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Contents
Some algebraic properties of Sierpiński-type graphs
Mohammad Farrokhi Derakhshandeh Ghouchan, Ebrahim Ghorbani,
Hamid Reza Maimani, Farhad Rahimi Mahid . . . . . . . . . . . . . . . . 171
The chromatic index of strongly regular graphs
Sebastian M. Cioabă, Krystal Guo, Willem H. Haemers . . . . . . . . . . . 187
A simple and elementary proof of Whitney’s unique embedding theorem
Gunnar Brinkmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
The average genus for bouquets of circles and dipoles
Jinlian Zhang, Xuhui Peng, Yichao Chen . . . . . . . . . . . . . . . . . . . 199
Well-totally-dominated graphs
Selim Bahadır, Tınaz Ekim, Didem Gözüpek . . . . . . . . . . . . . . . . 209
Nordhaus-Gaddum type inequalities for the distinguishing index
Monika Pilśniak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Closed formulas for the total Roman domination number of lexicographic
product graphs
Abel Cabrera Martínez, Juan Alberto Rodríguez-Velázquez . . . . . . . . . 233
Wiener-type indices of Parikh word representable graphs
Nobin Thomas, Lisa Mathew, Sastha Sriram, K. G. Subramanian . . . . . . 243
A double Sylvester determinant
Darij Grinberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Boundary-type sets of strong product of directed graphs
Bijo S. Anand, Manoj Changat, Prasanth G. Narasimha-Shenoi,
Mary Shalet Thottungal Joseph . . . . . . . . . . . . . . . . . . . . . . . . 275
Volume 20, Number 2, Spring/Summer 2021, Pages 171–288
xiii

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 171–186
https://doi.org/10.26493/1855-3974.2199.97e
(Also available at http://amc-journal.eu)
Some algebraic properties of
Sierpiński-type graphs*
Mohammad Farrokhi Derakhshandeh Ghouchan
Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS),
and the Center for Research in Basic Sciences and Contemporary Technologies, IASBS,
P. O. Box 45137-66731, Zanjan, Iran
Ebrahim Ghorbani †
Department of Mathematics, K. N. Toosi University of Technology,
P. O. Box 16765-3381, Tehran, Iran
Hamid Reza Maimani , Farhad Rahimi Mahid
Department of Basic Sciences, Shahid Rajaee Teacher Training University,
P. O. Box 16783-163, Tehran, Iran
Received 16 December 2019, accepted 4 November 2020, published online 21 October 2021
Abstract
This paper deals with some algebraic properties of Sierpiński graphs and a family of
regular generalized Sierpiński graphs. For the family of regular generalized Sierpiński
graphs, we obtain their spectrum and characterize those graphs that are Cayley graphs. As
a by-product, a new family of non-Cayley vertex-transitive graphs, and consequently, a
new set of non-Cayley numbers are introduced. We also obtain the Laplacian spectrum of
Sierpiński graphs in some particular cases, and make a conjecture on the general case.
Keywords: Sierpiński graph, spectrum, Laplacian, Cayley graph, non-Cayley number.
Math. Subj. Class. (2020): 05C50, 05C25, 05C75
*The authors would like to thank anonymous referees for constructive comments which led to improvement of
the presentation of the paper. The second author carried this work during a Humboldt Research Fellowship at the
University of Hamburg. He thanks the Alexander von Humboldt-Stiftung for financial support.
†Corresponding author.
E-mail addresses: farrokhi@iasbs.ac.ir, m.farrokhi.d.g@gmail.com (Mohammad Farrokhi Derakhshandeh
Ghouchan), ghorbani@kntu.ac.ir (Ebrahim Ghorbani), maimani@ipm.ir (Hamid Reza Maimani),
farhad.rahimi@sru.ac.ir (Farhad Rahimi Mahid)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
172 Ars Math. Contemp. 20 (2021) 171–186
1 Introduction
Sierpiński-type graphs show up in a wide range of areas; for instance, physics, dynamical
systems, probability, and topology, to name a few. The Sierpiński gasket graphs form one
of the most significant families of such graphs that are obtained by a finite number of
iterations that give the Sierpiński gasket in the limit. Several more families of Sierpiński-
type graphs have been introduced and studied in the literature (see Barrière, Comellas, and
Dalfó [1] and Hinz, Klavžar, and Zemljič [6]). In this paper, we deal with two families of
them, as described below.
For positive integers n, k, the Sierpiński graph S(n, k) is defined with vertex set [k]n,
where [k] := {1, . . . , k}, and two different vertices (u1, . . . , un) and (v1, . . . , vn) are ad-
jacent if and only if there exists a t ∈ [n] such that
• ui = vi for i = 1, . . . , t− 1,
• ut ̸= vt,
• uj = vt and vj = ut for j = t+ 1, . . . , n.
For instance, the graphs S(3, 3) and S(2, 4) are depicted in Figure 1.
222 223 232 233 322 323 332 333
221 231 321 331
212 213 312 313
211 311
122
123 132
133
121 131
112 113
111
44 43 34 33
41
42 31
32
14
13 24
23
11
12 21
22
Figure 1: The Sierpiński graphs S(3, 3) (left) and S(2, 4) (right).
Sierpinśki graphs S(n, k) were introduced in Klavžar and Milutinović [8]. The graph
S(n, 3) is indeed isomorphic to the graph of the Tower of Hanoi with n disks. The graph
S(n, k) has kn − k vertices of degree k and k vertices of degree k − 1 that are (i, . . . , i)
for i ∈ [k]. These vertices are called the extreme vertices.
In addition to S(n, k), we consider a ‘regularization’ of them as another family of
Sierpiński-type graphs. The graphs S++(n, k), introduced in Klavžar and Mohar [9],
are defined as follows. The graph S++(1, k) is the complete graph Kk+1. For n ≥ 2,
S++(n, k) is the graph obtained from the disjoint union of k + 1 copies of S(n− 1, k) in
which the extreme vertices in distinct copies of S(n− 1, k) are connected as the complete
graph Kk+1. See Figure 2 for an illustration of S++(3, 3).
Many properties of Sierpiński-type graphs, including those of S(n, k) and S++(n, k),
have been studied in the literature, for a survey see Hinz, Klavžar, and Zemljič [6]. In
M. Farrokhi D. G. et al.: Some algebraic properties of Sierpiński-type graphs 173
Figure 2: The graph S++(3, 3).
this paper, we investigate some algebraic properties of the two families of graphs, namely
the spectrum and the property of being a Cayley graph. More precisely, in Section 2,
we determine the spectrum of the graphs S++(n, k). The Laplacian spectrum of S(n, k)
is already known for k = 2, 3. We establish the case n = 2, in Section 3, and make
a conjecture on the Laplacian spectrum of S(n, k) in general. We also characterize the
graphs S++(n, k) that are Cayley graphs in Section 4. As a by-product, a new family of
non-Calyley vertex-transitive graphs are obtained. From this result, we conclude a new set
of square-free non-Cayley numbers in Section 5, and we discuss its distribution.
2 Spectrum of S++(n, k)
Let Γ be a simple graph with vertex set V (Γ) = {v1, . . . , vn} and edge set E(Γ). Its
adjacency matrix A(Γ) = [aij ] is an n × n symmetric matrix with aij = 1 if vi and vj
are adjacent, and aij = 0 otherwise. The multi-set of the eigenvalues of A(Γ) is called the
spectrum of Γ.
In this section, we determine the spectrum of S++(n, k). As we will see, the recursive
structure of these Sierpiński-type graphs also shows up in their spectrum. We first recall
some basic facts.
The incidence matrix of a graph Γ is a 0-1 matrix X(Γ) = [xve], with rows indexed by
the vertices and columns indexed by the edges of Γ, where xve = 1 if the vertex v is an
endpoint of the edge e. For a graph Γ, L(Γ) denotes the line graph of Γ, in which V (L(Γ))
corresponds with E(Γ), and two vertices of L(Γ) are adjacent if and only if they have a
common vertex as edges of Γ. The subdivision graph S(Γ) of Γ is the graph obtained by
inserting a new vertex into every edge of Γ. It is easy to verify that
X(Γ)⊤X(Γ) = 2I +A(L(Γ)), (2.1)
and, moreover, if Γ is k-regular, then
X(Γ)X(Γ)⊤ = kI +A(Γ). (2.2)
174 Ars Math. Contemp. 20 (2021) 171–186
The following lemma gives a recursive relation for the graphs S++(n, k).
Lemma 2.1. The graph S++(n+ 1, k) is isomorphic to L(S(S++(n, k))).
Proof. Let k be fixed. The graph Γn := S++(n, k) can be obtained by the union of S(n, k)
and S(n − 1, k) by adding a matching between the extreme vertices of the two graphs. If
we consider {0} × [k]n−1 as the vertex set of S(n− 1, k) (to make them compatible with
the length n of the vertices of S(n, k)), then ({0} ∪ [k])× [k]n−1 is the vertex set of Γn. It
follows that any edge e = {u,v} of Γn is of one of the following types:
(1) u = (u1, . . . , ur, u, v, . . . , v), v = (u1, . . . , ur, v, u, . . . , u) for some r ≤ n− 2 and
u ̸= v;
(2) u = (u1, . . . , un−1, u), v = (u1, . . . , un−1, v) with u ̸= v;
(3) u = (0, u, . . . , u), v = (u, u, . . . , u).
Each e = {u,v} ∈ E(Γn) is divided into two new edges eu and ev in S(Γn), where we
assume that u ∈ eu and v ∈ ev. We define a map ψ : E(S(Γn)) → ({0} ∪ [k]) × [k]n
based on the type of e as follows:
(i) If e is of type (1) or (2), then ψ(eu) = (u, v) and ψ(ev) = (v, u);
(ii) If e is of type (3), then ψ(eu) = (u, u) and ψ(ev) = (v, u).
It is easily seen that ψ is a one-to-one map. We show that ψ is an isomorphism from
L(S(Γn)) to Γn+1. Let e and e′ be two edges that share a vertex x of S(Γn). If x =
(x1, . . . , xn) is an ‘old’ vertex of S(Γn), then ψ(e) = (x, y) and ψ(e′) = (x, z) for some
y ̸= z. Then, it is clear that ψ(e) and ψ(e′) are adjacent in Γn+1. If x is a ‘new’ vertex
of S(Γn), then from (i) and (ii), it is clear that ψ(e) and ψ(e′) are adjacent in Γn+1. This
shows that ψ is indeed a one-to-one homomorphism. As L(S(Γn)) and Γn+1 have the
same number of edges, it follows that ψ is an isomorphism.
We recall that if A is a non-singular square matrix, then∣∣∣∣A BC D
∣∣∣∣ = |A| · ∣∣D − CA−1B∣∣ , (2.3)
where | · | denotes the determinant of a matrix. Also, recall that if M is a p× q matrix, then
|xI −MM⊤| = xp−q|xI −M⊤M |. (2.4)
(Note that (2.4) might not be valid if p ≤ q and x = 0, but this has no effect in our argument
since two polynomials that agree in all but finitely many points, agree everywhere.)
Let
f(x) = x2 + (2− k)x− k, (2.5)
and let f j(x) denote the polynomial of degree 2j obtained by j times composition of f
with itself. As a convention, we let f0(x) = x.
We now give the main result of this section.
M. Farrokhi D. G. et al.: Some algebraic properties of Sierpiński-type graphs 175
Theorem 2.2. Let k be an integer and Pn(x) denote the characteristic polynomial of the
adjacency matrix of S++(n, k). Then, Pn satisfies the recursion relation
Pn(x) = (x(x+ 2))
kn−2((k2)−1) Pn−1(f(x)), n ≥ 2, (2.6)
with P1(x) = (x − k)(x + 1)k. Moreover, for n ≥ 2, the spectrum of S++(n, k) consists
of the following eigenvalues:
(i) k with multiplicity 1,
(ii) the zeros of fn−1(x) + 1 each with multiplicity k,
(iii) the zeros of f j(x) each with multiplicity kn−2−j
((
k
2
)
− 1
)
for j = 0, 1, . . . , n− 2,
(iv) the zeros of f j(x)+2 each with multiplicity kn−2−j
((
k
2
)
− 1
)
+1 for j = 0, 1, . . . ,
n− 2.
Proof. Let Γn := S++(n, k). Suppose that X and Y are the incidence matrices of Γn−1
and S(Γn−1), respectively. By Lemma 2.1, Γn is isomorphic to L(S(Γn−1)). It follows
that
Y Y ⊤ =
[
kIp X
X⊤ 2Iq
]
,
where the matrix is divided according to the partition of the vertices into p = kn−1+kn−2
‘old’ vertices of Γn−1 and q = 12 (k
n + kn−1) ‘new’ vertices (which have degree 2) added
to Γn−1 to obtain S(Γn−1). Therefore, from (2.3),∣∣xI − Y Y ⊤∣∣ = |(x− k)Ip| · ∣∣∣(x− 2)Iq −X⊤ ((x− k)Ip)−1X∣∣∣
= (x− k)p
∣∣∣∣(x− 2)Iq − 1x− kX⊤X
∣∣∣∣
= (x− k)p−q
∣∣(x− 2)(x− k)Iq −X⊤X∣∣
= (x− k)p−q((x− 2)(x− k))q−p
∣∣(x− 2)(x− k)Ip −XX⊤∣∣ (by (2.4))
= (x− 2)q−p |((x− 2)(x− k)− k)Ip −A(Γn−1)| (by (2.2))
= (x− 2)q−pPn−1((x− 2)(x− k)− k). (2.7)
On the other hand, by (2.1) and (2.2), we have
Pn(x) =
∣∣(x+ 2)I2q − Y ⊤Y ∣∣ = (x+ 2)q−p ∣∣(x+ 2)Ip+q − Y Y ⊤∣∣ .
Now, from (2.7) it follows that
Pn(x) = (x(x+ 2))
q−pPn−1(x(x+ 2− k)− k),
implying (2.6).
To prove the second part of the theorem, note that as Γ1 = Kk+1, we have P1(x) =
(x− k)(x+ 1)k. From (2.6), we conclude that
P2(x) = (x(x+ 2))
(k2)−1(f(x)− k)(f(x) + 1)k,
176 Ars Math. Contemp. 20 (2021) 171–186
and since
f(x)− k = (x+ 2)(x− k), (2.8)
the assertion follows for n = 2. Now assume that n ≥ 3 and the assertion holds for n− 1.
So, we have
Pn−1(x) = (x− k)
(
fn−2(x) + 1
)k n−3∏
j=0
(
f j(x)
)mn−3−j (
f j(x) + 2
)1+mn−3−j
,
in which mi = ki
((
k
2
)
− 1
)
. It follows that
Pn−1(f(x)) = (f(x)− k)
(
fn−1(x) + 1
)k n−2∏
j=1
(
f j(x)
)mn−2−j (
f j(x) + 2
)1+mn−2−j
.
This, together with (2.6) and (2.8), implies the result.
Remark 2.3. It is straightforward to see that the zeros of f j(x) and f j(x) + 2 for j = 1
are 12 (k − 2 ±
√
k2 + 4) and 12 (k − 2 ±
√
k2 − 4), respectively, and for j ≥ 2 are of the
form
1
2
(k − 2)± 1
2
√
k(k + 2)± 2
√
k(k + 2)± 2
√
· · · ± 2
√
k2 + 4
and
1
2
(k − 2)± 1
2
√
k(k + 2)± 2
√
k(k + 2)± 2
√
· · · ± 2
√
k2 − 4,
respectively, each of them consisting of j nested radicals in iterative forms. Moreover, the
zeros of fn−1(x) + 1 are
− 1, k − 1, 1
2
(k − 2)± 1
2
√
k2 + 4k,
1
2
(k − 2)± 1
2
√
k(k + 2)± 2
√
k2 + 4k, . . . ,
1
2
(k − 2)± 1
2
√
k(k + 2)± 2
√
k(k + 2)± 2
√
· · · ± 2
√
k2 + 4k,
where the last one consists of n− 2 nested radicals.
3 Laplacian spectrum of S(n, k)
For a graph Γ, the matrix L(Γ) = D(Γ)−A(Γ) is the Laplacian matrix of Γ, where D(Γ)
is the diagonal matrix of vertex degrees. The multi-set of eigenvalues of L(Γ) is called the
Laplacian spectrum of Γ. In this section, we deal with the Laplacian spectrum of S(n, k).
This is trivial for n = 1 or k = 1. For k = 2, 3, the Laplacian spectrum of S(n, k) is
already known (see Remark 3.4 below). We establish the case n = 2, and put forward a
conjecture explicitly describing the Laplacian spectrum of S(n, k) in general.
Let Eij be a k × k matrix in which all entries are 0, except the (i, j) entry that is 1.
Consider the k2 × k2 matrix
C :=
k∑
i=1
k∑
j=1
(Eij ⊗ Eji),
M. Farrokhi D. G. et al.: Some algebraic properties of Sierpiński-type graphs 177
where ‘⊗’ denotes the Kronecker product. The matrix C is called the commutation matrix.
The main property of the commutation matrix (see Magnus and Neudecker [10]) is that it
commutes the Kronecker product: for any k × k matrices M,N ,
C(M ⊗N)C = N ⊗M.
Note that each row and each column of C corresponds with a pair (i, j) for 1 ≤ i, j ≤ k.
Moreover, C is indeed a permutation matrix in which the only 1 entry in the row (i, j) is
located at the column (j, i) for every 1 ≤ i, j ≤ k.
For n = 1, the Laplacian spectrum of S(1, k) = Kk is
{
0[1], k[k−1]
}
, where the
superscripts indicate multiplicities. In the following theorem, we determine the Laplacian
spectrum of S(2, k).
Theorem 3.1. The Laplacian spectrum of S(2, k) is the following:{
0[1], k[(
k
2)], (k + 2)[(
k−1
2 )],
(
1
2
(k + 2)± 1
2
√
k2 + 4
)[k−1]}
.
Proof. First, note that the graph S(2, k) consists of k copies ofKk together with a matching
M of size
(
k
2
)
; exactly one edge for each pair of copies of Kk. Let L denote the Laplacian
matrix of S(2, k), and L′ be the Laplacian matrix of the induced subgraph by the edges of
M . It is seen that L = Q − B, where Q = L(kKk) + I and B = I − L′. Note that B is
a permutation matrix with
(
k
2
)
+ k eigenvalues 1 and
(
k
2
)
eigenvalues −1. Observe that Q
has k eigenvalues 1 and k2 − k eigenvalues k + 1. We have the following bounds on the
dimensions of intersections of the eigenspaces of B and Q:
dim(E1(B) ∩ Ek+1(Q)) ≥ k2 − k +
(
k
2
)
+ k − k2 =
(
k
2
)
,
dim(E−1(B) ∩ Ek+1(Q)) ≥ k2 − k +
(
k
2
)
− k2 =
(
k
2
)
− k,
in which Eλ denotes the eigenspace corresponding to the eigenvalue λ. For x ∈ E1(B) ∩
Ek+1(Q), we have Lx = kx and for x ∈ E−1(B)∩Ek+1(Q), Lx = (k+2)x. This means
that L has eigenvalues k and k+2 with multiplicities at least
(
k
2
)
and
(
k
2
)
−k, respectively.
We also have
Q = Ik ⊗ ((k + 1)Ik − Jk), (3.1)
and from the eigenvalues of Q,
Q2 − (k + 2)Q+ (k + 1)I = O. (3.2)
Coming back to B, for each of the extreme vertices (1, 1), . . . , (k, k) of S(2, k), there is a
1 on all the entries of the diagonal of B. The off-diagonal 1’s correspond with the edges
of M . By the definition of S(2, k), the edges of M connect the vertices (i, j) and (j, i) for
i ̸= j. It turns out that B is the commutation matrix, and thus
BQB = ((k + 1)Ik − Jk)⊗ Ik. (3.3)
The right sides of (3.1) and (3.3) commute, and so
BQBQ = QBQB.
178 Ars Math. Contemp. 20 (2021) 171–186
Next, we see that
(L2 − (k + 2)L+ kI)(QB −BQ)
= ((Q−B)2 − (k + 2)(Q−B) + kI)(QB −BQ)
=
(
Q2 − (k + 2)Q+ kI +B2 + (k + 2)B −QB −BQ
)
(QB −BQ)
= ((k + 2)B −QB −BQ) (QB −BQ) (by (3.2) and since B2 = I)
= Q2 − (k + 2)Q−B
(
Q2 − (k + 2)Q
)
B − (QB)2 + (BQ)2
= (BQ)2 − (QB)2
= O.
