights, which we write as monomials in tai, bi, ci | i “
1, . . . , 2nu.
Suppose w1 is a simple walk on γn´1. If J “ t2u, then it is on the understrand at the
p2n´ 2q-nd crossing, and so w1 “ w2b2n´2. We set w “ w2a2n´2b2n and note that w is a
simple walk on γn with J “ t2u. If J “ t3u, then we set w “ w1a2n. If J “ t2, 3u, then
we set w “ w1b2n ¨ a2n´1c2n. Note that in this last case, the paths become inverted, but
this is allowable for the walks with J “ t2, 3u. Figure 8 shows how the simple walks are
extended. In all three cases it is clear that w is a simple walk on γn.
182 Ars Math. Contemp. 23 (2023) #P1.10 / 163–189
J = {2}
J = {3} J = {2, 3}
 
Figure 8: Extending simple walks from γn´1 to γn.
J = {2} J = {3} J = {2, 3} 
Figure 9: Extending simple walks from γn´2 to γn.
In a similar way, we can extend simple walks on γn´2. Let w1 be a simple walk on
γn´2. If J “ t2u, then we set w “ w1a2n´3c2n´2b2n. If J “ t3u, then we set w “
w1b2n´2a2n´1c2n. If J “ t2, 3u, then we set w “ w1a2n´2b2n ¨ b2n´3c2n´1c2n. In this
last case, notice that the paths become inverted. Figure 9 shows how the simple walks are
extended. In all three cases, it is clear that w is a simple walk on γn.
Lastly, let w1 be a simple walk on γn´3. Then since γn “ γn´3pσ´11 σ´12 q3 with
pσ´11 σ´12 q3 inducing the identity permutation, we can extend w1 to a simple walk on γn
by remaining on the same strands. If J “ t2u, then we set w “ w1b2n´5c2n´3c2n´2b2n.
If J “ t3u, then we set w “ w1b2n´4b2n´3c2n´1c2n. If J “ t2, 3u, then we set w “
w1b2n´5c2n´3c2n´2b2n ¨ b2n´4b2n´3c2n´1c2n. Figure 10 shows how the simple walks are
extended. In all three cases, it is clear that w is a simple walk on γn.
It is not difficult to see that the sets of extended simple walks from γn´1, γn´2, and
γn´3 are all disjoint. For instance, this follows by comparing their weights, and noting
they are pairwise unequal.
The last step is to show that this accounts for all simple walks on γn. We will see that
every simple walk on γn is an extension of a simple walk on γn´1, γn´2 or γn´3.
Suppose w is a simple walk on γn. If J “ t2u, then it must traverse the understrand
at the 2n-th crossing and is on the overstrand just below the p2n ´ 2q-nd crossing. If it
jumps down at the p2n ´ 2q-nd crossing, then w “ w2a2n´2b2n, and w is the extension of
the simple walk w1 “ w2n2n´2 on γn´1. Otherwise, if it remains on the overstrand at the
p2n ´ 2q-nd crossing, then either w “ w1a2n´3c2n´2b2n for w1 a simple walk on γn´2, or
w “ w1b2n´5c2n´3c2n´2b2n for w1 a simple walk on γn´3.
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 183
J = {2} J = {3} J = {2, 3} 
Figure 10: Extending simple walks from γn´3 to γn.
If instead J “ t3u, then it is on the overstrand just below the p2nq-th crossing. If it
jumps down at the p2nq-th crossing, then w “ w1a2n for w1 a simple walk on γn´1. Other-
wise, if it remains on the overstrand at the p2nq-th crossing, then either w “
w1b2n´2a2n´1c2n for w1 a simple walk on γn´2, or w “ w1b2n´4b2n´3c2n´1c2n for w1 a
simple walk on γn´3.
Lastly, suppose J “ t2, 3u. Then w consists of two paths, ends with b2nc2n, and is on
the overstrand below the p2n´ 1q-st crossing. If it jumps down at the p2n´ 1q-st crossing,
then w “ w1b2n ¨ a2n´1c2n for w1 a simple walk on γn´1. If it remains on the overstrand
at the p2n ´ 1q-st crossing, then it is also on the overstrand at the p2n ´ 2q-nd crossing. If
it jumps down at the p2n ´ 2q-nd crossing, then w “ w1a2n´2b2n ¨ b2n´3c2n´1c2n for w1
a simple walk on γn´2. If it remains on the overstrand at the p2n ´ 2q-nd crossing, then it
approaches the p2n ´ 4q-th and p2n ´ 5q-th crossings on understrands, and it follows that
w “ w1b2n´5c2n´3c2n´2b2n ¨ b2n´4b2n´3c2n´1c2n for w1 a simple walk on γn´3.