The above equality shows that every vector in the column space of QB − BQ is an
eigenvector forLwith eigenvalue λ, where λ2−(k+2)λ+k = 0. To obtain the multiplicity
of such λ, we compute the rank of QB −BQ:
rank(QB −BQ) = rank(Q−BQB)
= rank(Ik ⊗ ((k + 1)Ik − Jk)− ((k + 1)Ik − Jk)⊗ Ik)
= rank(Jk ⊗ Ik − Ik ⊗ Jk)
= 2k − 2. (3.4)
To show (3.4), suppose P is a k × k matrix whose first column is 1√
k
(1, . . . , 1)⊤ and that
PP⊤ = Ik. Then, Jk = P (kE11)P⊤, and so
Jk ⊗ Ik − Ik ⊗ Jk = (P ⊗ P )
(
(kE11 ⊗ Ik)− (Ik ⊗ kE11)
)
(P⊤ ⊗ P⊤).
Since (kE11 ⊗ Ik) − (Ik ⊗ kE11) is a diagonal matrix having precisely 2k − 2 non-zero
entries in the columns 2, 3, . . . , k, k + 1, 2k + 1, 3k + 1, . . . , (k − 1)k + 1, (3.4) follows.
As x2 − (k + 2)x + k is an irreducible polynomial, each of its roots is an eigenvalue
of L with multiplicity at least k − 1. The matrix L has a 0 eigenvalue. Thus, we have
obtained so far
(
k
2
)
+
(
k
2
)
− k + 2(k − 1) + 1 = k2 − 1 eigenvalues of L. As the sum of
the eigenvalues of L is twice the number of edges of S(2, k), it follows that the remaining
eigenvalue is k + 2. So the proof is complete.
Based on empirical evidence, we put forward the following conjecture.
Conjecture 3.2. For n, k ≥ 2, the Laplacian spectrum of S(n, k) consists of the following
eigenvalues:
(i) 0 with multiplicity 1.
(ii) The zeros of f j(k − x), each with multiplicity 12 (k
n−j − 2kn−j−1 + k) for j =
0, 1, . . . , n− 1, where f is given in (2.5).
(iii) The zeros of f j(k − x) + 2, each with multiplicity 12 (k
n−j−1 − 1)(k − 2) for j =
0, 1, . . . , n− 2.
Remark 3.3. The zeros of f j(k−x) and f j(k−x)+2 for j = 1 are 12 (k+2±
√
k2 + 4)
and 12 (k + 2±
√
k2 − 4), respectively, and for j ≥ 2 are of the form
1
2
(k + 2)± 1
2
√
k(k + 2)± 2
√
k(k + 2)± 2
√
· · · ± 2
√
k2 + 4
M. Farrokhi D. G. et al.: Some algebraic properties of Sierpiński-type graphs 179
and
1
2
(k + 2)± 1
2
√
k(k + 2)± 2
√
k(k + 2)± 2
√
· · · ± 2
√
k2 − 4,
respectively, each of them consisting of j nested radicals.
Remark 3.4. The graph S(n, 2) is the path graph on 2n vertices. Proposition 3.5 below
shows that Conjecture 3.2 holds for S(n, 2). In Grigorchuk and Šunić [5], the spectrum
of the Schreier graph Γn was determined. The graph Γn is, in fact, the graph obtained
from S(n, 3) by adding a loop on each extreme vertex. By the way the adjacency matrix of
A(Γn) is defined in [5], for each loop a 1 entry on the diagonal is considered, so that each
row and column of A(Γn) has constant sum 3. It is then observed that the Laplacian spec-
trum of S(n, 3) can be deduced from the spectrum of Γn, which agrees with Conjecture 3.2.
In summary, Conjecture 3.2 holds for n = 2 and for k = 2, 3.
It is known in the literature that the characteristic polynomial of the Laplacian matrix
of a path can be expressed in terms of the Chebyshev polynomials. From this fact, for a
path with 2n vertices, we obtain the iterated form according to Conjecture 3.2. For the sake
of completeness, we give its complete argument here.
Proposition 3.5. The characteristic polynomial of the Laplacian matrix of the path graph
on 2n vertices is equal to x
∏n−1
j=0 g
j(2− x), where g(x) = x2 − 2.
Proof. Let ϕm be the characteristic polynomial of the Laplacian matrix of the path graph on
m vertices. Let Tm and Um be the Chebyshev polynomials of degree m of the first and the
second kind, respectively. Then Tm is the only polynomial satisfying Tm(cos θ) = cosmθ
and Um(x) = sin((m + 1) arccosx)/ sin(arccosx) (Snyder [18]). From the identities
given in Cvetković, Doob, and Sachs [2, p. 220], it follows that ϕm(x) = xUm−1(x/2−1).
By successive use of the identity U2k−1(x) = 2Tk(x)Uk−1(x) (see [18, p. 98]), we get
U2n−1(x) = 2
n−1T2n−1(x)T2n−2(x) · · ·T2(x)U1(x).
Note that U1(x) = 2x and T2(x) = 2x2 − 1. It is seen that 2T2(x/2 − 1) = x2 −
4x + 2 = g(2 − x). This, together with the identity T2k(x) = T2(Tk(x)), implies that
2T2j (x/2− 1) = gj(2− x). The proof is now complete.
4 What S++(n, k) are Cayley graphs?
Recall that a graph Γ is vertex-transitive if for any two vertices u, v of Γ, there exists an
automorphism σ of Γ such that σ(u) = v. Let G be a group and C ⊂ G such that 1 ̸∈ C
and c ∈ C implies that c−1 ∈ C. The Cayley graph Cay(G,C) with the group G and the
‘connection set’ C is the graph with vertex set G in which vertex u is connected to v if and
only if vu−1 ∈ C.
It is known that any Cayley graph is vertex-transitive. In the other way around, at least
for small orders, it seems that the great majority of vertex-transitive graphs are Cayley
graphs, see McKay and Praeger [12]. It is expected to continue to be this way for larger or-
ders. In fact, it is conjectured in Praeger, Li, and Niemeyer [15] that most vertex-transitive
graphs are Cayley graphs. In this section, we first determine what S++(n, k) are vertex-
transitive and, then, classify S++(n, k) that are Cayley graphs.
180 Ars Math. Contemp. 20 (2021) 171–186
Proposition 4.1. The graph S++(n, k) is vertex-transitive if and only if either n ≤ 2 or
k ≤ 2.
Proof. We have S++(n, 1) ∼= K2, S++(1, k) ∼= Kk+1, and S++(n, 2) is the cycle graph
on 2n−1 · 3 vertices, which are all vertex-transitive graphs. By Lemma 2.1, S++(2, k) is
isomorphic to L(S(Kk+1)). In the graph S(Kk+1), the ‘new’ vertices are in one-to-one
correspondence with 2-subsets of [k+1]. It is then easy to see that that any permutation of
[k+ 1] induces an automorphism of S(Kk+1). Now, for a given pair of edges of S(Kk+1)
which can be represented as e = {i, {i, j}} and e′ = {i′, {i′, j′}}, the automorphism
induced by a permutation σ that σ(i) = i′ and σ(j) = j′, maps e to e′. It follows that
S(Kk+1) is edge-transitive, and so L(S(Kk+1)) ∼= S++(2, k) is vertex-transitive. Hence,
assume that n ≥ 3 and k ≥ 3. We observe that an extreme vertex of a copy ∆ of S(n−1, k)
in Γ = S++(n, k) cannot be mapped to a non-extreme vertex of ∆ by any automorphism
of Γ. To be more precise, let u = (1, . . . , 1, 1) and v = (1, . . . , 1, 2). It can be seen that u
is a cut vertex for the induced subgraph by the vertices at distance at most 3 form u, while
v is not a cut vertex for the induced subgraph by the vertices at distance at most 3 form
v. It follows that u cannot be mapped to v by any automorphism of Γ, and thus Γ is not
vertex-transitive.
From Proposition 4.1, it follows that the graphs S++(n, k) for n ≥ 3 and k ≥ 3 cannot
be Cayley graphs. The graphs S++(n, 1), S++(n, 2), and S++(1, k) are all Cayley graphs.
It remains to characterize what S++(2, k) are Cayley graphs for k ≥ 3. This is our goal in
the rest of this section.
Definition 4.2. Let Γ be a graph and ∆ a subgraph of Γ. We say that Γ is strongly ∆-
partitioned if:
(i) The vertex set of Γ is partitioned by the vertex sets of copies ∆0, . . . ,∆k of ∆.
(ii) Apart from ∆0, . . . ,∆k, the graph Γ contains no further copies of ∆.
By the way S++(n, k) is defined, it is constructed based on k+1 copies of S(n−1, k).
The following proposition gives a structural property of S++(n, k) that it is indeed strongly
S(n − 1, k)-partitioned for n ≥ 2 and k ≥ 3. Note that this is not the case for k = 2
because S++(n, 2), that is a cycle with 3 · 2n−1 vertices, contains more than three copies
of S(n− 1, 2), which is a path on 2n−1 vertices. Although we only need the case n = 2 of
the proposition, we state it in its full generality because it could be of independent interest.
Proposition 4.3. Let n ≥ 2 and k ≥ 3. The graph S++(n, k) is strongly S(n − 1, k)-
partitioned.
Proof. Let Γ := S++(n, k) and Γ0, . . . ,Γk be the k + 1 copies of S(n − 1, k) used to
construct Γ by its definition. Clearly, V (Γ0), . . . , V (Γk) is a partition of V (Γ). We show
that Γ contains no more copies of S(n − 1, k). Let ∆ be a subgraph of Γ isomorphic to
S(n− 1, k).
First, assume that n = 2. Let u ∈ V (∆) ∩ V (Γt) for some t, with 0 ≤ t ≤ k.
Since u has at most one neighbor in V (Γ) \ V (Γt) and k ≥ 3, there exists another vertex
v ∈ V (∆)∩V (Γt) adjacent to u. Now, ifw is any vertex of ∆ other than u and v, then since
w is adjacent to two vertices u and v of Γt it must belong to V (Γt). Hence V (∆) ⊆ V (Γt)
and, consequently, ∆ = Γt.
M. Farrokhi D. G. et al.: Some algebraic properties of Sierpiński-type graphs 181
Now, let n ≥ 3. Note that S(n − 1, k) is connected and has no bridges since every
edge of S(n − 1, k) lies on a cycle (which can be seen by induction on n). If ∆ ̸= Γi
for i = 0, . . . , k, then ∆ shares its vertices with at least two Γs and Γt. By the definition,
exactly one extreme vertex, say u, of Γs is adjacent to exactly one extreme vertex, say v,
of Γt. Because of the connectivity, ∆ must contain the edge uv. Note that for any vertex w
outside Γs and Γt, the distance between w and either u or v is greater than the diameter of
S(n−1, k), and so w ̸∈ V (∆). It follows that ∆ is a subgraph of Γ′ := Γ[V (Γs)∪V (Γt)].
However, uv is a bridge for Γ′ and thus a bridge for ∆, a contradiction.
The following lemma reveals the structure of strongly ∆-partitioned Cayley graphs.
Lemma 4.4. Let Γ be a Cayley graph with a subgraph ∆ such that Γ is strongly ∆-
partitioned. Then, the vertex sets of the copies of ∆ are all the right cosets of a subgroup
of the underlying group of Γ.
Proof. Let Γ be a Cayley graph on a group G, and X ⊆ G be such that 1 ∈ X and Γ[X],
the subgraph of Γ induced by X , is isomorphic to ∆. Since for any x ∈ X , Γ[Xx−1]
is isomorphic to ∆ and 1 ∈ Xx−1, from the hypothesis of the lemma, it follows that
Xx−1 = X . Thus XX−1 = X and, hence, X is a subgroup of G. As for any g ∈ G,
Γ[Xg] is isomorphic to ∆ and the setsXg cover all elements ofG, it follows that the vertex
set of every induced subgraph of Γ isomorphic to ∆ is a right coset of X , as required.
Definition 4.5. Let Γ be a strongly ∆-partitioned graph. We say that Γ has connection
constant c if there are exactly c edges between any two copies of ∆ in Γ. We denote the set
of all strongly ∆-partitioned graphs with connection constant c by SPc(∆).
Remark 4.6. The family SP1(Kd) contains only one regular graph. However, this is not
the case in any regular graph ∆. If Γ ∈ SP1(∆) is a regular graph with ∆ being a d-
regular graph on k vertices, then Γ necessarily contains k + 1 copies of ∆ and thus Γ is
(d+1)-regular with k(k+1) vertices. For instance, in the case in which ∆ is C4, the cycle
on 4 vertices, Γ is a cubic graph on 20 vertices. By a computer search, we found all the
regular graphs in SP1(C4). It turned out that there are seven non-isomorphic such graphs,
among which only one is a Cayley graph.
Here we recall some notions from group theory that will be used in what follows. Let
G be a finite group and H be a nontrivial proper subgroup of G. The conjugate of H by
an element g of G is defined as Hg = {hg : h ∈ G}, where hg := g−1hg denotes the
conjugate of h by g. The group G is called a Frobenius group with Frobenius complement
H if H ∩ Hg = {1} for all g ∈ G \ H . A celebrated theorem of Frobenius states that
N := G \
⋃
g∈G(H \ {1})g is a normal subgroup of G, called the Frobenius kernel of G,
satisfying G = NH and N ∩H = {1}, that is, G = N ⋊H is a semidirect product of N
by H (see [17, 8.5.5]). The other concepts we use in the following are standard and can be
found in Robinson [17].
Theorem 4.7. Suppose that Γ and ∆ are two regular graphs and Γ ∈ SP1(∆). If Γ is a
Cayley graph Cay(G,C), then |∆|+1 = pm is a prime power, G = N ⋊H is a Frobenius
group with minimal normal Frobenius kernel N ∼= Zmp and Frobenius complement H ,
C = C ′ ∪ {c} with ∆ ∼= Cay(H,C ′) and c2 = 1, and either
(i) c ∈ N and H = ⟨C ′⟩, or
182 Ars Math. Contemp. 20 (2021) 171–186
(ii) c = hn for some h ∈ H \ {1} and n ∈ N \ {1}, and H = ⟨C ′, h⟩.
Conversely, if ∆ satisfies the above conditions, then Cay(G,C) ∈ SP1(∆).
Proof. Let Γ = Cay(G,C), and ∆0,∆1, . . . ,∆k be the copies of ∆ in Γ. As there is
exactly one edge between any two copies of ∆, it is observed that |∆| = k and Γ is (d+1)-
regular if ∆ is d-regular. Let H := V (∆0) and assume, without loss of generality, that
1 ∈ H . By Lemma 4.4, H is a subgroup of G. Let C ′ be the neighborhood of 1 in Γ[H].
Since Γ is (d + 1)-regular, besides the elements of C ′, the vertex 1 has exactly one other
neighbor, say c ∈ G \ H . So C = C ′ ∪ {c}. Since H is a subgroup of G, C ′−1 ⊆ H ,
which implies that C ′−1 = C ′. Thus, c = c−1 is an involution. Clearly, Hc ̸= H so that
Γ[Hc] = ∆i for some 1 ≤ i ≤ k. On the other hand,
1 = |E(∆0,∆i)| = |{{h, ch} : h ∈ H ∩Hc}| = |H ∩Hc|,
from which it follows that H ∩Hc = {1}. Now, a simple verification shows that Hch ∩
Hch′ = ∅ for all distinct elements h, h′ ∈ H . Since Γ[H] = ∆0 and Γ[Hch] (h ∈ H) are
equal to ∆1, . . . ,∆k in some order, we must have
G = H ∪
⋃
h∈H
Hch,
where the unions are disjoint. As a result, every element g ∈ G \ H can be written as
g = hch′ for some h, h′ ∈ H , from which it follows that
H ∩Hg = (Hh
′−1
∩ (Hh)c)h
′
= (H ∩Hc)h
′
= {1}h
′
= {1}.
Hence, G is a Frobenius group with complement H . Let N be the Frobenius kernel of G.
By [17, 10.5.1(i)], N is nilpotent. Let N0 be a nontrivial characteristic subgroup of N with
minimum order. Then N0 is a normal subgroup of G (see [17, 1.5.6(iii)]). Note that N0
is an elementary Abelian p-group for N0 is nilpotent and the subgroup of N0 generated
by central elements of a given prime order p dividing |Z(N0)| is a characteristic subgroup
of N0 and hence of N (see [17, 1.5.6(ii)]). If N ̸= N0, then N0H is a Frobenius group
for N0H is a subgroup of G and H ∩ Hg = 1 for all g ∈ N0H \ H . Moreover, as a
proper subgroup of N , |N0| ≤ |N |/2 ≤ (k + 1)/2 and hence |N0| − 1 is not divisible by
|H| = k contradicting [17, Exercises 8.5(6)]. Thus N = N0 so that k + 1 = |N | = pm
is a prime power for some m ≥ 1. Note that N is a minimal normal subgroup of G
for if N contains a nontrivial normal subgroup N0 of G properly, then N0H would be a
Frobenius group which leads us to the same contradiction as above. If c ∈ N , then since
G ⊆ N⟨C ′⟩ it follows that H = ⟨C ′⟩. Now assume that c /∈ N . Then cn ∈ H \ {1} for
some n ∈ N \ {1}. As G ⊆ N⟨C ′, cn⟩ it follows that H = ⟨C ′, cn⟩, as required. The
converse is straightforward.
We are now in a position to conclude the main result of this section.
Theorem 4.8. The graph S++(n, k) is a Cayley graph if and only if either
(i) n = 1,
(ii) k ≤ 2, or
M. Farrokhi D. G. et al.: Some algebraic properties of Sierpiński-type graphs 183
(iii) n = 2 and k + 1 = pm is a prime power.
Furthermore, in the case (iii), we have
S++(n, k) ∼= Cay(G, (H \ {1}) ∪ {c}),
for every Frobenius group G with complement H of order pm − 1, elementary Abelian
minimal normal Frobenius kernel of order pm, and involution c ∈ G \H .
Proof. By Proposition 4.1, S++(n, k) for n ≥ 3 and k ≥ 3 is not a Cayley graph. As
mentioned above, S++(n, 1), S++(n, 2), and S++(1, k) are all Cayley graphs. So, we
may assume that n = 2 and k ≥ 3.
First, we show that S++(2, q − 1) are Cayley graphs for all prime powers q. Let Fq
denote the finite field with q elements. Then G := F∗q ×Fq together with the multiplication
(x, a) · (y, b) = (xy, xb+ a),
forms a group known as one dimensional affine group. We show that S++(2, q − 1) ∼=
Cay(G,C), where C =
{
(x, 0) : 1 ̸= x ∈ F∗q
}
∪ {(−1,−1)}. To this end, let H :=
{(x, 0) : x ∈ F∗q} be a subgroup of G of order q − 1. Then H has q right cosets each
of which induces a complete subgraph in Cay(G,C) for h′g(hg)−1 = h′h−1 ∈ C for all
distinct elements hg and h′g of a right coset Hg of H . Since (x, 0)(1, ax−1) = (x, a)
covers all elements of G when x and a ranges over F∗q and Fq , respectively, it follows that
every right coset of H has a representative of the form (1, b) for some b ∈ Fq . Let Hg and
Hg′ be distinct right cosets of H with g = (1, a) and g′ = (1, a′). Then an element hg of
Hg is adjacent to an element h′g′ of Hg′ if and only if h′g′g−1h−1 = (h′g′)(hg)−1 ∈ C
or equivalently (h′g′)(hg)−1 = (−1,−1) as g′g−1 /∈ H . A simple verification shows that
this equation has a unique solution for (h, h′) so that there is a unique edge between any
two right cosets of H . Indeed, h = (x, 0) and h′ = (x′, 0) satisfy the equation if and only
if −x′ = x = (a′ − a)−1. Hence, from the definition, it follows that S++(2, q − 1) ∼=
Cay(G,C).
Now, assume that Γ := S++(2, k) ∼= Cay(G,C) be a presentation of S++(2, k) as a
Cayley graph. By Proposition 4.3, Γ is strongly Γ0-partitioned for some complete subgraph
Γ0 of Γ of order k. Let H := V (Γ0) and assume that 1 ∈ H . We know from Lemma 4.4
that H is a subgroup of G. By Theorem 4.7, k + 1 = pm is a prime power, G = N ⋊H
is a Frobenius group with Frobenius kernel N and Frobenius complement H such that
N ∼= Zmp is a minimal normal subgroup of G, C = C ′ ∪ {c}, C ′−1 = C ′ ⊆ H , c2 = 1,
and either
(a) c ∈ N and H = ⟨C ′⟩, or
(b) c = hn for some h ∈ H \ {1} and n ∈ N \ {1}, and H = ⟨C ′, h⟩.
Since Γ0 is a complete graph, we must have H \ {1} ⊆ C. Then, (a) and (b) together are
equivalent to say that c ∈ G \H . The proof is now complete.
As a generalization of Theorem 4.7, we pose the following problem.
Problem 4.9. Let ∆ be a regular graph. Classify all Cayley graphs in SPc(∆) for c ≥ 2.
184 Ars Math. Contemp. 20 (2021) 171–186
5 New non-Cayley numbers
In this final section, we give a partial answer to a famous rather old open problem in alge-
braic graph theory. A positive integer n is called a Cayley number if all vertex-transitive
graphs of order n are Cayley graphs. Marušič [11] in 1983 posed the problem of charac-
terizing the set NC of all non-Cayley numbers. Since disjoint unions of copies of vertex-
transitive (non-Cayley) graphs are again vertex-transitive (non-Cayley) graphs, it follows
that every multiple of a non-Cayley number is again a non-Cayley number. Hence the
problem of determining NC reduces to finding ‘minimal’ non-Cayley numbers. It is well-
known that all primes are Cayley numbers. Following a series of papers by various authors,
McKay and Praeger [13] and Iranmanesh and Praeger [7] provided necessary and sufficient
conditions under which the product of two and three distinct primes is a Cayley number,
respectively. In the same paper, McKay and Praeger established the following remarkable
result determining all non-square-free Cayley numbers.
Theorem 5.1 (McKay and Praeger [13]). Let n be a positive integer that is divisible by the
square of a prime p. Then n ∈ NC unless n = p2, n = p3, or n = 12.
It follows that, for determining NC, it is enough to consider only square-free positive
integers. While the problem is yet open for the products of at least four distinct primes,
there are partial results worth to mention here.
Theorem 5.2 (Dobson and Spiga [3]). There exists an infinite set of primes every finite
product of its distinct elements is a Cayley number.
As a consequence of Theorem 4.8, the graph S++(2, k) that has k(k + 1) vertices is
not a Cayley graph if k + 1 is not a prime power. Therefore, we obtain a new infinite class
of square-free non-Cayley numbers as follows.
Theorem 5.3. Let k be any positive integer such that k(k+ 1) is square-free, and k+ 1 is
not a prime. Then, k(k + 1) ∈ NC.
As mentioned in Dobson and Spiga [3], it is straightforward by making use of the
group-theoretic and the number-theoretic results already available in the literature to prove
that Cayley numbers have density zero in the set of natural numbers, and hence the density
of non-Cayley numbers is 1. In the light of this fact, one might wonder about the dis-
tribution of the numbers k satisfying the conditions of Theorem 5.3 in the set of positive
integers. The following theorem shows that for large enough N , more than one third of
positive integers less than or equal to N satisfies the conditions of Theorem 5.3.
Theorem 5.4. The density of the set
{k : k(k + 1) is square-free, and k + 1 is not a prime}
is about 0.3226.
Proof. Let f ∈ Z[t] be a primitive polynomial (that is, the greatest common divisor of
its coefficients is 1) without multiple roots such that its image on N has k-free greatest
common divisor. Recall that a number that is not divisible by any proper k-th power is
called k-free. Let Skf (x) denote the number of all positive integers n ≤ x such that f(n) is
k-free, and consider
δf,k :=
∏
p prime
(
1− ϱ(p
k)
pk
)
,
M. Farrokhi D. G. et al.: Some algebraic properties of Sierpiński-type graphs 185
where ϱ(d) denotes the number of roots of f in Zd. Ricci [16] (see also Pappalardi [14])
proved that
Skf (x) ∼ δf,kx
provided that deg f ≤ k. Clearly, the function f(t) := t(t + 1) satisfies the above re-
quirements of Ricci’s theorem for k = 2. Also, it is obvious that ϱ(p2) = 2 for all primes
p. Thus, by Ricci’s theorem, the density of all positive integers k, for which k(k + 1) is
square-free, in the set of all positive integers, is equal to
δf,2 =
∏
p prime
(
1− 2
p2
)
= 2CFeller-Tornier − 1 ≈ 0.3226340989,
where CFeller-Tornier is the Feller-Tornier constant (see Finch [4, §2.4.1]). Since primes have
zero density in the set of all positive integers, the result follows.
To date, all the numbers whose membership in NC is known are determined based on
the results of [7, 12, 13]. Using a computer search, we see that the list of the numbers
whose membership in NC is not yet determined begins with
9982, 12958, 18998, 19646, 20398, 21574, 24662, 25438, 25606, . . . .
A simple computation reveals that among the numbers less than or equal to 108, there
are 2763 square-free integers of the form k(k + 1), with k + 1 not a prime of which the
following eight integers are new non-Cayley numbers:
1386506, 2668322, 15503906, 23985506, 38359442, 74261306, 89898842, 95912642.
ORCID iDs
Mohammad Farrokhi Derakhshandeh Ghouchan https://orcid.org/0000-0002-5850-969X
Ebrahim Ghorbani https://orcid.org/0000-0001-7195-8601
Hamid Reza Maimani https://orcid.org/0000-0001-9020-0871
Farhad Rahimi Mahid https://orcid.org/0000-0001-8996-2078
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 187–194
https://doi.org/10.26493/1855-3974.2435.3db
(Also available at http://amc-journal.eu)
The chromatic index of strongly regular graphs
Sebastian M. Cioabă *
Department of Mathematical Sciences, University of Delaware, United States
Krystal Guo †
Korteweg-De Vries Institute, University of Amsterdam, The Netherlands
Willem H. Haemers
Department of Econometrics and Operations Research, Tilburg University,
The Netherlands
Received 16 September 2020, accepted 20 January 2021, published online 27 October 2021
Abstract
We determine (partly by computer search) the chromatic index (edge-chromatic num-
ber) of many strongly regular graphs (SRGs), including the SRGs of degree k ≤ 18 and
their complements, the Latin square graphs and their complements, and the triangular
graphs and their complements. Moreover, using a recent result of Ferber and Jain, we
prove that an SRG of even order n, which is not the block graph of a Steiner 2-design or its
complement, has chromatic index k, when n is big enough. Except for the Petersen graph,
all investigated connected SRGs of even order have chromatic index equal to k, i.e., they
are class 1, and we conjecture that this is the case for all connected SRGs of even order.
Keywords: Strongly regular graph, chromatic index, edge coloring, 1-factorization.
Math. Subj. Class. (2020): 05C15, 05E30
1 Introduction
An edge-coloring of a graph G is a coloring of its edges such that intersecting edges have
different colors. Thus a set of edges with the same colors (called a color class) is a match-
ing. The edge-chromatic number χ′(G) (also known as the chromatic index) of G is the
*Research supported by NSF grants DMS-1600768 and CIF-1815922.
†This research was done while K. Guo was a postdoctoral fellow at Université Libre de Bruxelles, supported
by ERC Consolidator Grant 615640-ForEFront.
E-mail addresses: cioaba@udel.edu (Sebastian M. Cioabă), k.guo@uva.nl (Krystal Guo), haemers@uvt.nl
(Willem H. Haemers)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
188 Ars Math. Contemp. 20 (2021) 187–194
minimum number of colors in an edge-coloring. By Vizing’s famous theorem [24], the
chromatic index of a graph G of maximum degree ∆ is ∆ or ∆ + 1. A graph with maxi-
mum degree ∆ is called class 1 if χ′(G) = ∆ and is called class 2 if χ′(G) = ∆ + 1. It is
also known that determining whether a graph G is class 1 is an NP-complete problem [18].
If G is regular of degree k, then G is class 1 if and only if G has an edge coloring such that
each color class is a perfect matching. A perfect matching is also called a 1-factor, and a
partition of the edge set into perfect matchings is called a 1-factorization. So being regular
and class 1 is the same as having a 1-factorization (being 1-factorable), and requires that
the graph has even order.
A graph G is called a strongly regular graph (SRG) with parameters (n, k, λ, µ) if it
has n vertices, is k-regular (0 < k < n− 1), any two adjacent vertices of G have exactly λ
common neighbors and any two distinct non-adjacent vertices of G have exactly µ common
neighbors. The complement of a strongly regular graph with parameters (n, k, λ, µ) is
again strongly regular, and has parameters (n, n − k − 1, n − 2k + µ − 2, n − 2k + λ).
An SRG G is called imprimitive if G or its complement is disconnected, and primitive
otherwise. A disconnected SRG is ℓKm (ℓ,m ≥ 2), the disjoint union of ℓ cliques of
order m (indeed, µ = 0 implies that no two vertices have distance two, so every connected
component is a clique). It is well-known that Km (m ≥ 2), and hence also ℓKm, is class 1
if and only if m is even. The complement of ℓKm is a regular complete multipartite graph
which is known to be class 1 if and only if the order is even [17].
A vertex coloring of G is a coloring of the vertices of G such that adjacent vertices have
different colors. The chromatic number χ(G) of G is the minimum number of colors in a
vertex coloring. For the chromatic number there exist bounds in terms of the eigenvalues
of the adjacency matrix, which turn out to be especially useful for strongly regular graphs
(see for example [6]). These bounds imply that there exist only finitely many primitive
SRGs with a given chromatic number, and made it possible to determine all SRGs with
chromatic number at most four (see [15]). Motivated by these results, Alex Rosá asked the
third author whether eigenvalue techniques can give information on the chromatic index
of an SRG. There exist useful spectral conditions for the existence of a perfect matching
(see [5, 9]), and Brouwer and Haemers [5] have shown that every regular graph of even
order, degree k and second largest eigenvalue ϑ2 contains at least ⌊(k − ϑ2 + 1)/2⌋ edge
disjoint perfect matchings. From this it follows that every connected SRG of even order has
a perfect matching. Moreover, Cioabă and Li [10] proved that any matching of order k/4
of a primitive SRG of valency k and even order, is contained in a perfect matching. These
authors conjectured that k/4 can be replaced by ⌈k/2⌉ − 1 which would be best possible.
Unfortunately, we found no useful eigenvalue tools for determining the chromatic index.
However, the following recent result of Ferber and Jain [13] gives an asymptotic condition
for being class 1 in terms of the eigenvalues.
Theorem 1.1. There exist universal constants n0 and k0, such that the following holds. If G
is a connected k-regular graph of even order n with eigenvalues k = ϑ1 > ϑ2 ≥ · · · ≥ ϑn,
and n > n0, k > k0 and max{ϑ2,−ϑn} < k0.9, then G is class 1.
If the maximum distance in G is 2 (as is the case for a connected SRG), then n ≤
1 + k + k(k − 1) = k2 + 1. This implies that for an SRG we do not need to require
that k > k0 when we take n0 ≥ k20 + 1. Theorem 1.1 enables us to show that, except for
one family of SRGs, all connected SRGs of even order n are class 1, provided n is large
enough. In addition, we present a number of sufficient conditions for an SRG to be class 1.
S. M. Cioabă, K. Guo and W. H. Haemers: The chromatic index of strongly regular graphs 189
By computer, using SageMath [22], we verified that all primitive SRGs of even order and
degree k ≤ 18 and their complements are class 1, except for the Petersen graph, which has
parameters (10, 3, 0, 1) and edge-chromatic number 4 (see [20, 25] for example). We also
determine the chromatic index of several other primitive SRGs of even order, and all are
class 1. Therefore we believe:
Conjecture 1.2. Except for the Petersen graph, every connected SRG of even order is
class 1.
2 Sufficient conditions for being class 1
A well known conjecture (first stated by Chetwynd and Hilton [8], but attributed to Dirac)
states that every k-regular graph of even order n with k ≥ n/2 is 1-factorable. Cariolaro
and Hilton [7] proved that the conclusion holds when k ≥ 0.823n, and Csaba, Kühn, Lo,
Osthus, and Treglown [11], proved the following result.
Theorem 2.1. There exists a universal constant n0, such that if n is even, n > n0 and if
k ≥ 2⌈n/4⌉ − 1, then every k-regular graph of order n has chromatic index k.
König [19] proved that every regular bipartite graph of positive degree has a 1-factor-
ization. We need the following generalization of König’s result.
Lemma 2.2. Let G = (V,E) be a connected regular graph of even order n, and let
{V1, V2} be a partition of V such that |V1| = |V2| = n/2.
(i) If the graphs induced by V1 and V2 are 1-factorable, then so is G.
(ii) If V1 (and hence V2) is a clique or a coclique, then G is class 1.
Proof. Partition the edge set E into two classes E1 and E2, where E1 contains all edges
with both endpoints in the same vertex set V1 or V2, and the edges of E2 have one endpoint
in V1 and the other endpoint in V2.
(i): If the graphs induced by V1 and V2 are 1-factorable, then both have the same degree,
and therefore also (V,E1) is 1-factorable. By König’s theorem (V,E2) is 1-factorable,
therefore G is class 1.
(ii): If V1 is a coclique, then so is V2 and we have the theorem of König. If V1 is a
clique, then so is V2. If n/2 is even, then the result is proved in (i). If n/2 is odd, then we
choose a 1-factor F in (V,E2) (here we use that G is connected), and define E′2 = E2 \ F
and E′1 = E1 ∪ F . Then (V,E′2) is 1-factorable (or has no edges), and (V,E′1) consists of
two cliques of order n/2 and the 1-factor F . Thus F gives a bijection ϕ (say) between V1
and V2. By Vizing’s theorem the edges of both cliques can be colored with n/2 colors. We
do this coloring such that ϕ preserves colors, which means that {v, w} and {ϕ(v), ϕ(w)}
get the same color. Then for each edge {v, ϕ(v)} of F , the two sets of colored edges that
intersect at v and ϕ(v) use the same n/2 − 1 colors. So we can color {v, ϕ(v)} with the
remaining color.
There exist several SRGs that have the partition of case (i). The Gewirtz graph is the
unique SRG with parameters (56, 10, 0, 2), and admits a partition into two Coxeter graphs
(see [4]). The Coxeter graph is known to be 1-factorable (see [3]), therefore the Gewirtz
graph is class 1. The same holds for the point graph of the generalized quadrangle GQ(3, 9)
(the unique SRG(112, 30, 2, 10)), which admits a partition into two Gewirtz graphs, and for
190 Ars Math. Contemp. 20 (2021) 187–194
the Higman-Sims graph (the unique SRG(100, 22, 0, 6)), which can be partitioned into two
copies of the Hoffman-Singleton graph (the unique strongly regular graph with parameters
(50, 7, 0, 1)), which has chromatic index 7 (see Section 4).
Suppose ϑ1 ≥ ϑ2 ≥ · · · ≥ ϑn are the eigenvalues of a graph G of order n. Hoffman
(see [6, Theorem 3.6.2] for example) proved that the chromatic number of G is at least
1 − ϑ1/ϑn. A vertex coloring that meets this bound is called a Hoffman coloring. For
k-regular graphs, the color classes of a Hoffman coloring are cocliques of which the size
meets Hoffman’s coclique bound nϑn/(ϑn−k). This implies (see [6] for example) that all
the color classes have equal size, and any vertex v of G has exactly −ϑn neighbors in each
color class different from the color class of v.
Theorem 2.3. Suppose G = (V,E) is a k-regular graph with an even chromatic number
2t (say) that meets Hoffman’s bound. Then both G and its complement G are class 1, or G
is a disjoint union of cliques of odd order.
Proof. Let S1, . . . , S2t be the color classes in a Hoffman coloring of G. Since G is regular,
this implies that each Si is a coclique attaining equality in the Hoffman ratio bound, which
means that each vertex outside Si has exactly −ϑn neighbors in Si. Hence, each subgraph
induced by two distinct cocliques Si and Sj is a bipartite regular graph of valency −ϑn.
A 1-factorization of K2t corresponds to a partition E1, . . . , E2t−1 of E, such that each
(V,Ei) consists of t disjoint regular bipartite graphs of degree −ϑn = k/(2t − 1). By
König’s theorem it follows that each (V,Ei) is 1-factorable, and therefore G is class 1.
For the complement G = (V, F ) of G, a similar approach works. We can partition the
edge set F into the subsets F0, F1, . . . , F2t−1, such that for i = 1, . . . , 2t − 1 the graph
(V, Fi) is the disjoint union of t regular bipartite graphs of the same positive degree (if the
degree is 0, then G is a disjoint union of cliques). But now there is an additional graph
(V, F0) consisting of 2t disjoint cliques. We combine F0 and F1. Then (V, F0 ∪ F1) is
the disjoint union of t complements of regular incomplete bipartite graphs with the same
positive degree, and therefore has a 1-factorization by Lemma 2.2. Since (V, Fi) has a
1-factorization for i = 2, . . . , 2t− 1, it follows that G is 1-factorable.
For an SRG the color partition of a Hoffman coloring corresponds to a so-called spread
in the complement (see [16]). As a consequence of this result, it follows that any primitive
strongly regular graph with a spread with an even number of cliques, or a Hoffman coloring
with an even number of colors is class 1. Among such SRGs are the Latin square graphs.
Consider a set of t (t ≥ 0) mutually orthogonal Latin squares of order m (m ≥ 2). The
vertices of the Latin square graph are the m2 entries of the Latin squares, and two distinct
entries are adjacent if they lie in the same row, the same column, or have the same symbol
in one of the squares. If t = m− 1 we obtain the complete graph Km2 , and if t = m− 2
we have a complete multipartite graph. Otherwise the Latin square graph is a primitive
SRG with parameters (m2, (t+ 2)(m− 1),m− 2 + t(t+ 1), (t+ 1)(t+ 2)). If t = 0 we
only have the rows and columns, then the Latin square graph is better known as the Lattice
graph L(m). If m ̸= 4, the Lattice graph is determined by the parameters. The m rows of a
Latin square give a partition of the vertex set of the Latin square graph into cliques, which
is a spread. Thus we have:
Corollary 2.4. If G is a Latin square graph of even order, then both G and its complement
are 1-factorable.
S. M. Cioabă, K. Guo and W. H. Haemers: The chromatic index of strongly regular graphs 191
3 Asymptotic results
A Steiner 2-design (or 2-(m, ℓ, 1) design) consists of a point set P of cardinality m, to-
gether with a collection of subsets of P of size ℓ (ℓ ≥ 2), called blocks, such that every pair
of points from P is contained in exactly one of the blocks. The block graph of a Steiner
2-design is defined as follows. The blocks are the vertices, and two vertices are adjacent if
the blocks intersect in one point. If m = ℓ2 − ℓ + 1, the Steiner 2-design is a projective
plane, and the block graph is Km. Otherwise the block graph is an SRG with parameters
(m(m− 1)/ℓ(ℓ− 1), ℓ(m− ℓ)/(ℓ− 1), (ℓ− 1)2 + (m− 2ℓ+ 1)/(ℓ− 1), ℓ2).
Theorem 3.1. There exists an integer n0, such that every primitive strongly regular graph
of even order n > n0, which is not the block graph of a Steiner 2-design or its complement,
is class 1.
Proof. Suppose G is a primitive (n, k, λ, µ)-SRG of even order n. Then it is well-known
(see for example [6, Chapter 9]) that G has exactly three distinct eigenvalues k = ϑ1,
ϑ2 and ϑn. Moreover, the eigenvalues are nonzero integers and satisfy k + ϑ2ϑn = µ.
Assume that G nor its complement G is the block graph of a Steiner 2-design or a Latin
square graph. Using a result of Neumaier [21] (known as the claw bound), we get that
ϑ2 ≤ ϑn(ϑn + 1)(µ+ 1)/2− 1.
Another result of Neumaier [21] (the µ-bound) gives µ ≤ ϑ3n(2ϑn + 3). Combining these
inequalities, after some straightforward calculations, we obtain that ϑ2 < (−ϑn)6. Since
k + ϑ2ϑn = µ > 0, we deduce that
k6 > (−ϑ2ϑn)6 > ϑ62ϑ2 = ϑ72, so ϑ2 < k6/7.
Next we apply the same result to G, and obtain −1 − ϑn < (1 + ϑ2)6, which yields
−ϑn ≤ (2ϑ2)6 (since ϑ2 is a positive integer). Hence
k6 > (−ϑ2ϑn)6 > 2−6(−ϑn)(−ϑn)6 = 2−6(−ϑn)7, so − ϑn < (2k)6/7 < k0.9,
when k is large enough. Thus we get max{ϑ2,−ϑn} ≤ k0.9. Now we apply the result of
Ferber and Jain and conclude that G is class 1 when n is large enough.
If G is a Latin square graph of even order then by Corollary 2.4 both G and its comple-
ment G are class 1.