This shows that every simple walk on γn is obtained by extending a simple walk on
γn´1, γn´2 or γn´3. It follows that there is a bijection correspondence between the set
of simple walks on γn and the union of the simple walks on γn´1, γn´2 and γn´3. The
bijective correspondence implies that the sequence gpnq of simple walks on γn satisfy the
recurrence Relation (6).
8 Minimal braid representatives
In Table 5, we list knots up to 9 crossings with the braid representatives giving minimal
numbers of simple walks. More extensive tables of knots up to 13 crossings and braid
representatives for them can be found online at [7]. (In [7] and Table 1 below, we use the
notation for braid words from sagemath, meaning that a braid word σε1a1 ¨ ¨ ¨σ
εℓ
aℓ
is denoted
by rε1a1, . . . , εℓaℓs. Tables 4 and 5 use even more compactified notation similar to that at
the end of [16].)
These results are empirical. The braid words listed in Table 5 and [7] are the output of
a sagemath program developed by the second author. It takes as input braid representatives
184 Ars Math. Contemp. 23 (2023) #P1.10 / 163–189
Knot Braid Word |SWβ |
11a322 r´1,´1, 2,´3, 4,´3, 2,´3, 4, 1, 2,´3,´2,´2s 51
12a23 r´1,´3, 2,´3,´5, 2, 4, 1, 2,´3,´4, 5, 4,´3,´5, 4, 2s 153
12a155 r´1, 2, 2,´3, 4,´3, 4,´5,´4, 3, 2, 1,´4, 5, 2,´3, 2s 127
12a288 r´1, 2, 2,´3, 2,´3, 2, 1, 2,´3, 4, 2,´3, 4s 71
12a449 r´1, 2, 4,´3, 4, 5, 4, 2,´3,´4,´4,´5, 1, 2, 4,´3, 2s 125
12a494 r´1, 2,´3, 4, 5, 2,´3,´4,´4, 1, 2,´3, 4,´5, 4,´3, 2s 137
12a750 r´1, 2,´3, 2,´3, 5, 4,´3,´4,´4,´5,´4,´4,´3, 1, 4,´2, 3,´2s 183
12n546 r´1, 2, 3, 1,´2, 1, 1, 1, 1,´2,´3,´3, 2s 41
12n601 r´1,´1, 2, 3, 3, 3, 2,´4,´4,´3, 1, 2, 3, 3,´4, 2s 67
12n622 r´1,´1, 2, 2, 3, 3, 2,´4, 1,´2, 3,´4,´2, 3s 47
Table 1: Knots up to 12 crossings whose minimizing braid word begins with σ´11 .
for knots (given by the braids from [19] and [29]) and applies cyclic permutation, reflection,
and rotation. It then selects the braid word that minimizes the number of simple walks.
The output braid word may represent the knot K or its mirror image K˚, whichever has
fewest simple walks.
The braids listed have the fewest simple walks among all known braid representatives
for the given knots. In general, the question of finding a complete list of braid representa-
tives for a given knot is a delicate and open problem. As we shall see, it is not enough to
consider only braid representatives of minimal width. Even if it were, it is an open problem
to develop an algorithm for computing the braid width of a knot (see Open Problem 1 in
[6]). Nevertheless, these problems have been studied extensively, and much is known about
minimal braid representatives of low-crossing knots; see [13, 16] and [29].
Given a knot, one can look for braid representatives that minimize its braid width or the
braid length. For many knots, there is a braid representative that simultaneously minimizes
both the width and length, but in general, the braid representatives that minimize width
need not be the same as the ones that minimize length. The earliest known examples are
the knots 16472381 and 161223549, which were discovered by Stoimenow and have braid
width 4 but no minimal length braid representative of width 4, [28, Figure 7].
This interesting aspect has been further studied by Gittings [13] and Van Cott [30], and
the “smallest” example is the knot 10136. For all other knots with up to 10 crossings, there
is a braid representative that simultaneously minimizes the braid width and length. Further
examples of knots whose minimal width braid representatives are not minimal length are
listed in Table 4. (These examples come from [19].)