In many cases the complement of the block graph of a Steiner 2-design has k > n/2,
so it will have a 1-factorization by Theorem 2.1, provided n is even and large enough.
The following result follows straightforwardly from the mentioned result of Cariolaro and
Hilton [7].
Proposition 3.2. If G is the complement of the block graph of a 2-(m, ℓ, 1) design with
6ℓ2 ≤ m, then G is class 1, provided G has even order.
For every m ≥ 2 there is a unique 2-(m, 2, 1) design, and its block graph is the triangu-
lar graph T (m). It is isomorphic to the line graph of the complete graph Km, and if m ≥ 4
T (m) is an SRG with parameters (m(m− 1)/2, 2(m− 2),m− 2, 4). The triangular graph
is uniquely determined by its parameters if m ̸= 8. Alspach [1] has proved that T (m) has
192 Ars Math. Contemp. 20 (2021) 187–194
a 1-factorization if the order is even, which is the case if m ≡ 0, or 1 (mod 4). Proposi-
tion 3.2 implies that the complement of T (m) is class 1 if m ≥ 24 and the order is even.
The complement of T (5) is the Petersen graph, which is class 2. For 5 < m < 24 and
m ≡ 0, or 1 (mod 4) we found a 1-factorization in the complement of T (m) by computer
(see next section for more about the computer search). Thus we can conclude:
Theorem 3.3. For m ≡ 0, or 1 (mod 4) the triangular graph T (m) is class 1, and if
m ̸= 5 so is its complement.
If the block size equals 3, the design is better known as a Steiner triple system. The
chromatic index of the block graph of a Steiner triple system is investigated in [12]. The
paper contains several sufficient conditions for such a graph to be class 1, and the authors
conjecture that all these graphs are class 1 when the order is even. One of the results from
[12] (Theorem 2.2) can be generalized to arbitrary Steiner 2-designs. A set of m/ℓ disjoint
blocks of a 2-(m, ℓ, 1) design is called a parallel class, and a partition of the block graph
into parallel classes is a parallelism. A parallelism of a Steiner 2-design gives a Hoffman
coloring in the block graph, so we have:
Proposition 3.4. If a Steiner 2-design has a parallelism with an even number of parallel
classes, then the block graph and its complement are class 1.
4 SRGs of degree at most 18
According to the list of Brouwer [2] all primitive SRGs of even order and degree at most
18 are known (one only has to check the parameter sets up to n = 182 + 1 = 325). The
parameters together with the number of nonisomorphic SRGs with k < n/2 are given in
Table 1 (the ones with k ≥ n/2 are the complements of a to e). The graph with parameter
Table 1: Primitive SRGs with n even and k ≤ 18, k < n/2.
a (10, 3, 0, 1) 1 f (36, 10, 4, 2) 1 k (50, 7, 0, 1) 1
b (16, 5, 0, 2) 1 g (36, 14, 4, 6) 180 l (56, 10, 0, 2) 1
c (16, 6, 2, 2) 2 h (36, 14, 7, 4) 1 m (64, 14, 6, 2) 1
d (26, 10, 3, 4) 10 i (36, 15, 6, 6) 32548 n (64, 18, 2, 6) 167
e (28, 12, 6, 4) 4 j (40, 12, 2, 4) 28 o (100, 18, 8, 2) 1
set a is the Petersen graph, which is class 2. The complement of the Petersen graph is the
triangular graph T (5) which is class 1 by Alspach’s result [1]. Also Case h and one of the
graphs of Case e is a triangular graph and therefore class 1. For the parameter sets f , m
and o there is a unique SRG, the so called Lattice graph. This SRG belongs to the Latin
square graphs, and by Corollary 2.4 the graph is class 1, and so is its complement. Case l is
the Gewirtz graph, which is class 1 by Lemma 2.2, as we saw in Section 2. All other graphs
are tested by computer (we actually tested all graphs in Table 1 and their complements).
Using SageMath [22], we wrote a computer program that searches for an edge coloring in
a k-regular graph with k colors. In each step we look (randomly) for a perfect matching,
remove all its edges and continue until the remaining graph has no perfect matching. If
there are still edges left we start again. We run this algorithm repeatedly until an edge
coloring is found. The code for this project is made freely available in a public GitHub
repository which can be found at [14]. By use of this approach we found a 1-factorization
S. M. Cioabă, K. Guo and W. H. Haemers: The chromatic index of strongly regular graphs 193
in all graphs of Table 1, and in their complements, except for the Petersen graph. Thus we
found:
Theorem 4.1. With the single exception of the Petersen graph, a primitive SRG of even
order and degree at most 18 is class 1 and so is its complement.
For the description of the graphs we used the website of Spence [23]. This website
also contains several SRGs with parameters (50, 21, 8, 9). We also ran the search for these
graphs. All are class 1.
It is surprising that in all cases our straightforward heuristic finds a 1-factorization. The
heuristic is fast. It took about one hour to find a 1-factorization in each of the 32548 SRGs
with parameter set i.
ORCID iDs
Sebastian M. Cioabă https://orcid.org/0000-0001-9983-0212
Krystal Guo https://orcid.org/0000-0001-8776-537X
Willem H. Haemers https://orcid.org/0000-0001-7308-8355
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3-edge-colorable – a new proof’ [Discrete Math. 268 (2003), 325–326], Discrete Math. 287
(2004), 193–194, doi:10.1016/j.disc.2004.07.008.
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 195–197
https://doi.org/10.26493/1855-3974.2334.331
(Also available at http://amc-journal.eu)
A simple and elementary proof of Whitney’s
unique embedding theorem
Gunnar Brinkmann *
Applied Mathematics, Computer Science and Statistics, Ghent University, Belgium
Received 13 May 2020, accepted 12 February 2021, published online 27 October 2021
Abstract
In this note we give a short and elementary proof of a more general version of Whit-
ney’s theorem that 3-connected planar graphs have a unique embedding in the plane. A
consequence of the theorem is also that cubic plane graphs cannot be embedded in a higher
genus with a simple dual. The aim of this paper is to promote a simple and elementary
proof, which is especially well suited for lectures presenting Whitney’s theorem.
Keywords: Polyhedra, graph, embedding.
Math. Subj. Class. (2020): 05C10, 57M60, 57M15
1 Introduction
Whitney’s famous unique embedding theorem has been formulated in various equivalent
forms. One form is that the facial cycles of 3-connected graphs embedded in the plane are
well determined, so that for any two embeddings there is a graph isomorphism between
the duals. Another is (implied by the Jordan-Schönflies Theorem) that any two topological
embeddings of a graph on the sphere can be mapped onto each other by a homeomorphism
of the sphere that maps the two images of a vertex onto each other.
We will formulate the theorem and describe the proof in the language of combinatorial
embeddings in oriented closed surfaces. For the translation to the language of topological
2-cell embeddings, methods from standard books like [1] or [3] can be used.
We interpret each edge {u, v} of an undirected embedded graph G as two directed
edges: e = (u, v) and its inverse e−1 = (v, u). An embedded graph in an oriented closed
surface is a graph where for every vertex u there is a cyclic order of all edges (u, .) (usually
called a rotation). The cyclic ordering defines the orientation around the vertex. We write
*I would like to thank Bojan Mohar for pointing me to the earlier uses of the crossing Jordan curves argument!
E-mail address: gunnar.brinkmann@ugent.be (Gunnar Brinkmann)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
196 Ars Math. Contemp. 20 (2021) 195–197
nx(e) for the next edge in the order around the starting point of a directed edge e. The
inverse graph or mirror image is the graph G−1 with all cyclic orders reversed.
A face in an embedded graph G is a directed cyclic walk e0, . . . , en−1, so that for
0 ≤ i < n we have that nx(e−1i ) = e(i+1) (mod n). We say that the set {e,nx(e)} forms an
angle of G and G−1 if one of them has a face containing e−1,nx(e) as a subsequence. In
this case the other has a face containing nx(e)−1, e. If a face is a simple cyclic walk, we call
the corresponding undirected cycle also a (simple) facial cycle. We consider an embedded
graph G and its mirror image G−1 as equivalent, as the faces have the same sequences of
underlying undirected edges – only in reversed order. The genus of an embedded graph can
be computed by the Euler formula using the number v of vertices, e of (undirected) edges,
and f of faces as γ(G) = 2−(v−e+f)2 . We refer to a (not necessarily embedded) graph that
can be embedded with genus 0 as planar and to an embedded graph with genus 0 as plane.
With this notation and concept of equivalence Whitney’s famous theorem [5] can be
shortly stated as:
A 3-connected planar graph has an – up to equivalence – unique embedding
in the plane.
We will prove a stronger theorem using the concept of polyhedral embedding that re-
quires some important properties of polyhedra – that is plane 3-connected graphs – but
allows higher genera. It is an easy consequence of the Jordan Curve Theorem that polyhe-
dra are polyhedral embeddings.
Definition 1.1. A polyhedral embedding of a graph G = (V,E) in an oriented closed
surface is an embedding so that each facial walk is a simple cycle and the intersection of
any two faces is either empty, a single vertex or a single edge.
For cubic embedded graphs this is equivalent to the dual graph being simple.
The argument of crossing Jordan curves that we will use in the proof was first published
by Thomassen in [4], but also known to Robertson and later used by Mohar and Robertson
in [2]. See also Theorem 5.7.1 in [3]. In fact, in [4] the argument was used to prove that
3-connected planar graphs embedded with genus g > 0 have facewidth at most 2. Together
with Whitney’s theorem, this implies Theorem 1.2. We will give every detail of the proof in
order to make it well suited for lectures presenting Whitney’s theorem, but the arguments
are exactly the same arguments of crossing Jordan curves that Thomassen used – only that
here the planar case, that is: Whitney’s theorem – is included too.
Theorem 1.2. A 3-connected planar graph has an – up to equivalence – unique polyhedral
embedding.
Proof. Let G be a plane embedding of a 3-connected planar graph with mirror image G−1
and let G′ be an embedding different from these two. We say that a vertex of G′ has type 1
if the order is the same as in G, type −1 if it is the same as in G−1 and type 2 otherwise.
As G′ is neither G nor G−1, G′ has a vertex of type 2 or an edge with one vertex of type 1
and one vertex of type −1.
Assume first that there is a vertex v of type 2. Let e0, . . . , ed−1 be the order of edges
around v in G′. If {e0, e1} is not an angle of G, we take this set of edges. Otherwise
assume w.l.o.g. that e1 = nx(e0) in G and let j be minimal so that in G we have nx(ej) ̸=
e(j+1) (mod d). As in G−1 we have nx(ej) = ej−1, the edge e(j+1) (mod d) follows ej
neither in G nor in G′, so {ej , e(j+1) (mod d)} is not an angle in G. W.l.o.g. assume j = 0.
G. Brinkmann: A simple and elementary proof of Whitney’s unique embedding theorem 197
So the order around v in G is e0, ei1 , . . . , eij , e1, eij+1 , . . . , eid−2 with 1 ≤ j < d − 2
and assume w.l.o.g. that ed−1 ∈ {eij+1 , . . . , eid−2}. Let y = max{i1, . . . , ij}, so y < d−1
and (y+1) ∈ {ij+1, . . . , id−2}, which implies that {ey, ey+1} is an angle of G′ with ey ∈
{ei1 , . . . , eij} and ey+1 ∈ {eij+1 , . . . , eid−2}. Let F be the facial cycle in G′ containing
the angle {e0, e1} and F ′ be the facial cycle containing {ey, ey+1}. We have F ̸= F ′ as
otherwise the faces would not be simple cycles. In G these cycles are not facial cycles, but
two Jordan curves crossing each other in v. Due to the Jordan curve theorem, there must be
a second crossing, so F, F ′ are two facial cycles that have at least two vertices in common
that are not endpoints of a common edge – a contradiction to G′ being polyhedral.
Assume now that all vertices are of type 1 or type −1. Then there is an edge e0 with
one vertex of type 1 and one of type −1. Assume that in G the orientation around the type
1 vertex of e0 is e0, e1, . . . , ed and around the type −1 vertex it is e−10 , e′1, . . . , e′d′ , so in G′
it is e0, e1, . . . , ed resp. e′d′ , e
′
d′−1, . . . , e
−1
0 . In G
′ there is a face F containing e−1d , e0, e
′
d′
and another face F ′ containing e′1
−1
, e−10 , e1. In G the corresponding cycles are again no
facial cycles but Jordan curves crossing each other (with one common edge), so like in
the first case we get a contradiction from the fact that there must be a second intersection
between F and F ′.
As plane embeddings of 3-connected graphs are all polyhedral, this also implies Whit-
ney’s theorem, but there are also other consequences that are worth mentioning. They
follow already from Theorem 8.1 in [4]. Note that for graphs with 1- or 2-cut there are no
polyhedral embeddings in any closed orientable surface.
Corollary 1.3.
• There are no polyhedral embeddings of planar graphs in any orientable surface but
the plane.
• There are no embeddings of cubic planar graphs with a simple dual in any orientable
surface but the plane.
ORCID iD
Gunnar Brinkmann https://orcid.org/0000-0003-4168-0877
References
[1] J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley-Interscience Series in Discrete
Mathematics and Optimization, John Wiley & Sons, New York, 1987.
[2] B. Mohar and N. Robertson, Planar graphs on nonplanar surfaces, J. Comb. Theory Ser. B 68
(1996), 87–111, doi:10.1006/jctb.1996.0058.
[3] B. Mohar and C. Thomassen, Graphs on Surfaces, Johns Hopkins Studies in the Mathematical
Sciences, Johns Hopkins University Press, Baltimore, Maryland, 2001.
[4] C. Thomassen, Embeddings of graphs with no short noncontractible cycles, J. Comb. Theory Ser.
B 48 (1990), 155–177, doi:10.1016/0095-8956(90)90115-g.
[5] H. Whitney, 2-isomorphic graphs, Amer. J. Math. 55 (1933), 245–254, doi:10.2307/2371127.

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 199–208
https://doi.org/10.26493/1855-3974.2043.fbb
(Also available at http://amc-journal.eu)
The average genus for bouquets of
circles and dipoles
Jinlian Zhang
School of Mathematics and Statistics, Hunan University of Finance and Economics,
Changsha, P. R. China
Xuhui Peng *
MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University,
Changsha, P. R. China
Yichao Chen †
Department of Mathematics and Physics, Suzhou University of Science and Technology,
Suzhou, P. R. China
Received 9 July 2019, accepted 6 February 2021, published online 29 October 2021
Abstract
The bouquet of circles Bn and dipole graph Dn are two important classes of graphs in
topological graph theory. For n ≥ 1, we give an explicit formula for the average genus
γavg(Bn) of Bn. By this expression, one easily sees γavg(Bn) = n−lnn−γ+1−ln 22 + o(1),
where γ is the Euler-Mascheroni constant. Similar results are obtained for Dn. Our method
mainly depends on the technique of generating series and the knowledge in ordinary differ-
ential equations.
Keywords: Average genus, bouquet of circles, dipole, ordinary differential equation.
Math. Subj. Class. (2020): 05C10
*Corresponding author. Xuhui Peng was supported by NNSFC (No. 12071123), the Scientific Research
Project of Hunan Province Education Department (No. 20A329) and the Construct Program of the Key Disci-
pline in Hunan Province.
†Yichao Chen was supported by NNSFC under Grant No. 11471106.
E-mail addresses: jinlian916@hnu.edu.cn (Jinlian Zhang), xhpeng@hunnu.edu.cn (Xuhui Peng),
ycchen@hnu.edu.cn (Yichao Chen)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
200 Ars Math. Contemp. 20 (2021) 199–208
1 Introduction and main results
A graph G = (V (G), E(G)) is permitted to have both loops and multiple edges. An
embedding of a graph G into an orientable surface Ok is a cellular embedding, i.e., the
interior of every face is homeomorphic to an open disc. The subscript in Ok is the genus of
the orientable surface Ok, for k ≥ 0. We denote the number of cellular embeddings of G
on the surface Ok by gk(G), where, by the number of embeddings, we mean the number of
equivalence classes under ambient isotopy. The genus polynomial of a graph G is given by
ΓG(x) =
∑
k≥0
gk(G)x
k.
This sequence {gk(G), k = 0, 1, 2, . . .} is called the genus distribution of the graph
G. For a graph G, it is well known that the total number of cellular embeddings is∏
v∈V (G)(dG(v) −1)!, where dG(v) is the degree of the vertex v in G. For example,
see [13, Chapter 3]. Hence,
ΓG(1) =
∑
k≥0
gk(G) =
∏
v∈V (G)
(dG(v)− 1)!. (1.1)
The average genus γavg(G) of the graph G is the expected value of the genus random
variable, over all labeled 2-cell orientable embeddings of G, using the uniform distribution.
In other words, the average genus of G is
γavg(G) =
Γ′G(1)
ΓG(1)
=
∞∑
k=0
k · gk(G)
ΓG(1)
.
The study of the average genus of a graph began by Gross and Furst [9], and was much
further developed by Chen and Gross [1, 2, 3]. Two lower bounds were obtained in [4] for
the average genus of two kinds of graphs. In [19], Stahl gave the asymptotic result for the
average genus of linear graph families. The exact values for the average genus of small-
order complete graphs, closed-end ladders, and cobblestone paths were derived by White
[22]. More references are the following: [5, 10, 15, 17, 20] etc. For a general background
in topological graph theory, we refer the reader to see Gross and Tucker [13] or White [21].
One of the purposes of the paper is to give an explicit expression of the average genus
for a bouquet of circles. By a bouquet of circles, or more briefly, a bouquet, we mean a
graph with one vertex and some self-loops. In particular, the bouquet with n self-loops is
denoted by Bn. Figure 1 shows the graphs B1, B2, B3. The bouquets {Bn, n ≥ 1} are
very important graphs in topological graph theory. First, since any connected graph can be
reduced to a bouquet by contracting a spanning tree to a point, bouquets are fundamental
building blocks of topological graph theory. Second, as shown in [8, 12], Cayley graphs
and many other regular graphs are covering spaces of bouquets.
For the genus distribution of Bn, Gross, Robbins and Tucker [11] proved that the num-
bers gk(Bn) of embeddings of the Bn in an oriented surface of genus k satisfy the following
recurrence for n > 2,
(n+ 1)gk(Bn) = 4(2n− 1)(2n− 3)(n− 1)2(n− 2)gk−1(Bn−2)
+ 4(2n− 1)(n− 1)gk(Bn−1)
(1.2)
J. Zhang et al.: The average genus for bouquets of circles and dipoles 201
1B 2B 3B
Figure 1: The bouquets B1, B2, and B3.
with initial conditions
gk(B0) = 1 for k = 0 and gk(B0) = 1 for k > 0,
gk(B1) = 1 for k = 0 and gk(B1) = 1 for k > 0.
(1.3)
With the aid of an edge-attaching surgery technique, the total embedding polynomial of Bn
was computed in [14]. Stahl [18] also did some research on the average genus of Bn. By
[18, Theorem 2.5] and the definition of Euler-Mascheroni constant, one easily sees that
lim
n→∞
(
γavg(Bn)−
(
n+ 1
2
− 1
2
2n∑
k=1
1
k
))
= 0. (1.4)
To achieve this, Stahl made many accurate estimates on the unsigned Stirling numbers
s(n, k) of the first kind. In this paper, using knowledge in ordinary differential equations
and Taylor’s formula, we derive an explicit expression of γavg(Bn). By this expression,
(1.4) follows immediately. Our methods are totally different from that in [18] and we do
not need to make estimates on s(n, k). In Section 2, we will give the computation of
γavg(Bn) in detail.
A dipole with n edges, denoted by Dn, has two vertices joined by n edges. Figure 2
shows the graphs D1, D2, D3.
1D 2D 3D
Figure 2: The dipoles D1, D2, and D3.
Another purpose of this paper is to give an explicit expression of the average genus
for dipole Dn. The dipole, like the bouquet, is useful as a voltage graph. See [21] for
example. Moreover, hypermaps correspond with the 2-cell embeddings of the dipole. The
genus distribution of Dn is given by [14] and [16].
In Lemma 2.1 below, we obtain the following recurrence relation for γavg(Bn)
(n+ 1)γavg(Bn) = 2γavg(Bn−1) + (n− 1)
(
γavg(Bn−2) + 1
)
. (1.5)
The most popular way to deal with sequences of numbers is to manipulate infinite
series that “generate” those sequences. For instance, see [6, 7]. We apply this method to
202 Ars Math. Contemp. 20 (2021) 199–208
the calculation of γavg(Bn). Multiplying both sides of (1.5) by tn and summing on n ≥
1, the generating function u(t) =
∑
n≥1 γavg(Bn)t
n will satisfy an ordinary differential
equation. We solve this differential equation with the aid of a computer system and find an
explicit expression for γavg(Bn) by expanding u(t) as a power series in t. The calculation
of γavg(Dn) is similar to that in γavg(Bn). But the processes are more complicated, so we
still give their details in Section 3.
2 The average genus of Bn
We begin by proving the following lemma.
Lemma 2.1. The following recurrence relation holds for the average genus γavg(Bn)
of Bn
(n+ 1)γavg(Bn) = 2γavg(Bn−1) + (n− 1)
(
γavg(Bn−2) + 1
)
(2.1)
with initial conditions γavg(B1) = 0, γavg(B2) = 13 .
Proof. Multiplying both sides of (1.2) by xk and summing on k ≥ 0, it holds that∑
k≥0
(n+ 1)gk(Bn)x
k =
∑
k≥0
4(2n− 1)(2n− 3)(n− 1)2(n− 2)gk−1(Bn−2)xk
+
∑
k≥0
4(2n− 1)(n− 1)gk(Bn−1)xk.
(2.2)
Hence, the genus polynomial ΓBn(x) satisfies the following recurrence
(n+ 1)ΓBn(x) = 4(2n− 1)(2n− 3)(n− 1)2(n− 2) · x · ΓBn−2(x)
+ 4(2n− 1)(n− 1)ΓBn−1(x).
(2.3)
Differentiating both sides of (2.3) and taking x = 1 lead to
(n+ 1)Γ′Bn(1) = 4(2n− 1)(2n− 3)(n− 1)
2(n− 2) · Γ′Bn−2(1)
+ 4(2n− 1)(2n− 3)(n− 1)2(n− 2) · ΓBn−2(1) + 4(2n− 1)(n− 1)Γ′Bn−1(1).
Applying (1.1) to the graph Bn yields ΓBn(1) = (2n − 1)!. Dividing both sides of the
above equality by ΓBn(1), by the definition of average genus, one arrives at
(n+ 1)γavg(Bn) = 2γavg(Bn−1) + (n− 1)
(
γavg(Bn−2) + 1
)
.
A direct calculation gives rise to γavg(B1) = 0 and γavg(B2) = 13 . The proof is com-
pleted.
The main purpose of this section is to prove the following theorem.
Theorem 2.2. The average genus of Bn is given by
γavg(Bn) =
n+ 1
2
−
n−1∑
m=0
1 + (−1)m
2(m+ 1)
− 1 + (−1)
n
4(n+ 1)
. (2.4)
In particular, we have
γavg(Bn) =
n− lnn− γ + 1− ln 2
2
+ o(1),
where γ ≈ 0.5772 is the Euler-Mascheroni constant.
J. Zhang et al.: The average genus for bouquets of circles and dipoles 203
Proof. For n ≤ 0, we define γavg(Bn) = 0 so that (2.1) holds for any integer n ≥ 1. For
the simplicity of writing, we use an to denote γavg(Bn) in the proof. Multiplying both
sides of (2.1) by tn and summing on n ≥ 1, we obtain∑
n≥1
(n+ 1)ant
n = 2
∑
n≥1
an−1t
n +
∑
n≥1
(n− 1)(an−2 + 1)tn. (2.5)
Let u(t) =
∑
n≥1 ant
n. Then, with the help of (2.5), we obtain(
t ·
∑
n≥1
ant
n
)′
= 2t ·
∑
n≥1
an−1t
n−1 +
∑
n≥1
(n− 2)an−2tn +
∑
n≥1
an−2t
n +
∑
n≥1
(n− 1)tn
= 2tu(t) + t3
∑
n≥1
(n− 2)an−2tn−3 + t2u(t) + t2 ·
(∑
n≥2
tn−1
)′
,
that is
(tu(t))′ = 2tu(t) + t3
∑
n≥3
(n− 2)an−2tn−3 + t2u(t) + t2
( t
1− t
)′
= 2tu(t) + t3
∑
n≥1
nant
n−1 + t2u(t) + t2
( t
1− t
)′
= 2tu(t) + t3u′(t) + t2u(t) + t2
( t
1− t
)′
,
which implies that u(t) satisfies the following equation
(t− t3)u′(t) + (1− 2t− t2)u(t) = t
2
(1− t)2
(2.6)
with initial condition u(0) = 0. Since the above equation is a first order linear differential
equation, we can solve it directly and obtain its solution:
u(t) =
−
(
t2 − 1
)
ln(1− t) +
(
t2 − 1
)
ln(t+ 1) + 2t
4(t− 1)2t
.
Denote
u1(t) =
1
2(t− 1)2
, u2(t) = −
(t+ 1) ln(1− t)
4(t− 1)t
, u3(t) =
(t+ 1) ln(t+ 1)
4(t− 1)t
.
Then, clearly, u(t) = u1(t) + u2(t) + u3(t). Using Taylor’s formula, we get
u1(t) =
∑
n≥0
n+ 1
2
tn (2.7)
and
u2(t) =
1
4
(1 + t) · 1
1− t
· ln(1− t)
t
=
1
4
(1 + t) ·
∑
ℓ≥0
tℓ ·
∑
m≥0
(
− 1
m+ 1
tm
)
204 Ars Math. Contemp. 20 (2021) 199–208
=
1
4
(1 + t) ·
∑
n≥0
n∑
m=0
(
− 1
m+ 1
)
tn =
∑
n≥0
bnt
n, (2.8)
where b0 = − 14 and bn =
1
4
[∑n
m=0(−
1
m+1 ) +
∑n−1
m=0(−
1
m+1 )
]
, n ≥ 1. Also by the
Taylor’s formula,
u3(t) = −
1
4
(1 + t) · 1
1− t
· ln(1 + t)
t
= −1
4
(1 + t) ·
∑
ℓ≥0
tℓ ·
∑
m≥0
(−1)m
m+ 1
tm
= −1
4
(1 + t) ·
∑
n≥0
n∑
m=0
(−1)m
m+ 1
tn =
∑
n≥0
cnt
n, (2.9)
where c0 = − 14 and
cn = −
1
4
[ n∑
m=0
(−1)m
m+ 1
+
n−1∑
m=0
(−1)m
m+ 1
]
, n ≥ 1.
It follows from (2.7) – (2.9) that
an =
n+ 1
2
+ bn + cn =
n+ 1
2
+
1
4
[ n∑
m=0
(
− 1
m+ 1
)
+
n−1∑
m=0
(
− 1
m+ 1
)]
− 1
4
[ n∑
m=0
(−1)m
m+ 1
+
n−1∑
m=0
(−1)m
m+ 1
]
,
which yields (2.4). In view of
γ = lim
n→+∞
[ n∑
m=0
1
m+ 1
− lnn
]
and lim
n→+∞
n−1∑
m=0
(−1)m
m+ 1
= ln 2, (2.10)
we complete the proof of (2.2).
3 The average genus of Dn
Our first purpose is to show the following lemma.
Lemma 3.1. The following recurrence relation holds for the average genus γavg(Dn)
of Dn
n(n+ 2)γavg(Dn+1) = (2n+ 1)γavg(Dn) + (n
2 − 1) · γavg(Dn−1) + n2 (3.1)
with initial conditions γavg(D1) = γavg(D2) = 0.
Proof. By [16, Theorem 5.2], we obtain
(n+2)gk(Dn+1) = n(2n+1)gk(Dn) +n
3(n− 1)2gk−1(Dn−1)−n(n− 1)2gk(Dn−1).
Applying (1.1) to the graph Dn+1 yields ΓDn+1(1) = (n!)
2. Following the lines in the
proof of Lemma 2.1, we derive the recurrence relation (3.1).
The initial conditions γavg(D1) = γavg(D2) = 0 are due to a direct calculation. The
proof is finished.
J. Zhang et al.: The average genus for bouquets of circles and dipoles 205
The main purpose of this section is to prove the following theorem.
Theorem 3.2. γavg(D1) = γavg(D2) = 0 and for n ≥ 3, we have
γavg(Dn) = n
[
1
2
n+1∑
m=4
(−1)m(4m2 − 12m+ 6)
(m− 3)(m− 2)(m− 1)m
+
1
6
]
− 1
2
n+1∑
m=1
1
m
−
n+1∑
m=4
(−1)m(2m2 − 6m+ 3)
(m− 3)(m− 1)m
+
7
12
.
(3.2)
In particular, we have
γavg(Dn) =
n− lnn− γ
2
+ o(1), (3.3)
where γ ≈ 0.5772 is the Euler-Mascheroni constant.
Proof. First, we give a proof of (3.2). For the simplicity of writing, we use an to denote
γavg(Dn) in the proof. Let u(t) =
∑
n≥1 ant
n−3 =
∑
n≥2 an+1t
n−2. Multiplying both
sides of (3.1) by tn−2 and summing on n ≥ 2, we obtain∑
n≥2
n(n+ 2)an+1t
n−2 =
∑
n≥2
(2n+ 1)ant
n−2
+
∑
n≥2
(n2 − 1)an−1tn−2 +
∑
n≥2
n2tn−2.
(3.4)
Since
u′(t) =
∑
n≥2
(n− 2)an+1tn−3,
u′′(t) =
∑
n≥2
(n− 2)(n− 3)an+1tn−4,
it follows that∑
n≥2
n(n+ 2)an+1t
n−2 =
∑
n≥2
[
(n− 2)(n− 3) + 7(n− 2) + 8
]
an+1t
n−2
= t2u′′(t) + 7tu′(t) + 8u(t),∑
n≥2
(2n+ 1)ant
n−2 =
∑
n≥2
(2n+ 3)an+1t
n−1 =
∑
n≥2
(
2(n− 2) + 7
)
an+1t
n−1
= 2t2u′(t) + 7tu(t),∑
n≥2
(n2 − 1)an−1tn−2 =
∑
n≥4
(n2 − 1)an−1tn−2 =
∑
n≥2
(n2 + 4n+ 3)an+1t
n
=
∑
n≥2
[
(n− 2)(n− 3) + 9(n− 2) + 15
]
an+1t
n
= t4u′′(t) + 9t3u′(t) + 15t2u(t),∑
n≥2
n2tn−2 =
∑
n≥2
n(n− 1)tn−2 +
∑
n≥2
ntn−2 = v′′(t) +
∑
n≥0
ntn−2 − t−1
= v′′(t) +
v′(t)
t
− t−1 = 3t− 4− t
2
(t− 1)3
,
206 Ars Math. Contemp. 20 (2021) 199–208
where v(t) =
∑
n≥0 t
n, v′(t) =
∑
n≥0 nt
n−1, v′′(t) =
∑
n≥0 n(n− 1)tn−2. Substituting
the above equalities into (3.4), u(t) satisfies the following second order linear differential
equation
(t2 − t4)u′′(t) + (7t− 2t2 − 9t3)u′(t) + (8− 7t− 15t2)u(t) = 3t− 4− t
2
(t− 1)3
with initial conditions u(0) = a3 = γavg(D3) = 12 , u
′(0) = a4 = γavg(D4) =
5
6 .
With the help of a computer algebra systems, the solution of the above equation is
u(t) =
1
4(t− 1)t2
+
w(t)
4(t− 1)2t4
, (3.5)
where
w(t) = −t3 + 2t3 ln(t+ 1) + 3t2 − 2t2 ln(t+ 1)
− 2t ln(1− t)− 2t ln(t+ 1) + 2 ln(1− t) + 2 ln(t+ 1).
By Taylor’s formula, we get
1
4(t− 1)t2
=
∑
m≥−2
(
− 1
4
)
tm,
w(t) = t2 − t3
+
∑
m≥4
2
(
4(−1)mm2 +m2 − 12(−1)mm− 5m+ 6(−1)m + 6
)
(m− 3)(m− 2)(m− 1)m
tm,
1
4(t− 1)2t4
=
∑
m≥−4
m+ 5
4
tm.
Therefore, comparing the coefficients of tn−3 of the both sides of (3.5) gives
an = −
1
4
+
n
4
− n− 1
4
+
n+1∑
m=4
2
(
4(−1)mm2 +m2 − 12(−1)mm− 5m+ 6(−1)m + 6
)
(m− 3)(m− 2)(m− 1)m
· n−m+ 2
4
=
n
2
n+1∑
m=4
[
(−1)m(4m2 − 12m+ 6)
(m− 3)(m− 2)(m− 1)m
+
(m2 − 5m+ 6)
(m− 3)(m− 2)(m− 1)m
]
−
n+1∑
m=4
(−1)m(4m2 − 12m+ 6) + (m2 − 3m) + (−2m+ 6)
(m− 3)(m− 2)(m− 1)m
· m− 2
2
=
n
2
n+1∑
m=4
(−1)m(4m2 − 12m+ 6)
(m− 3)(m− 2)(m− 1)m
+
n
2
n+1∑
m=4
1
(m− 1)m
− 1
2
n+1∑
m=4
(−1)m(4m2 − 12m+ 6)
(m− 3)(m− 1)m
− 1
2
n+1∑
m=4
1
m− 1
+
n+1∑
m=4
1
m(m− 1)
J. Zhang et al.: The average genus for bouquets of circles and dipoles 207
=
n
2
n+1∑
m=4
(−1)m(4m2 − 12m+ 6)
(m− 3)(m− 2)(m− 1)m
+
n
2
(1
3
− 1
n+ 1
)
− 1
2
n+1∑
m=4
(−1)m(4m2 − 12m+ 6)
(m− 3)(m− 1)m
− 1
2
n+1∑
m=1
1
m
+
3
4
+
1
2(n+ 1)
+
(1
3
− 1
n+ 1
)
which yields the desired result (3.2).
Now we are in a position to prove (3.3). Using the software Mathematica or series
theory, one has
n+1∑
m=4
(−1)m(4m2 − 12m+ 6)
(m− 3)(m− 2)(m− 1)m
=
2
3
+ o
( 1
n
)
(3.6)
and
n+1∑
m=4
(−1)m(2m2 − 6m+ 3)
(m− 3)(m− 1)m
=
7
12
+ o(1). (3.7)
Combining (3.6) – (3.7), (2.10) and (3.2), we complete the proof of (3.3).
4 Some remarks
Bouquets and dipoles are two important classes of graphs in topological graph theory. Their
average genera are of independent interest. In this paper, we obtain explicit formulas for
γavg(Bn) and γavg(Dn). By Theorems 2.2 and 3.2, we have the following relation between
γavg(Bn) and γavg(Dn),
γavg(Bn) = γavg(Dn) +
1− ln 2
2
+ o(1).
It follows that the difference of γavg(Bn) and γavg(Dn) tends to the constant 1−ln 22 when
n tends to infinity.
Since both Bn and Dn are upper-embeddable, the maximum genera of Bn and Dn are⌊
n
2
⌋
and
⌊
n−1
2
⌋
, respectively. Recall that the minimum genera of Bn and Dn equal 0.
Therefore, also by Theorems 2.2 and 3.2, we have
lim
n→∞
γavg(Bn)
⌊n2 ⌋
= 1 and lim
n→∞
γavg(Dn)
⌊n−12 ⌋
= 1.
This implies that the average genus of Bn (Dn) is closer to the maximum genus than to the
minimum genus.
ORCID iDs
Xuhui Peng https://orcid.org/0000-0002-2443-4896
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 209–222
https://doi.org/10.26493/1855-3974.2465.571
(Also available at http://amc-journal.eu)
Well-totally-dominated graphs*
Selim Bahadır
Department of Mathematics, Ankara Yıldırım Beyazıt University, Ankara, Turkey
Tınaz Ekim
Department of Industrial Engineering, Boğaziçi University, İstanbul, Turkey
Didem Gözüpek
Department of Computer Engineering, Gebze Technical University, Kocaeli, Turkey
Received 22 October 2020, accepted 19 February 2021, published online 29 October 2021
Abstract
A subset of vertices in a graph is called a total dominating set if every vertex of the
graph is adjacent to at least one vertex of this set. A total dominating set is called minimal
if it does not properly contain another total dominating set. In this paper, we study graphs
whose all minimal total dominating sets have the same size, referred to as well-totally-
dominated (WTD) graphs. We first show that WTD graphs with bounded total domination
number can be recognized in polynomial time. Then we focus on WTD graphs with total
domination number two. In this case, we characterize triangle-free WTD graphs and WTD
graphs with packing number two, and we show that there are only finitely many planar
WTD graphs with minimum degree at least three. Lastly, we show that if the minimum
degree is at least three then the girth of a WTD graph is at most 12. We conclude with
several open questions.
Keywords: Total domination, well-totally-dominated graphs, minimal total dominating sets.
Math. Subj. Class. (2020): 05C69
*This work has been supported by the Scientific and Technological Research Council of Turkey (TUBITAK)
under grant no. 118E799. The work of Didem Gözüpek was also supported by the BAGEP Award of the Science
Academy of Turkey.
E-mail addresses: sbahadir@ybu.edu.tr (Selim Bahadır), tinaz.ekim@boun.edu.tr (Tınaz Ekim),
didem.gozupek@gtu.edu.tr (Didem Gözüpek)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
210 Ars Math. Contemp. 20 (2021) 209–222
1 Introduction
Total domination in graphs has been extensively studied in the literature (see [15]) and has
numerous applications. For instance, consider a computer network where a core group of
file servers has the ability to communicate directly with every computer outside the core
group. Moreover, each file server is directly linked to at least one other backup file server
where duplicate information is stored. This core group of servers corresponds to a total
dominating set in the graph representing the computer network. Another application area
is a specific committee selection mechanism such that every non-member of the committee
knows at least one member of the committee and every member of the committee knows
at least one other member of the committee to avoid feelings of isolation and thus enhance
cooperation (see [14]).
Let G be a graph with no isolated vertices. A subset S of V (G) is called a total dom-
inating set (TDS) of G if every vertex in G is adjacent to at least one element in S. A
total dominating set is minimal if it contains no other TDS of G. The minimum size of a
total dominating set of a graph G is called the total domination number and denoted by
γt(G), while the maximum size of a minimal total dominating set is called the upper total
domination number and denoted by Γt(G). G is called well-totally-dominated (WTD) if
every minimal TDS of G is of the same size, that is, γt(G) = Γt(G). WTD graphs with
γt = k are denoted by WTD(k).
Given a graph, computing its total domination number and its upper total domination
number are NP-hard in general [6, 18] and already NP-hard even in specific graph classes
such as bipartite graphs, comparability graphs and claw-free graphs [15]. One way to deal
with such a problem is to consider “trivial” instances where these two paramaters have the
same value. Examples of graph classes defined in this way in the literature include well-
covered graphs (whose all maximal independent sets have the same size), well-dominated
graphs (whose all minimal dominating sets have the same size), and equimatchable graphs
(whose all maximal matchings have the same size). Structural properties of each one of
these graph classes have been studied extensively in the literature. In this paper, we take
the same approach for the total dominating sets. Works on total domination in the literature
mostly focused on the relation of the total domination number with other graph parame-
ters and characterized graphs with total domination number being equal to an upper bound
(e.g. [3, 4]). Inequalities relating the total domination number to other domination param-
eters and characterization of graphs that tightly attain these bounds have also been studied
(see [1, 16]).
Clearly, if the total domination number and the upper total domination number are
polynomial time solvable for a given class of graphs, then the recognition of WTD graphs
belonging to this class of graphs is polynomial. However, the complexity of recognizing
WTD graphs in general is unknown. In such a situation, a classical approach consists in
studying the structure of WTD graphs in restricted graph classes and providing structural
characterizations along with efficient recognition algorithms whenever possible.
WTD graphs were initially introduced in [12], where WTD cycles and paths are char-
acterized and several constructions of WTD trees are given. They also proved that a WTD
graph with minimum degree at least two has girth at most 14. The work in [7] focused
on the composition and decomposition of WTD trees and proved that any WTD tree can
be constructed from a family of three small trees. To the best of our knowledge, [12] and
[7] are the only work on WTD graphs. A graph class resembling WTD graphs is well-
dominated graphs, which are graphs whose minimal dominating sets have the same size. It
S. Bahadır et al.: Well-totally-dominated graphs 211
is known that well-dominated graphs form a proper subset of well-covered graphs [8]. We
note that well-covered graphs are graphs whose maximal independent sets have the same
size and there is a rich literature about them (see [13, 19]). Well-dominated graphs were in-
troduced by Finbow et al. [8], who provided a characterization of bipartite well-dominated
graphs and well-dominated graphs with girth at least 5. Characterizations of these graphs
within other graph classes were also obtained [9, 10, 17, 20]. Although their definitions re-
semble each other, there is not a containment relationship between WTD graphs and well-
dominated graphs. For instance, a cycle on six vertices is WTD but not well-dominated,
whereas the graph T10 described in [17] is well-dominated but not WTD.