Interestingly, the braid representative that minimizes the number of simple walks is not
always a minimal length braid, nor is it always a minimal width braid either. For example,
consider the knots 10136 and 11n8 and their braid representatives in Table 4. For 10136, the
number of simple walks is minimized on a braid representative of minimal length but not
one of minimal width, whereas for 11n8, the number of simple walks is minimized on a
braid representative of minimal width but not one of minimal length. Similar examples can
be found among the other knots listed in Table 4.
Our computations suggest that, for any knot, one can always minimize the number
of simple walks on a braid representative of minimal width or minimal length. This is
an interesting problem for future investigation. In order to make progress, we need more
information about the minimal width and minimal length braid representatives for knots.
At present, we do not have complete information on the 13-crossing knots. In particular,
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 185
we do not know which 13-crossing knots have minimal length braid representatives that
are not minimal width. The braid representatives for the 13-crossing knots from [29] are
known to be of minimal width, but they are not known to be of minimal length.
Notice that for every knot in Table 5, the braid representative that minimizes the number
of simple walks begins with σ1. This is actually true for all knots up to 10 crossings, but
not immediately true for knots with 11 or more crossings (see [7]).
In general, by Lemma 5.4, the minimizing braid representative can always be chosen to
begin with either σ1 or σ´11 . For knots with 11 and 12 crossings, there are only a handful
of examples whose minimizing braid representative begins with σ´11 and not σ1. They are
listed in Table 1. (There are in addition 82 examples among the knots with 13 crossings,
see [7].) In each case, we can find a minimizing braid representative that begins with σ1 by
reversing the braid word and applying cyclic permutation. We explain these steps in more
detail.
Take, for example, the first knot in Table 1, namely 11a322. Its minimizing braid rep-
resentative is the braid word σ´21 σ2σ
´1
3 σ4σ
´1
3 σ2σ
´1
3 σ4σ1σ2σ
´1
3 σ
´2
2 . The reversed braid
word will have the same number of simple walks, so it follows that σ´22 σ3σ2σ1
σ4σ
´1
3 σ2σ
´1
3 σ4σ
´1
3 σ2σ
´2
1 is also a minimizing braid word for 11a322. Now repeated ap-
plication of Lemma 5.4 shows that the braid word σ1σ4σ´13 σ2σ
´1
3 σ4σ
´1
3 σ2σ
´2
1 σ
´2
2 σ3σ2
is also minimizing for 11a322.
3 4 5 6 7 8 9 10 11 12 13
0
20
40
60 Average Number of Simple Walks
Table 2: Average number of simple walks by crossing number.
The same method applies to the other knots in Table 1. Each one admits a minimizing
braid word that starts with σ1. A similar argument applies to the braid representatives for
the 13 crossing knots that begin with σ´11 . This follows by a routine but somewhat tedious
exercise.
Table 2 shows the growth rate of the number of simple walks as a function of the
crossing number of the knot. Table 3 shows the growth rate of the number of simple walks
as a function of the braid length. Note that Table 3 contains information for knots with up
to 12 crossings but not the 13-crossing knots. The reason is that we do not have definitive
information about the braid representatives of minimal length for the 13-crossing knots.
We end this paper with a few questions and open problems for future investigation. One
is whether braid words that minimize the number of simple walks have a preferred shape
or form. By Proposition 4.4, we can assume the braid word is reduced and irreducible, and
by Lemma 5.4, we can assume it begins with σ1 or σ´11 . We conjecture the braid word
can always be chosen to begin with σ1. Further results as to the shape of minimizing braid
words would be helpful for developing efficient search algorithms.
186 Ars Math. Contemp. 23 (2023) #P1.10 / 163–189
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
0
50
100 Average Number of Simple Walks
Table 3: Average number of simple walks by braid length.
More generally, it would be extremely useful to automate the generation of minimal
width and/or minimal length braid representatives for a given knot. Such tools would allow
fast computation of the colored Jones polynomial and other quantum knot invariants, en-
abling calculations for higher crossing knots, including those in the knot tables of Burton
[8], who has recently extended the classification of knots to 19 crossings.
Auxilliary files
Sagemath programs and datasets are available online [7]. This includes a program that
generates all the simple walks as operators for a given braid and another that selects braid
words that minimize the number of simple walks. It also includes input datasets used to
create Tables 4 and 5, as well as output datasets of braid words that minimize the number
of simple walks for knots up to 13 crossings.