It follows from the previous studies on WTD graphs that we do not know much about
their structure. In this paper, we investigate the study of WTD graphs from a structural
point of view. We first study WTD graphs with bounded total domination number. We
prove in Section 2 that the recognition of WTD graphs with total domination number k is
solvable in polynomial time for every positive integer k. We then focus on WTD graphs
with total domination number 2, referred to as WTD(2) graphs in Section 3. We char-
acterize triangle-free WTD(2) graphs and WTD(2) graphs with packing number 2 (or
equivalently of diameter 3). We also show that there is a finite number of planar WTD(2)
graphs with minimum degree at least 3. Subsequently, we study the girth of WTD graphs in
Section 4. In particular, building on a result in [12], we prove that WTD graphs with mini-
mum degree at least three have girth at most 12. Finally, we discuss several open research
directions.
2 WTD graphs with bounded total domination number
Recall that the complexity of recognizing WTD graphs is unknown. In this section, we
show that for any positive integer k, WTD(k) graphs can be recognized in polynomial time.
To this end, we will use an equivalent description of WTD(k) graphs using transversal
hypergraphs. Let us first introduce necessary definitions. A hypergraph H is a pair H =
(X,E) where X is a set of elements called vertices, and E is a set of nonempty subsets
of X called hyperedges. Therefore, a hypergraph might have a vertex which belongs to
none of the hyperedges, but cannot have multiple hyperedges. A transversal (or hitting set)
of a hypergraph H = (X,E) is a set T ⊆ X that has nonempty intersection with every
hyperedge of H . A transversal of a collection of sets is a transversal of the hypergraph
whose hyperedges are the given collection. A transversal T is called minimal if no proper
subset of T is a transversal. The transversal hypergraph of H = (X,E) is the hypergraph
H∗ = (X,F ) whose hyperedge set F consists of all minimal transversals of H .
Let G be a graph with no isolated vertex. Let HG be the hypergraph whose vertex set
is V (G) and hyperedges are open neighborhoods of the vertices of G. Let also MTDS(G)
denote the set of all minimal total dominating sets of G.
Lemma 2.1. MTDS(G) consists of hyperedges of the transversal hypergraph of HG.
Proof. Let T be a hyperedge of H∗G, that is a minimal transversal of the set of open neigh-
borhoods of G. This means that T contains a neighbor of every vertex in G, thus it is a total
dominating set. By minimality of the transversal T , it is also a minimal total dominating set
of G. Conversely, let S be a minimal total dominating set of G. Then, every vertex in G is
adjacent to at least one vertex in S. That is, S has a nonempty intersection with every open
neighborhood in G. Therefore, S is a transversal of the hypergraph HG and minimality of
S implies that it is a minimal transversal. Thus, S is a hyperedge of H∗G.
212 Ars Math. Contemp. 20 (2021) 209–222
Proposition 2.2. Let G be a graph. Then, for any minimal transversal T of MTDS(G),
there exists a vertex v in G such that N(v) = T .
Proof. Let MTDS(G) = {A1, . . . , Am}. Since T has nonempty intersection with each
Ai, V (G) \ T contains none of the minimal total dominating sets A1, . . . , Am. Therefore,
V (G) \ T is not a TDS of G, and hence there exists at least one vertex v ∈ V (G) such that
N(v) ∩ (V (G) \ T ) = ∅. Thus, we see that N(v) ⊆ T . Suppose that N(v) ̸= T . Then
T \N(v) ̸= ∅ and let u ∈ T \N(v). Since T is a minimal transversal, T \ {u} is disjoint
with at least one of A1, . . . , Am, say A1. As u ∈ T \ N(v), we have N(v) ⊆ T \ {u},
and hence N(v) ∩ A1 = ∅. That is, v is not dominated by A1, which is a contradiction.
Therefore, N(v) = T .
A hypergraph H is said to be Sperner if no hyperedge of H contains another hyperedge.
The following result shows that any finite collection of finite sets which forms a Sperner
hypergraph corresponds to the set of all minimal total dominating sets of a graph.
Proposition 2.3. Let H be a Sperner hypergraph. Then there exists a graph G such that
E(H) = MTDS(G).
Proof. Let E(H) = {A1, . . . , Am} and A = ∪mi=1Ai. Consider a graph with vertex set A
and draw edges between its vertices such that each vertex is adjacent to at least one vertex
in Ai for all i = 1, . . . ,m (for example, draw all possible edges). Then, in accordance with
Proposition 2.2, for each minimal transversal T of H , add a vertex vT to the graph such
that N(vT ) = T . Let G be the resulting graph.
We first show that each Ai is a TDS of G. By construction, every vertex of A is adjacent
to at least one vertex in Ai. Moreover, for every minimal transversal T of A1, . . . , Am we
have T ∩ Ai ̸= ∅, and hence, each vT is dominated by Ai. Therefore, Ai is a TDS for
i = 1, . . . ,m.
We next show that every TDS of G contains at least one of A1, . . . , Am. Let S be a
TDS of G and suppose that Ai ⊈ S for i = 1, . . . ,m. Then, V (G) \ S is a transversal
of A1, . . . , Am, and hence, there exists a minimal transversal T of A1, . . . , Am such that
T ⊆ V (G) \ S. On the other hand, we have N(vT ) = T and thus, we get N(vT )∩ S = ∅,
which contradicts that S is a TDS of G.
Consequently, a set other than A1, . . . , Am can not be a minimal TDS of G. We finally
show that each Ai is a minimal TDS of G. Suppose that Ai is not minimal for some i.
Then, Ai \{x} is still a TDS of G for some x ∈ Ai, and therefore, Aj ⊆ Ai \{x} for some
j, which implies Aj ⊆ Ai contradicting that H is Sperner. Therefore, minimal TDSs of G
are exactly A1, . . . , Am.
Remark 2.4. One can extend G to another graph whose minimal TDSs are A1, . . . , Am
as follows: Let G′ be a graph disjoint from G. Draw edges between the vertices of G′
and A in such a way that every vertex of G′ is adjacent to at least one vertex of Ai for
i = 1, . . . ,m. By following the same arguments, it is easy to check that minimal TDSs of
the resulting graph are A1, . . . , Am.
Notice that any finite collection consisting of distinct sets of size k corresponds to a
Sperner hypergraph and therefore, Proposition 2.3 implies the following result.
Corollary 2.5. For every integer k ≥ 2, WTD(k) is an infinite graph family.
S. Bahadır et al.: Well-totally-dominated graphs 213
The HYPERGRAPH TRANSVERSAL problem is the decision problem that takes as input
two Sperner hypergraphs H and H ′ and asks whether H ′ is the transversal hypergraph H∗
of H .
Theorem 2.6 ([2, 5]). For every positive integer k, the HYPERGRAPH TRANSVERSAL
problem is solvable in polynomial time if all hyperedges of one of the two hypergraphs H
and H ′ are of size at most k.
Theorem 2.6 has the following consequence:
Corollary 2.7 ([11]). For every positive integer k, the following problem is solvable in
polynomial time: Given a Sperner hypergraph H , determine whether all minimal transver-
sals of H are of size k.
The complexity of recognition of WTD graphs with bounded total domination number
can now be derived from Corollary 2.7.
Theorem 2.8. For every positive integer k, the problem of recognizing WTD(k) graphs
can be solved in polynomial time.
Proof. Let G be a graph with no isolated vertices. Consider the hypergraph HG = (V, E),
where E contains the inclusion-minimal elements of {N(v) : v ∈ V }. Observe that HG is
Sperner and that the minimal transversals of HG are exactly the minimal total dominating
sets of G by Lemma 2.1. It follows that G is WTD if and only if all minimal transversals
of HG are of size k. By Corollary 2.7, this condition can be tested in polynomial time.
3 WTD graphs with total domination number two
In this section, we study WTD graphs whose total domination number is 2. We give
complete characterizations of WTD(2) graphs with packing number 2 and triangle-free
WTD(2) graphs. We also show that planar WTD(2) graphs with minimum degree at least
3 have at most 16 vertices.
Let G be a WTD(2) graph. Note that every minimal TDS of G is a pair consisting
of endpoints of an edge of G. Consequently, every WTD(2) graph is connected. We
will call an edge of G whose endpoints is a TDS of G a dominating edge of G. Let Gde
be the graph with vertex set ∪S∈MTDS(G)S (i.e., vertices of G serve as an endpoint of a
dominating edge) and edge set which consists of dominating edges of G. In other words,
Gde is the edge-induced subgraph of G obtained by the dominating edges. See Figure 1 for
an example.
G
x
y
z
t
w
y
z
t
w
Gde
Figure 1: A WTD(2) graph G and the graph Gde obtained by the dominating edges of G.
214 Ars Math. Contemp. 20 (2021) 209–222
Remark 3.1. Notice that the graph Gde and the subgraph of G induced by V (Gde) are not
necessarily the same. In general, Gde is a subgraph of G but not necessarily an induced
subgraph of G with respect to a set of vertices.
A set S is a vertex cover of a graph G if every edge of G has an endpoint from S. Let
MVC(G) denote the set of all minimal vertex covers of the graph G.
Proposition 3.2. Let G be a WTD(2) graph. For every minimal vertex cover S of Gde
there exists a vertex vS in G such that N(vS) = S.
Proof. We notice that every minimal vertex cover S of Gde is a minimal transversal of
MTDS(G). Therefore, by Proposition 2.2 there exists a vertex in G whose neighborhood
is exactly S.
3.1 Characterization of WTD(2) graphs with packing number 2
A set S ⊆ V (G) is called a packing of G if N [u] ∩N [v] = ∅ for every distinct u, v ∈ S.
The packing number ρ(G) is the maximum size of a packing of G. It is well-known that
for any graph G we have ρ(G) ≤ γ(G) ≤ γt(G). Therefore, if γt(G) = 2, then ρ(G) is
either 1 or 2. In this subsection, we provide a characterization of WTD(2) graphs G with
ρ(G) = 2. In particular, this characterization allows us to construct any WTD(2) graph
with ρ(G) = 2.
Let W2 be the set of graphs obtained as follows:
Step 1: Choose a bipartite graph H with no isolated vertices.
Step 2: For every S ∈ MVC(H), choose a new vertex vS and draw edges from vS to
every vertex in S.
Step 3: For each edge uv in H and every w ∈ V (H) \ {u, v}, add the edges wu and/or
wv if needed to make sure w is adjacent to at least one of u and v.
Step 4: Choose a new graph H ′ (might be the empty graph) which is disjoint from the
current graph. Then for each edge uv in H and every w ∈ V (H ′), draw at least
one of the edges wu and wv.
A graph in W2 is given in Figure 2.
H
Step 2
v{x,z}
v{x,t} v{y,z}
v{y,t}
x y
z t
x y
z t
Step 3
v{x,z}
v{x,t} v{y,z}
v{y,t}
x y
z t
Step 4
v{x,z}
v{x,t} v{y,z}
v{y,t}
x y
z t
u1 u2
u3
Figure 2: A graph in W2 obtained by the given process. Bold edges represent the dominat-
ing edges.
S. Bahadır et al.: Well-totally-dominated graphs 215
Lemma 3.3. If a graph G is in W2, then G is a WTD(2) graph with ρ(G) = 2.
Proof. Let G ∈ W2 and H = (U, V,E) be the bipartite graph in the first step of the
construction of G. We first show that the packing number of G is 2. As H has no isolated
vertices, both U and V are minimal vertex covers of H . Thus, the vertices vU and vV
have disjoint closed neighborhoods since N(vU ) = U and N(vV ) = V and hence, we get
ρ(G) ≥ 2. Clearly, by construction, every edge of H is a dominating edge of G. Therefore,
we get γt(G) = 2. Since ρ(G) ≤ γt(G), we obtain ρ(G) ≤ 2 and hence, ρ(G) = 2.
Now let T be a minimal TDS of G other than the edges of H . Then T contains at
most one endpoint of an edge of H because otherwise T contains a TDS, which contradicts
that T is minimal. Therefore, V (H) \ T is a vertex cover of H and hence, it contains a
minimal vertex cover S of H . By construction there exists a vertex vS with N(vS) = S.
As S ⊆ V (H) \ T , we obtain N(vS) ∩ T = ∅, which contradicts that T is a TDS of G.
Consequently, edges of H are the only minimal TDSs of G and hence, G is a WTD(2)
graph and Gde = H .
Lemma 3.4. Let G be a WTD(2) graph with ρ(G) = 2. Then, G is in W2.
Proof. Let {x, y} be a packing with minimum |N [x]|+ |N [y]|. Note that every dominating
edge of G has one endpoint from N(x) and one from N(y) and hence, Gde is a bipartite
graph, say with parts X and Y where X ⊆ N(x) and Y ⊆ N(y).
We next show that X = N(x) and Y = N(y). By symmetry, it suffices to prove
X = N(x). Notice that Gde has no isolated vertices and therefore, X is a minimal vertex
cover of Gde. By Proposition 3.2 there exists a vertex vX satisfying N(vX) = X . Suppose
that X ̸= N(x). Then, we get X ⊂ N(x). Clearly vX ̸= y. Moreover, vX /∈ N(y)
since y /∈ X = N(vX). Thus, we get N [vX ] ∩N [y] = ∅ and hence {vX , y} is a packing
of G. However, we obtain |N [vX ]| + |N [y]| < |N [x]| + |N [y]| since X ⊂ N(x), which
contradicts the definition of the packing {x, y}. Consequently, we get X = N(x) and
hence, we may take vX = x. Similarly, we have Y = N(y) and we may assume vY = y.
Now let S be a minimal vertex cover of Gde. By Proposition 3.2 there exists a vertex vS
satisfying N(vS) = S. If S = X or S = Y , we can take vS to be x or y, respectively, and
in both cases, we have vS /∈ V (Gde). Otherwise, suppose that vS ∈ V (Gde) = X ∪ Y .
Without loss of generality, let vS ∈ X . Then, as X = N(x), we get x ∈ N(vS) =
S ⊆ N(x) ∪N(y), which is a contradiction. Therefore, vS is not a vertex of Gde, that is,
vS ∈ V (G) \ V (Gde).
Finally, we see that one can obtain the graph G by following the procedure in the
definition of W2 with the initial bipartite graph H = Gde.
Combining the results in Lemma 3.3 and Lemma 3.4 gives the following structural
characterization of WTD(2) graphs with ρ(G) = 2. Moreover, by definition of the class
W2, this provides us with a procedure to construct any WTD(2) graph with ρ(G) = 2.
Theorem 3.5. A graph G is WTD(2) with ρ(G) = 2 if and only if G ∈ W2.
Given a graph G, the diameter of G, denoted by diam(G) is the maximum length of
a shortest path between any pair of vertices of G. Let G be a graph such that γt(G) = 2.
Then, it is easy to see that diam(G) ≤ 3. Moreover, whenever γt(G) = 2, we have
diam(G) = 3 if and only if ρ(G) = 2 and therefore, in all the statements in Lemma 3.3,
Lemma 3.4 and Theorem 3.5, the condition ρ(G) = 2 can be replaced with diam(G) = 3.
216 Ars Math. Contemp. 20 (2021) 209–222
Corollary 3.6. A graph G is WTD(2) with diam(G) = 3 if and only if G ∈ W2.
One may attempt to modify the description of W2 graphs in order to describe all
WTD(2) graphs with ρ(G) = 1. In the first step of the process of building a graph in
W2, if one starts with a non-bipartite graph H with no isolated vertices, then the resulting
graph is still WTD(2) but has packing number 1. However, not every WTD(2) graph G
with ρ(G) = 1 can be obtained in this way. For example, consider the graph presented in
Figure 1. To obtain this graph G, in Step 1 one should definitely choose H to be the graph
with vertex set {z, y, t, w} and edge set {zy, yt, tw} which is indeed Gde. However, in
Step 2 if one chooses a new vertex vS for S = {y, w} (which is a minimal vertex cover of
Gde), then the graph G can not be obtained. So, the complete characterization of WTD(2)
graphs with ρ(G) = 1 is left as an open question.
3.2 Triangle-free WTD(2) graphs
In this subsection, we provide characterization of triangle-free WTD(2) graphs.
Lemma 3.7. If G is a triangle-free graph with γt(G) = 2, then G is a bipartite graph and
we have
ρ(G) =
{
1, if G is complete bipartite;
2, otherwise.
Proof. Let uv be a dominating edge of G. Then we have N(u) ∪ N(v) = V (G). As
G is triangle-free, none of two adjacent vertices have a common neighbor. Therefore,
we have N(u) ∩ N(v) = ∅ and also see that both N(u) and N(v) are independent sets.
We consequently obtain that G is a bipartite graph with parts N(u) and N(v). Since
ρ(G) ≤ γt(G) = 2, we have ρ(G) ∈ {1, 2}. Moreover, it is clear that ρ(G) = 1 if and
only if each vertex in N(u) is adjacent to all the vertices in N(v), i.e., G is a complete
bipartite graph.
For a bipartite graph with parts X and Y , define Xu = {x ∈ X : N(x) = Y } and
Yu = {y ∈ Y : N(y) = X}. In other words, Xu (resp. Yu) is the set of vertices in X (resp.
Y ) which are adjacent to every vertex in Y (resp. X). The following result characterizes
all triangle-free WTD(2) graphs.
Theorem 3.8. The following three statements are equivalent:
(i) G is a triangle-free WTD(2) graph.
(ii) G is a bipartite WTD(2) graph.
(iii) G is complete bipartite graph or G is a bipartite graph with parts X and Y such
that there exist vertices a ∈ X \Xu and b ∈ Y \ Yu satisfying N(a) = Yu ̸= ∅ and
N(b) = Xu ̸= ∅.
Proof. By Lemma 3.7 we see that (i) implies (ii). On the other hand, the implication
(iii) ⇒ (i) can be easily verified and hence, the proof finishes if we show that (ii) implies
(iii). Now let G be a bipartite WTD(2) graph, say with parts X and Y . Clearly we will
only consider the case when G is not a complete bipartite graph. By definition of Xu and
Yu, note that every dominating edge of G has one endpoint in Xu ̸= ∅ and one endpoint
in Yu ̸= ∅. Moreover, any edge xy where x ∈ Xu and y ∈ Yu is a dominating edge of G.
S. Bahadır et al.: Well-totally-dominated graphs 217
Therefore, Gde is the subgraph of G induced by Xu∪Yu and it is complete bipartite. Thus,
Gde has only two minimal vertex covers, namely Xu and Yu. Then, definition of a graph
in W2 and Theorem 3.5 imply the existence of the vertices a ∈ X \ Xu and b ∈ Y \ Yu
with N(a) = Yu and N(b) = Xu.
Although a polynomial time recognition algorithm for WTD(2) graphs follows from
Theorem 2.8, the characterization in Theorem 3.8 provides us with a simple linear time
recognition algorithm.
Corollary 3.9. Triangle-free WTD(2) graphs can be recognized in linear time.
Proof. Given a graph G, one can check whether it is a connected bipartite graph and if so,
find its unique bipartition (X,Y ) in linear time (in the number of vertices and edges of G).
Then, sets Xu and Yu can be identified simply by assigning every vertex x ∈ X such that
d(x) = |Y | into Xu, and y ∈ Y such that d(y) = |X| into Yu. According to Theorem 3.8,
G is triangle-free WTD(2) if and only if either Xu = X and Yu = Y (thus, G is complete
bipartite), or the removal of Xu and Yu leaves at least one isolated vertex in each one of X
and Y . Clearly, all these checks take only linear time.
3.3 Planar WTD(2) graphs
In this subsection, we study planar WTD(2) graphs whose minimum degree is at least
three and show that such graphs can have at most sixteen vertices. Throughout this section,
we frequently use the fact that a graph obtained by an edge contraction of a planar graph is
also planar. Recall also that a planar graph contains no K5 or K3,3.
Observation 3.10. Let G be a WTD(2) graph. The vertex obtained by edge contraction
of a dominating edge is a universal vertex in the new graph.
Let ν(G) denote the matching number of a graph G.
Lemma 3.11. Let G be a planar WTD(2) graph. If ν(Gde) ≥ 3, then |V (G)| ≤ 8.
Proof. Suppose that ν(Gde) ≥ 3 and G has at least 9 vertices. Then, G has three inde-
pendent dominating edges, say u1v1, u2v2 and u3v3, and three vertices other than u1, u2,
u3, v1, v2, v3, say w1, w2 and w3. Now contract the edges u1v1, u2v2 and u3v3. In the
resulting graph, new three vertices and w1, w2, w3 contain a K3,3, which contradicts the
planarity.
Lemma 3.12. If G is a WTD(2) graph with δ(G) ≥ 3, then ν(Gde) ≥ 2.
Proof. Let G be a WTD(2) graph with δ(G) ≥ 3. It suffices to show that G has two
independent dominating edges. Let xy be a dominating edge of G. Since the minimum
degree is at least three, each vertex of G has at least one neighbor in V (G) \ {x, y}. There-
fore, V (G) \ {x, y} is a TDS of G and hence, it contains a dominating edge ab since G is
WTD(2). As the dominating edges xy and ab share no vertex, we get ν(Gde) ≥ 2.
Combining the results in Lemmas 3.11 and 3.12 gives the following result.
Proposition 3.13. If G is a planar WTD(2) graph with δ(G) ≥ 3, then ν(Gde) = 2 or
|V (G)| ≤ 8.
218 Ars Math. Contemp. 20 (2021) 209–222
We next study planar WTD(2) graphs whose minimum degree is at least 3 and match-
ing number is 2.
Proposition 3.14. If G is a planar WTD(2) graph with δ(G) ≥ 3 and ν(Gde) = 2, then
|V (G)| ≤ 16.
Proof. Let ab and xy be two independent dominating edges of G and H = G−{a, b, x, y}.
Let H1, . . . ,Hm be the connected components of H and order of Hi be hi for i =
1, . . . ,m. Note that it suffices to show that h1 + · · ·+ hm ≤ 12.
We first prove that each Hi is a path or a singleton. Note that it suffices to show that
maximum degree of H is at most 2 and H contains no cycle. Suppose that a vertex v of
H has three neighbors, say v1, v2, v3, in H . Then contraction of the edges ab and xy gives
rise to a K3,3 with parts {ab, xy, v} and {v1, v2, v3}, which is a contradiction. Therefore,
every vertex in H has at most two neighbors in H . Suppose that H has a cycle, say
v1, v2, . . . , vk. Contract the edge vkvk−1 and denote the new point by vk−1. Then contract
the edge vk−1vk−2 and denote the new point by vk−2 and so on. Follow this process until
we get a triangle v1, v2, v3. Then contracting the edges ab and xy yields a K5 with vertices
ab, xy, v1, v2, v3, which is a contradiction. Thus, H has no cycle and hence, H is a disjoint
union of paths and singletons.
We next show that for every vertex u ∈ H we have |N(u) ∩ {a, b, x, y}| ≥ 3 or
|(N(u) ∪N(v)) ∩ {a, b, x, y}| ≥ 3 for some neighbor v ∈ V (H) of u. Since both ab and
xy are dominating edges, the intersection N(u) ∩ {a, b, x, y} has at least two elements:
one from {a, b} and one from {x, y}. Consider the case when |N(u) ∩ {a, b, x, y}| = 2.
Without loss of generality, let N(u) ∩ {a, b, x, y} = {a, x}. Since the minimum degree of
G is at least 3, there is no vertex v ∈ G such that N(v) = {a, x}. Hence, by Proposition 3.2
the set {a, x} is not a vertex cover of Gde. Then, there exists an edge wv of Gde such that
{w, v} ∩ {a, x} = ∅. Thus, as ν(Gde) = 2 and ab, xy ∈ Gde, we have wv = by or
w ∈ {b, y} and v ∈ V (H). Recall that wv is a dominating edge in G and hence, u is
adjacent to w or v. Therefore, the case wv = by is impossible and we see that v is adjacent
to u. Consequently, we get |(N(u) ∪ N(v)) ∩ {a, b, x, y}| ≥ 3 since w ∈ {b, y} is a
neighbor of v. Note that this result implies that if {u} is a component of H , then u has at
least three neighbors among a, b, x, y; otherwise, contraction of the edge uv gives rise to a
vertex adjacent to at least three of a, b, x, y.
We then apply the following process for each i = 1, . . . ,m: If hi ≤ 3, contract the
edges of Hi and obtain a singleton. If hi ≥ 4, let Hi be the path v1, v2, . . . , vk where
k = hi. First, contract v1v2 and vk−1vk. Then contract the paths v3v4v5, v6v7v8, . . .
and so on. Note that for every i we obtain at least 2 + ⌊(hi − 4)/3⌋ = ⌊(hi + 2)/3⌋
vertices adjacent to at least three of a, b, x, y. Therefore, each such vertex is adjacent to
both a and b or adjacent to both x and y. Assume that the number of vertices having
at least three neighbors among a, b, x, y in the resulting graph is more than 4. Then, by
pigeonhole principle, there will be three distinct vertices u1, u2 and u3 each of which is
adjacent, without loss of generality, to both a and b. Then, contraction of the edge xy gives
a K3,3 with parts {a, b, xy} and {u1, u2, u3}, contradicting the planarity of G. Thus, there
are at most 4 vertices having at least three neighbors among a, b, x, y once the contraction
process is terminated, that is,
∑m
i=1⌊(hi +2)/3⌋ ≤ 4. Since hi is an integer, the inequality
hi/3 ≤ ⌊(hi + 2)/3⌋ holds, implying that
∑m
i=1 hi/3 ≤ 4 which yields
∑m
i=1 hi ≤ 12 as
desired.
S. Bahadır et al.: Well-totally-dominated graphs 219
Propositions 3.13 and 3.14 imply that, unlike the general case stated in Corollary 2.5,
there is a finite number of planar WTD(2) graphs with δ(G) ≥ 3.
Theorem 3.15. If G is a planar WTD(2) graph with δ(G) ≥ 3, then |V (G)| ≤ 16.
In contrast, there is no upper bound on the number of vertices for planar WTD(2)
graphs with minimum degree 1 or 2. For example, consider a star with arbitrarily many
leaves and a graph with arbitrarily many triangles sharing a common edge, respectively.
4 Girth of WTD graphs
In this section, we provide a relation between the minimum degree and the girth for WTD
graphs. We show that if the minimum degree is more than two in a WTD graph, then the
graph contains a cycle of length at most twelve. It is shown in [12] that if G is a WTD
graph with δ(G) ≥ 2, then the girth of G, g(G), is at most 14.
Theorem 4.1 ([12, Theorem 4.1]). Suppose G is a connected graph with no leaves such
that G has girth at least fifteen. Then γt(G) < Γt(G).
By following the idea in the proof of Theorem 4.1 in [12], one can find other relations
between δ(G) and g(G) of a WTD graph G. Before presenting such extensions, we need
the following useful lemma, which is also given in [12]:
Lemma 4.2. Let G be a WTD graph, u1v1, . . . , umvm be a subset of the edges of G and
A = ∪mi=1{ui, vi}. If the subgraph of G induced by A is disjoint union of m complete
graphs of order 2 and G−N [A] has no isolated vertices, then G−N [A] is also WTD.
Proof. Let S be a minimal TDS of G −N [A]. We claim that S ∪ A is a minimal TDS of
G. It is easy to see that it is a TDS of G. Suppose that S ∪ A contains another TDS of G,
say T . Then T ∩S is a TDS of G−N [A] and hence, since S is minimal we get T ∩S = S.
Therefore, we obtain T = S ∪A′ where A′ ⊆ A. If A \A′ is nonempty, then without loss
of generality we assume that u1 ∈ A \ A′. But then, v1 is not dominated by T , which is a
contradiction. Therefore, we have A′ = A, which implies that T = S ∪ A, that is, S ∪ A
is minimal.
As every minimal TDS of G has the same size, |S|+2m is independent of S and hence,
G−N [A] is a WTD graph as well.
Theorem 4.3. If G is a WTD graph with δ(G) ≥ 3, then g(G) ≤ 12.
Proof. Assume that G is a WTD graph with δ(G) ≥ 3 and g(G) ≥ 13. Let P =
v1, v2, v3, v4, v5 be a path in G. For any vertex v in G, let dP (v) = min1≤i≤5 dist(v, vi).
Define Nk to be the set of vertices v with dP (v) = k for k = 1, 2, . . . .
First note that every vertex in Nk has a neighbor in Nk−1 for every k ≥ 2. Moreover,
for k = 1, 2, 3, Nk is an independent set since otherwise we obtain a cycle of length at most
11. We will now show that for k = 1, . . . , 4, any vertex in Nk has at least one neighbor in
Nk+1. Suppose that there exist k ≤ 4 and v ∈ Nk such that v is adjacent to no vertex in
Nk+1. By definition, it is clear that v has no neighbor in Nl for any l ≥ k + 2. Therefore,
all the neighbors of v are in ∪1≤i≤kNi. Thus, as v has at least three neighbors, there exist
three paths from v to P such that one of them has length k and two of them have length at
most k + 1. By a simple case analysis, considering the vertices of these paths on P gives
that there exist a cycle of length at most 2k + 3 ≤ 11, which is a contradiction.
220 Ars Math. Contemp. 20 (2021) 209–222
Now, let N2 = {w1, . . . , wm}. For every i = 1, . . . ,m, choose a neighbor of wi in N3,
say ui. Let A = ∪mi=1{wi, ui}. For any i ̸= j, wi is not adjacent to uj because otherwise
we obtain a cycle of length at most 10. Therefore, the induced subgraph of G induced by
A is a disjoint union of m complete graphs of order 2.
Next, consider the graph H = G −N [A]. Note that N [A] consists of N1, N2, N3 and
some vertices in N4. Therefore, P is a connected component of H . As any vertex in N4
has a neighbor in N5, no vertex v ∈ N4 ∩ V (H) is isolated in H . Clearly, no vertex in
Nk with k ≥ 6 is isolated in H since it has a neighbor in Nk−1. Suppose to the contrary
that a vertex v in N5 is isolated in H . Then v has no neighbor in N5 and N6, and thus, all
its neighbors are in N4. Therefore, since there exist three paths from v to P , this yields
a cycle of length at most 12, which is a contradiction. Consequently, H has no isolated
vertices and we can apply Lemma 4.2 and conclude that H is a WTD graph. However, P
is a component of H and hence, it should be WTD as well. Nevertheless, a path of length 4
is not a WTD graph (both {v1, v2, v4, v5} and {v2, v3, v4} are minimal TDSs of P ), which
is a contradiction.
5 Conclusion
In this work, we studied graphs whose all minimal total dominating sets have the same size.
We say these graphs are well-totally-dominated. We proved that well-totally-dominated
graphs with bounded total domination number can be recognized in polynomial time. We
then analyzed well-totally-dominated graphs with total domination number two for the
special cases of triangle-free graphs and planar graphs. Finally, we focused on the girth
of well-totally-dominated graphs. In particular, we proved that a well-totally-dominated
graph with minimum degree at least three has girth at most 12. We now conclude with
several future research directions.
Although we proved in this paper that the problem of recognizing well-totally-dominated
graphs with bounded total domination number can be solved in polynomial time, the com-
plexity of the general case is an open research problem. Hence, we pose the following
question:
Problem 5.1. What is the computational complexity of recognizing well-totally-dominated
graphs?
We have already characterized WTD(2) graphs with packing number ρ(G) = 2 in The-
orem 3.5. Since WTD(2) graphs have ρ(G) ≤ 2, in order to complete the characterization
of all WTD(2) graphs, it remains to answer the following question:
Problem 5.2. What are WTD(2) graphs with ρ(G) = 1?
Along the same line, one may consider to generalize our result in Theorem 3.5. It
is well known that ρ(G) ≤ γt(G) ≤ Γt(G); hence graphs with ρ(G) = Γt(G) form a
subclass of WTD graphs. This suggests our next open problem:
Problem 5.3. What are WTD(k) graphs with ρ(G) = k?
Lastly, we have shown in Theorem 3.15 that planar WTD(2) graphs with δ(G) ≥ 3
have at most 16 vertices. Our intuition is that 16 is not a tight bound. Thus, we pose the
following question:
Problem 5.4. Is the upper bound of 16 for the number of vertices of a planar WTD(2)
graph with δ(G) ≥ 3 tight? Can we determine all (finitely many) planar WTD(2) graphs?
S. Bahadır et al.: Well-totally-dominated graphs 221
ORCID iDs
Selim Bahadır https://orcid.org/0000-0003-1533-7194
Tınaz Ekim https://orcid.org/0000-0002-1171-9294
Didem Gözüpek https://orcid.org/0000-0001-8450-1897
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 223–231
https://doi.org/10.26493/1855-3974.2173.71a
(Also available at http://amc-journal.eu)
Nordhaus-Gaddum type inequalities
for the distinguishing index
Monika Pilśniak
AGH University, Department of Discrete Mathematics,
al. Mickiewicza 30, 30-059 Krakow, Poland
Received 6 November 2019, accepted 1 March 2021, published online 3 November 2021
Abstract
The distinguishing index of a graph G, denoted by D′(G), is the least number of colours
in an edge colouring of G not preserved by any nontrivial automorphism. This invariant
is defined for any graph without K2 as a connected component and without two isolated
vertices, and such a graph is called admissible. We prove the Nordhaus-Gaddum type
relation:
2 ≤ D′(G) +D′(G) ≤ ∆(G) + 2
for every admissible connected graph G of order |G| ≥ 7 such that G is also admissible.
Keywords: Symmetry breaking in graphs, distinguishing index, Nordhaus-Gaddum type bounds.
Math. Subj. Class. (2020): 05C15, 05C25
1 Introduction and main result
We consider finite graphs and their edge colourings, not necessarily proper. Such a colour-
ing breaks an automorphism of a graph if there exists an edge that is mapped into an edge
with a different colour by that automorphism. A colouring is called asymmetric (or dis-
tinguishing), if it breaks all non-trivial automorphisms. The minimum number of colours
in an asymmetric colouring of a graph G is called the distinguishing index of G and is
denoted by D′(G). Obviously, the distinguishing index is defined only for graphs without
K2 as a component and with at most one isolated vertex. We call such graphs admissible.
The following general upper bound for the distinguishing index of connected graphs
with respect to the maximum degree was proved by Kalinowski and Pilśniak in [8].
E-mail address: pilsniak@agh.edu.pl (Monika Pilśniak)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
224 Ars Math. Contemp. 20 (2021) 223–231
Theorem 1.1 ([8]). If G is a connected graph with at least three vertices, then
D′(G) ≤ ∆(G)
unless G is C3, C4 or C5.
This result was improved in [10], where all connected graphs with the distinguishing
index equal to the maximum degree were characterized. In particular, a tree is called sym-
metric (respectively, bisymmetric) if it contains a central vertex v0 (resp. a central edge e0),
all leaves are at the same distance from v0 (resp. e0) and all vertices that are not leaves have
the same degree.
Theorem 1.2 ([10]). Let G be a connected graph of order at least three. Then
D′(G) ≤ ∆(G)− 1
unless G is a cycle, a symmetric or a bisymmetric tree, K4 or K3,3.
In the same paper, the conjecture for 2-connected graphs was formulated, and quite
recently was confirmed in [7] in a bit stronger form, as follows.
Theorem 1.3 ([7]). If G is a connected graph with minimum degree δ(G) ≥ 2, then
D′(G) ≤
⌈√
∆(G)
⌉
+ 1.
The main goal of the paper is a proof of a Nordhaus-Gaddum type inequalities for
the distinguishing index of a graph. Our investigation was motivated by the renowned result
of Nordhaus-Gaddum [9] who determined in 1956 lower and upper bounds for the sum
of the chromatic numbers of a graph and its complement (actually, the upper bound was
first proved by Zykov [12] in 1949). Since then, Nordhaus-Gaddum type bounds were
obtained for many graph invariants. An exhaustive survey is given in [1]. In particular,
it was considered by Collins and Trenk in [3] for the distinguishing chromatic number
χD(G), which is the minimum number of colours in an asymmetric proper vertex colouring
of a graph G. The Nordhaus-Gaddum type inequalities were also investigated in [10] for
the chromatic distinguishing index χ′D(G) of a graph G defined for asymmetric proper
edge colourings. It was proved therein that if G is an admissible graph of order n ≥ 3
distinct from K1,4, then
n− 1 ≤ χ′D(G) + χ′D(G) ≤ 2(n− 1).
Both upper and lower bounds are similar to Vizing bounds proved for the chromatic index
of a graph [11] but in the proof for the chromatic distinguishing index we have to be more
careful.
It was also conjectured in [10] that
2 ≤ D′(G) +D′(G) ≤ max{∆(G),∆(G)}+ 2
if both graphs G and G are admissible and of order n ≥ 7. It was confirmed for some
classes of graphs, in particular for trees, claw-free graphs, 3-connected graphs and traceable
graphs. Here, we prove the stronger version of Nordhaus-Gaddum type inequality for
the distinguishing index.
M. Pilśniak: Nordhaus-Gaddum type inequalities for the distinguishing index 225
Theorem 1.4 (Main Theorem). If both G and G are admissible graphs of order n ≥ 7,
and G is connected, then
2 ≤ D′(G) +D′(G) ≤ ∆(G) + 2.
The lower bound by 2 is obvious. Indeed, if a graph G is asymmetric, that means it
has a trivial automorphism group, then the distinguishing index of both G and G equals 1.
Moreover, Theorem 1.4 is tight. To see this, consider a symmetric (or bisymmetric) tree
T of order n ≥ 7. Then by [5] T contains an asymmetric spanning tree if T is different
from a star, so D′(T ) = 2 by Proposition 2.2. So, it follows from Theorem 1.2 that
D′(T ) +D′(T ) = ∆(T ) + 2 for symmetric and bisymmetric trees.
It has to be noted that there exist graphs of order less than 7 violating the right inequal-
ity in Theorem 1.4. For example, D′(K3,3) = 3, D′(K3,3) = 4, whence D′(K3,3) +
D′(K3,3) = ∆(K3,3) + 4. Also, D′(C5) + D′(C5) = ∆(C5) + 4, and D′(K1,i) +
D′(K1,i) = ∆(K1,i) + 3 for i = 3, 4, 5.
In Section 2 we recall some results useful in the proof of Theorem 1.4, which is given
in Section 3.
2 Known bounds for D′
Let us recall some useful results, before we start to prove the Main Theorem. A graph
that contains a Hamiltonian path, i.e. a path that visits each vertex of the graph, is called
traceable. Following [2], we define the k-th Bondy-Chvátal closure clk(G) of a graph G
as the graph obtained from G by recursively joining pairs of non-adjacent vertices with
degree-sum at least k. By the well-known theorem of Bondy and Chvátal [2], a graph G of
order n is traceable whenever cln−1(G) is traceable.
We begin with the distinguishing index of complete graphs and of traceable graphs.
Proposition 2.1 ([8]).
D′(Kn) =
{
2, if n ≥ 6,
3, if n = 3, 4, 5.
The following simple observation is very useful in some proofs. A subgraph H of a
graph G is called almost spanning if H is a spanning subgraph of a graph G− v for some
v ∈ V (G).
Proposition 2.2 ([10]). If H is a spanning or almost spanning subgraph with at least three
vertices of a graph G, then D′(G) ≤ D′(H) + 1.
In particular, the spanning path in a traceable graph needs two colours to break its non-
trivial automorphism, so a traceable graph has the distinguishing index at most 3. Actually
we have a stronger result.
Theorem 2.3 ([10]). Let G be a traceable graph of order at least 3. If G has at least 7
vertices, then D′(G) ≤ 2. Moreover, if G is of order at most 6, then D′(G) ≤ 3.
The distinguishing index of complete bipartite graphs was determined independently by
Fisher and Isaak [4] and by Imrich, Jerebic and Klavžar [6]. Actually, they formulated their
result for the distinguishing number D(Kp □Kq) of the Cartesian product of complete
graphs, but D′(Kp,q) = D(Kp □Kq).
226 Ars Math. Contemp. 20 (2021) 223–231
Theorem 2.4 ([4, 6]). Let p, q, d be integers such that d ≥ 2 and (d− 1)p < q ≤ dp. Then
D′(Kp,q) =
{
d, if q ≤ dp − ⌈logd p⌉ − 1,
d+ 1, if q ≥ dp − ⌈logd p⌉+ 1.
If q = dp − ⌈logd p⌉, then the distinguishing index D′(Kp,q) is either d or d + 1 and can
be computed recursively in O(log∗(q)) time.
In the rest of the paper, we make use of the following immediate corollary.
Corollary 2.5. If p ≤ q, then D′(Kp,q) ≤ ⌈ p
√
q⌉+ 1.
The following simple observation is used later in this section.
Proposition 2.6. If H is an admissible disconnected graph of order at least 7, then
D′(H) ≤ |H| − 2.