Knot Minimal width braid |SW| Minimal length braid |SW|
10136 123´121´1223´12´21 21 12´13´12243´1412´1 17
11n8 123´12´112´11´1232221 20 121´1213´12243´141 23
11n121 12322´112´11´1223´121 20 12123´121´1243´141 21
11n131 12322´2122´13´1212´1 17 123´121322´13´143´14 21
12n17 2´112´134´232´23´143´141 63 p12´1q23´1224325´145´1 52
12n20 3´112´1p32´1q21´123´1221 26 p12´1q2p3´12q243´141 32
12n24 12´132´232´13´11´123´121 35 2´112´13´12243´341 32
12n65 2´1123´14243´123´14´123´11 26 121´13´123´14´13254´151 25
12n119 2´1122´1313p12´1q23´11 31 2´113´1243´14122´112 23
12n284 p12´2q23132´112´13´1 36 2´112´13´123´143´123´141 32
12n311 4121´123´142´13´1432´13´11 37 2´11324´13´123´154´1351 30
12n314 2´13´1p12´1q33132´11 34 p12´1q23´12143´12´143 24
12n358 2´13´1121´1212´13132´11 24 3´12´11432´134´13421 20
12n362 123´221´1223´12´132´11 42 3´12´112´243´141221 24
12n403 12´1342´13´11´123´121 37 1213´12´112´143´1241 20
12n482 2´13´113´1221´2232´112 29 2´112´13´1243´123´241 32
Table 4: Simple walks for representatives of minimal braid width and length.
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 187
Knot Braid Word |SW|
31 13 1
41 p12´1q2 2
51 15 3
52 12312´1 2
61 121´13´123´11 3
62 132´112´1 4
63 122´112´2 5
71 17 8
72 12331´1213´1 3
73 1421´121 6
74 12232´13212´1 5
75 1321´1221 5
76 122´1132´13 8
77 p12´1q232´13 8
81 1234´12´13212´14´1 4
82 152´112´1 9
83 121´13´123´14´134´11 9
84 123´123´312´1 7
85 p132´1q2 8
86 1321´13´123´11 7
87 142´112´2 10
88 1221´13´123´21 10
89 132´112´3 9
810 132´1122´2 9
811 121´1223´123´11 7
812 p12´134´1q2 14
813 1223´123´212´1 8
814 1221´1p23´1q21 8
815 123132´132 9
816 122´112´1122´1 9
817 1p12´1q32´1 9
818 p12´1q4 10
819 p123q2 5
820 12312´3 5
821 1221´2221 6
91 19 21
92 12343´142´132212´1 5
93 1621´121 14
94 1232243´112 9
95 121´12342´13´14321 10
96 1521´1221 12
97 1234223´112´1 9
Knot Braid Word |SW|
98 1412´13´1423´112´1 13
99 1231´1214 12
910 1232´13231´121 11
911 12´113312´13 15
912 123´141´123´1414 13
913 1232´1321´1231 10
914 121´13´123´143´141 11
915 1223´141´123´141 16
916 1231´12213 10
917 p12´1q22´1p32´1q2 17
918 122322312´13´1 10
919 12´23´1243´1412´1 14
920 132´13132´13 15
921 121´123´1243´141 12
922 12´1312´332´1 17
923 1221´123´122132 15
924 1312´1312´3 17
925 12334´112´134´1 14
926 12´112312´132´1 13
927 122´112´232´13 15
928 12´11312´232 17
929 p12´132´1q22´1 16
930 122´212´132´13 16
931 12´11312´1322´1 15
932 1p12´1q2312´13 14
933 p12´1q22´1312´13 16
934 12´13p12´1q232´1 13
935 1234´1341´142´132123´1 17
936 132´11312´13 14
937 p12´13q243´123´12´14´1 29
938 123221´123´1221 13
939 123´12143´12´14´1342 18
940 12´1312´1132´13 14
941 13´1423´123´212´134 20
942 123´1212´33´1 7
943 1233´1123´12 8
944 123´12212´13´12´1 7
945 123312´132´1 8
946 123´1212´132´13 8
947 p123´12q23´1 8
948 12322´112´13´11´121 9
949 1221312´13212´1 9
Table 5: Knots up to 9 crossings and braid words minimizing the number of simple walks.
188 Ars Math. Contemp. 23 (2023) #P1.10 / 163–189
ORCID iDs
Hans U. Boden https://orcid.org/0000-0001-5516-8327
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Author Guidelines
Before submission
Papers should be written in English, prepared in LATEX, and must be submitted as a PDF
file. The title page of the submissions must contain:
• Title. The title must be concise and informative.
• Author names and affiliations. For each author add his/her affiliation which should
include the full postal address and the country name. If avilable, specify the e-mail
address of each author. Clearly indicate who is the corresponding author of the paper.
• Abstract. A concise abstract is required. The abstract should state the problem stud-
ied and the principal results proven.
• Keywords. Please specify 2 to 6 keywords separated by commas.
• Mathematics Subject Classification. Include one or more Math. Subj. Class. (2020)
codes – see https://mathscinet.ams.org/mathscinet/msc/msc2020.html.
After acceptance
Articles which are accepted for publication must be prepared in LATEX using class file
amcjoucc.cls and the bst file amcjoucc.bst (if you use BibTEX). If you don’t use BibTEX,
please make sure that all your references are carefully formatted following the examples
provided in the sample file. All files can be found on-line at:
https://amc-journal.eu/index.php/amc/about/submissions/#authorGuidelines
Abstracts: Be concise. As much as possible, please use plain text in your abstract and
avoid complicated formulas. Do not include citations in your abstract. All abstracts will be
posted on the website in fairly basic HTML, and HTML can’t handle complicated formulas.
It can barely handle subscripts and greek letters.
Cross-referencing: All numbering of theorems, sections, figures etc. that are referenced
later in the paper should be generated using standard LATEX \label{. . . } and \ref{. . . }
commands. See the sample file for examples.
Theorems and proofs: The class file has pre-defined environments for theorem-like state-
ments; please use them rather than coding your own. Please use the standard \begin{proof}
. . . \end{proof} environment for your proofs.
Spacing and page formatting: Please do not modify the page formatting and do not use
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formatting and spacing as needed for publication.
Figures: Any illustrations included in the paper must be provided in PDF format, or
via LATEX packages which produce embedded graphics, such as TikZ, that compile with
PdfLATEX. (Note, however, that PSTricks is problematic.) Make sure that you use uniform
lettering and sizing of the text. If you use other methods to generate your graphics, please
provide .pdf versions of the images (or negotiate with the layout editor assigned to your
article).
vii
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viii
Petra Šparl Award 2024: Call for Nominations
The Petra Šparl Award was established in 2017 to recognise in each even-numbered year
the best paper published in the previous five years by a young woman mathematician in
one of the two journals Ars Mathematica Contemporanea (AMC) and The Art of Discrete
and Applied Mathematics (ADAM). It was named after Dr Petra Šparl, a talented woman
mathematician who died mid-career in 2016.
The award consists of a certificate with the recipient’s name, and invitations to give a
lecture at the Mathematics Colloquium at the University of Primorska, and lectures at the
University of Maribor and University of Ljubljana. The first award was made in 2018 to
Dr Monika Pilśniak (AGH University, Poland) for a paper on the distinguishing index of
graphs, and then two awards were made for 2020, to Dr Simona Bonvicini (Università di
Modena e Reggio Emilia, Italy) for her contributions to a paper giving solutions to some
Hamilton-Waterloo problems, and Dr Klavdija Kutnar (University of Primorska, Slovenia),
for her contributions to a paper on odd automorphisms in vertex-transitive graphs. The
award for 2022 was made to Dr Jelena Sedlar (University of Split, Croatia) for resolving
two open conjectures regarding the Wiener index of trees.
The Petra Šparl Award Committee is now calling for nominations for the next award.
Eligibility: Each nominee must be a woman author or co-author of a paper published in
either AMC or ADAM in the calendar years 2019 to 2023, who was at most 40 years old at
the time of the paper’s first submission.
Nomination Format: Each nomination should specify the following:
(a) the name, birth-date and affiliation of the candidate;
(b) the title and other bibliographic details of the paper for which the award is recom-
mended;
(c) reasons why the candidate’s contribution to the paper is worthy of the award, in at
most 500 words; and
(d) names and email addresses of one or two referees who could be consulted with regard
to the quality of the paper.
Procedure: Nominations should be submitted by email to any one of the three members
of the Petra Šparl Award Committee (see below), by 31 October 2023.
Award Committee:
• Marston Conder, m.conder@auckland.ac.nz
• Asia Ivić Weiss, weiss@yorku.ca
• Aleksander Malnič, aleksander.malnic@guest.arnes.si
Marston Conder, Asia Ivić Weiss and Aleksander Malnič
Members of the 2024 Petra Šparl Award Committee
ix
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