Proof. Theorem 1.2 implies that the only connected graph H with D′(H) ≥ |H| − 1 is
K3,K4, C4 or a star K1,n−1 for n ≥ 3. If all components H1, . . . ,Hs of H are pairwise
non-isomorphic, then D′(H) = max{D′(Hi) : i = 1, . . . , s}, so D′(H) ≤ |H| − 2. If H
contains t ≥ 2 copies of a graph H1 as its components, so |H| ≥ t|H1|, then we colour
one of them distinguishingly and use one extra colour for each other copy. Hence,
D′(tH1) ≤ D′(H1) + t− 1 ≤ |H1|+ t− 1 ≤
|H|
t
+ t− 1 ≤ |H| − 2.
We easily extend a result of [10] for trees to forests. First, recall a result of Hedetniemi
et al. [5] on packing two trees into Kn.
Theorem 2.7 ([5]). A complete graph Kn contains edge disjoint copies of any two trees of
order n distinct from a star K1,n−1.
Proposition 2.8. Let G be an admissible graph of order n such that G is an admissible
forest. Then D′(G) ≤ 2 if n ≥ 7, and D′(G) ≤ 3 otherwise.
Proof. The case when G is a tree was proved in [10]. Otherwise, it easily follows from
Theorem 2.7 that G contains a Hamiltonian path Pn. Indeed, we can consider a tree F ′
spanned by G, and every tree distinct from a star is included in a subgraph F ′ of G. Thus
D′(G) ≤ 2 if n ≥ 7, and D′(G) ≤ 3 if 3 ≤ n ≤ 6 by Theorem 2.3.
Additionally, let us note the following observation for small graphs. Denote by W1 and
W2 the two graphs from Figure 1 called windmills.
W1
b b
bb b
b b
W2
b
b
bb b
b b
Figure 1: Two windmills.
M. Pilśniak: Nordhaus-Gaddum type inequalities for the distinguishing index 227
Proposition 2.9. If G is a connected graph of order at most 7 different from windmills W1
and W2, and from a star K1,n for n = 4, 5, 6, then D′(G) ≤ 3.
Proof. Observe that if ∆(G) ≤ 4, then the claim holds by Theorem 1.2. So let ∆(G) ≥ 5.
First assume that G does not have pendant edges. If the longest cycle in G is of order at
least 6, then G is traceable and D′(G) ≤ 3 by Theorem 2.3. If the longest cycle C in G
is of order 5, then we colour the edges of this cycle with 0, 1, 0, 2, 0 (in this order) and all
chords with 1. Thus, each vertex of C has an incident edge coloured with 0. If there exists
a vertex of G with noncoloured incident edges, then all of them we colour with 2 and all
the remaining edges with 1. It is an asymmetric colouring of G, because outside the initial
cycle C we create a monochromatic vertex with colour 2 and a monochromatic vertex with
colour 1 or a bichromatic one with colours 1 and 2.
If the longest cycle in G is of order 4, then a 2-connected G is isomorphic to K2,r
or K2,r + e with r ≤ 5 where e is an edge between two vertices of maximum degree.
Three colour suffice for an asymmetric colouring of r paths of length two between the two
vertices of maximum degree. If such a 2-connected graph is joined with a triangle (in a cut
vertex), then we can colour every edge of the triangle with a different colour. If two cycle
of length 4 (with chords) meet in only one vertex, then an asymmetric colouring of one of
them uses colour 1 twice, while the other one uses 2 twice (apart from one edge coloured
with 0).
If the longest cycle is a triangle C3, then G has a cut vertex v where three cycles meet.
Then a pair of edges including v in every triangle is coloured with 1 and 2, while every
remaining edge obtains a distinct colour 1, 2 or 3.
Now assume that there exists a leaf v in G. First suppose there exists an induced
subgraph B of G with minimal degree at least 2. Then we can colour it with three colours
as above. So we have to distinguish now only pendant trees (in particular paths and edges)
with a common vertex just fixed in B. It is always possible to do this with three colours if
|G| ≤ 7 unless B has a pendant star K1,4. Then we obtain an exceptional graph W1.
Finally, let G be a tree of order at most 7 different from a star K1,n with n = 3, . . . , 6.
Then D′(G) ≤ ∆(G) − 1 by Theorem 1.2. Clearly, D′(G) ≤ 3 unless G = W2, which
needs four colours like W1.
3 Proof of the Main Theorem
Proof of Theorem 1.4. Let G be a connected graph of order n ≥ 7 such that both G and its
complement G are admissible. Denote ∆ = ∆(G). Clearly, ∆ ≥ ∆(G) + 1 whenever G
is disconnected. Next, if ∆ ≤ n−12 , then
δ(G) = n− 1−∆ ≥ n− 1
2
,
and G is connected. Otherwise, n−12 ≥ ∆ ≥ ∆(G) + 1 ≥
n−1
2 + 1. Hence, G satisfies
the well-known Dirac’s condition for traceability, so G is traceable. Hence, D′(G) +
D′(G) ≤ ∆+ 2 by Theorem 1.1 and Theorem 2.3.
Assume then that ∆ ≥ ⌈n2 ⌉. Analogously, we can assume that ∆(G) ≥ ⌈
n
2 ⌉. Indeed,
our theorem holds if ∆(G) ≤ n−12 . Then D
′(G) ≤ 2 and D′(G) ≤ ∆(G) ≤ n−12 ≤ ∆ if
G is connected, or D′(G) ≤ ⌈n2 ⌉ ≤ ∆ if G is disconnected, by the following reasons.
Theorem 1.1 implies that the only connected graphs H with D′(H) > ∆(H) are
C3, C4 and C5. If all components H1, . . . ,Hs of G are pairwise non-isomorphic, then
228 Ars Math. Contemp. 20 (2021) 223–231
D′(G) = max{D′(Hi) : i = 1, . . . , s} ≤ max{3,∆(Hi)}, so D′(G) ≤ n2 . If G contains
ti ≥ 2 copies of a graph Hi as its components, then let t = max{ti : i = 1, . . . , s} and let
H = Hl where D′(Hl) = max{D′(Hi) : i = 1, . . . , s}, and we colour one copy of Hl
distinguishingly and use one extra colour for each other copy. So, for H different from C3,
C4 and C5 we have
D′(G) ≤ D′(tHl) ≤ D′(Hl) + t− 1 ≤ ∆(Hl) + t− 1 ≤
|G|
t
+ t− 2 ≤ n
2
,
and for i ∈ {3, 4, 5} we obtain D′(tCi) ≤ t+ 2 ≤ ⌈ ti2 ⌉.
We distinguish two cases.
Case A: Both G and G are connected graphs without pendant edges.
Then D′(G) ≤ ⌈
√
∆⌉+1 and D′(G) ≤ ⌈
√
∆(G)⌉+1. Hence, the inequality D′(G)+
D′(G) ≤ ∆+ 2 is weaker than
∆−
⌈√
∆
⌉
≥
⌈√
∆(G)
⌉
. (3.1)
First assume that ∆ ≥ ∆(G). It is easy to see that the inequality (3.1) is satisfied unless
∆ = ∆(G) = 5. In the latter case 8 ≤ n ≤ 10 and δ(G) = δ(G) = n − 6. We want to
show that either G or its complement G is traceable.
We say that a sequence (ai) is minorized by a sequence (bi) if bi ≤ ai for any i.
If n = 8, then the degree sequence, ordered non-increasingly, of G (or G) is minorized
by (5, 5, 4, 4, 2, 2, 2, 2) or by (5, 4, 4, 4, 3, 2, 2, 2). Indeed, we know by assumptions that
b1 = 5, b8 = 2, two terms of bi have to be odd since the sum of degrees is even in every
graph, and the sum of the fourth term of the sequence of G and the fifth term of the sequence
of G equals n − 1 = 7, so one of them cannot be smaller than ⌈n−12 ⌉ = 4. Now, by
definition of the (n − 1)-th Bondy-Chvátal closure cln−1(G), a vertex of degree five in G
has degree n − 1 = 7 in cl7(G), so we have to add two new edges incident to it. Observe
that adding new edges yields another vertex that has degree n − 1 in cl7(G), and this is
the case at each step of creating cl7(G). Finally, cl7(G) = K8. Hence, G is traceable, by
the Bondy-Chvátal theorem [2].
Similarly, if n = 9, we may assume that the degree sequence of G is minorized
by (5, 4, 4, 4, 4, 4, 3, 3, 3) or by (5, 5, 4, 4, 4, 3, 3, 3, 3), and it is not difficult to see that
cl8(G) = K9. For n = 10, the degree sequence of G is minorized by (5, 5, 4, . . . , 4) and
here it is clear that cl9(G) = K10. For brevity, the details are left to the reader.
Now assume that ∆(G) > ∆. Then it is easily seen that the inequality (3.1) holds for
any ∆, ∆(G) and n unless either n = 8,∆ = 4,∆(G) = 5, or n = 9,∆ = 5,∆(G) = 6,
or n = 10,∆ = 5,∆(G) ∈ {6, 7}, since ⌈n2 ⌉−⌈
√
⌈n2 ⌉⌉ ≥ ⌈
√
n− 3⌉. The same argument
as above confirms that G is traceable.
Case B: A graph G is disconnected or δ(G) = 1.
If G is a forest, then the conclusion follows from Proposition 2.8. Hence, assume that
G contains a 2-connected block. We now consider decompositions of G into two subgraphs
F1, F2 such that E(F1) ∪ E(F2) = E(G) and the vertex sets V (F1), V (F2) share at most
one vertex which is a cut-vertex of G. Denote p = |F1|−1, q = |F2|−1 if F1 and F2 share
a common vertex, and p = |F1|, q = |F2| if F1 and F2 are disjoint. Assume that p ≤ q
and the difference q − p is smallest possible. Observe that ∆(G) ≥ q and G is spanned or
almost spanned by a complete bipartite graph Kp,q .
M. Pilśniak: Nordhaus-Gaddum type inequalities for the distinguishing index 229
First, suppose that q ≤ 2p − p. Then D′(G) ≤ 2, since G is (almost) spanned by
an asymmetric spanning subgraph of Kp,q for p + q ≥ 7 by Proposition 3.10 in [10] (see
also [6]).
If G is connected, then ∆ = ∆(G) = n − 2 since δ(G) = 1 by the assumption of
Case B. Moreover, D′(G) ≤ n− 2 by Theorem 1.1 as ∆(G) ≤ n− 2. Hence
D′(G) +D′(G) ≤ 2 + n− 2 = ∆+ 2.
If G is disconnected, then either δ(G) = 1 or δ(G) ≥ 2. If δ(G) = 1, then D′(G) ≤
n− 2 by Proposition 2.6 and D′(G) +D′(G) ≤ 2 + n− 2 = ∆+ 2. Now, let δ(G) ≥ 2,
and assume (in the worst case) that G contains s components isomorphic to a connected
graph H . Recall that ∆(G) ≥ n2 , as we assumed at the beginning of the proof. So,
3 ≤ |H| < n2s . If we use a distinct additional colour for every component H , then D
′(G) ≤
⌈
√
∆(G)⌉+ s ≤ ⌈
√
n− 2⌉+ s, by Theorem 1.3. So, we would like to show that⌈√
n− 2
⌉
+ s+ 2 ≤ n− |H|+ 2, (3.2)
since ∆(G) ≥ n− |H|. To confirm the inequality (3.2), we estimate⌈√
n− 2
⌉
+ s ≤
⌈√
n− 2
⌉
+
n
6
≤ n− n
4
≤ n− |H|.
It is easy to verify that the second inequality is always true. The last inequality is obvious
if s is at least 2. If s = 1 we do not need to distinguishing connected component one from
another, hence we use the same colours in every component and D′(G) ≤ ⌈
√
∆(G)⌉+1 ≤
∆. So, this subcase is completed.
Now, assume that q ≥ 2p − p+ 1. Then D′(G) ≤ p√q + 2 for p ≥ 2 by Corollary 2.5
and Proposition 2.2. In this case the graph G can be obtained from a 2-connected graph B
by attaching a number of independent pendant subgraphs (of order at most p) to it or there
is a component of order p and a 2-connected component B of order q. Hence, ∆(B) ≤ q
and D′(G) ≤ ⌈√q⌉+ 1, by Theorem 1.3 for the block B, and by the observation that then
every subgraph attached to B has one vertex fixed and the order at most p ≤ √q + 2. We
obtain D′(G) + D′(G) ≤ ⌈ p√q⌉ + 2 + ⌈√q⌉ + 1. Recall that ∆ ≥ q, so it is enough to
check whether
⌈ p√q⌉+ ⌈√q⌉+ 3 ≤ q + 2.
Consequently, we obtain the inequality
q − ⌈ p√q⌉ − ⌈√q⌉ − 1 ≥ 0. (3.3)
For p = 3, we have q ≥ 23 − 3 + 1 = 6. For q = 6, p = 3 we have D′(G) ≤ 4
by Proposition 2.2, and D′(G) ≤ 3 by Proposition 2.9 for F2. Observe now that the
inequality (3.3) is satisfied, because it holds for q = 7, and q − ⌈ 3√q⌉ − ⌈√q⌉ − 1 is an
increasing function of q. The inequality (3.3) also holds for larger values of p since its
left-hand side is non-decreasing with respect to p.
If p = 2, then inequality (3.3) is satisfied for q ≥ 7. For q = 5 or q = 6, G is
(almost) spanned by K2,5 or K2,6, so D′(G) ≤ 4 by Proposition 2.2, and D′(G) ≤ 3 by
Proposition 2.9 for F2. For q = 4 we have n = 7 and our theorem is true by Proposition 2.9.
Now let p = 1. If G is disconnected, then G contains a 2-connected block B of order
n − 1 and an isolated vertex v. Hence, ∆ = n − 1, of course. Then D′(G) ≤ D′(B) ≤
230 Ars Math. Contemp. 20 (2021) 223–231
⌈
√
∆(G)⌉ + 1. Moreover, D′(G) ≤ ⌈
√
∆⌉ + 1 whenever δ(G) ≥ 2, by Theorem 1.3.
Hence,
D′(G) +D′(G) ≤ 2
⌈√
∆
⌉
+ 2,
and 2⌈
√
∆⌉+ 2 ≤ ∆+ 2 for ∆ ≥ 6, since ∆ = n− 1 ≥ 6.
If the graph G is connected, then G can be obtained from a 2-connected block B by
attaching a number (maybe zero) of independent pendant edges to it. It is enough to break
all nontrivial automorphisms of B. Then D′(G) ≤ D′(B) ≤ ⌈
√
∆(G)⌉ + 1. Moreover,
∆ = n−2 and D′(G) ≤ ⌈
√
∆⌉+1 whenever δ(G) ≥ 2, by Theorem 1.3. Then 2⌈
√
∆⌉+
2 ≤ ∆+2 for ∆ ≥ 6, unless ∆ = 5. But in the latter case n = 7 and D′(G)+D′(G) ≤ 6
by Proposition 2.9.
Finally, assume that p = 1 and δ(G) = 1. Then we consider decompositions of G into
two subgraphs F ′1, F
′
2 such that E(F
′
1)∪E(F ′2) = E(G) and the vertex sets V (F ′1), V (F ′2)
share one vertex which is a cut-vertex of G. Denote p′ = |F ′1| − 1, q′ = |F ′2| − 1. Assume
that p′ ≤ q′ and the difference q′ − p′ is smallest possible. Recall that ∆(G) = n− 2 and
G is spanned or almost spanned by a complete bipartite graph Kp′,q′ .
If q′ ≤ 2p′ − p′, then D′(G) ≤ 2 like above (for p′ + q′ ≥ 7), and D′(G) ≤ ∆ by
Theorem 1.1. So, we are done. If q′ ≥ 2p′ − p′ + 1, then D′(G) ≤ p′
√
q′ + 2 for p′ ≥ 2,
and we obtain
D′(G) +D′(G) ≤
⌈√
q′
⌉
+
⌈
p′
√
q′
⌉
+ 3 ≤ p′ + q′ + 1,
since ∆ ≥ n − 2 = p′ + q′ − 1. For p′ = 2, we have q′ ≥ 22 − 2 + 1 = 3, and
the inequality 2⌈
√
q′⌉ ≤ q′ is satisfied since n ≥ 7. Hence q′ ≥ 4. The inequality
p′ + q′ − ⌈
√
q′⌉ − ⌈ p′
√
q′⌉ − 2 ≥ 0 also holds for larger values of p′ since its left-hand side
is non-decreasing with respect to p′.
Let p′ = 1. Then there exists a 2-connected block B′ in G with a number of indepen-
dent pendant edges attached to it (at least one). Hence D′(G) ≤ D′(B′) ≤ ⌈
√
∆⌉ + 1.
Recall that also D′(G) ≤ ⌈
√
∆(G)⌉+ 1 ≤ ⌈
√
∆⌉+ 1, since ∆ = ∆(G) = n− 2. So we
verify the following inequality
2
⌈√
∆
⌉
+ 2 ≤ ∆+ 2
for ∆ ∈ {n − 2, n − 1}, which is true for n ≥ 7 unless ∆ = 5. But then n = 7 and
D′(G) +D′(G) ≤ 6 once more by Proposition 2.9.
ORCID iD
Monika Pilśniak https://orcid.org/0000-0002-3734-7230
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http://mi.mathnet.ru/eng/msb5974.

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 233–241
https://doi.org/10.26493/1855-3974.2284.aeb
(Also available at http://amc-journal.eu)
Closed formulas for the total Roman domination
number of lexicographic product graphs
Abel Cabrera Martı́nez , Juan Alberto Rodrı́guez-Velázquez
Universitat Rovira i Virgili, Departament d’Enginyeria Informàtica i Matemàtiques,
Av. Paı̈sos Catalans 26, 43007 Tarragona, Spain
Received 19 March 2020, accepted 6 January 2021, published online 6 November 2021
Abstract
Let G be a graph with no isolated vertex and f : V (G) → {0, 1, 2} a function. Let
Vi = {x ∈ V (G) : f(x) = i} for every i ∈ {0, 1, 2}. We say that f is a total Roman
dominating function on G if every vertex in V0 is adjacent to at least one vertex in V2
and the subgraph induced by V1 ∪ V2 has no isolated vertex. The weight of f is ω(f) =∑
v∈V (G) f(v). The minimum weight among all total Roman dominating functions onG is
the total Roman domination number of G, denoted by γtR(G). It is known that the general
problem of computing γtR(G) is NP-hard. In this paper, we show that if G is a graph with
no isolated vertex and H is a nontrivial graph, then the total Roman domination number of
the lexicographic product graph G ◦H is given by
γtR(G ◦H) =
{
2γt(G) if γ(H) ≥ 2,
ξ(G) if γ(H) = 1,
where γ(H) is the domination number of H , γt(G) is the total domination number of G
and ξ(G) is a domination parameter defined on G.
Keywords: Total Roman domination, total domination, lexicographic product graph.
Math. Subj. Class. (2020): 05C69, 05C76
1 Introduction
Let G be a graph with no isolated vertex and f : V (G) → {0, 1, 2} a function. Let Vi =
{x ∈ V (G) : f(x) = i} for every i ∈ {0, 1, 2}. We will identify f with the partition of
E-mail addresses: abel.cabrera@urv.cat (Abel Cabrera Martı́nez), juanalberto.rodriguez@urv.cat (Juan
Alberto Rodrı́guez-Velázquez)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
234 Ars Math. Contemp. 20 (2021) 233–241
V (G) induced by f and write f(V0, V1, V2). The weight of f is defined to be
ω(f) = f(V (G)) =
∑
v∈V (G)
f(v) = |V1|+ 2|V2|.
A function f(V0, V1, V2) is said to be total Roman dominating function onG if every vertex
in V0 is adjacent to at least one vertex in V2 and the subgraph induced by V1 ∪ V2 has no
isolated vertex [17]. The minimum weight among all total Roman dominating functions
on G is the total Roman domination number of G, denoted by γtR(G). In this article, we
continue the study initiated in [5] on the total Roman domination number of lexicographic
product graphs. In particular, we give closed formulas for the total Roman domination
number of lexicographic product graphs.
Let G and H be two graphs. The lexicographic product of G and H is the graph G ◦H
whose vertex set is V (G ◦H) = V (G)× V (H) and (u, v)(x, y) ∈ E(G ◦H) if and only
if ux ∈ E(G) or u = x and vy ∈ E(H). Notice that for any u ∈ V (G) the subgraph of
G ◦ H induced by {u} × V (H) is isomorphic to H . For simplicity, we will denote this
subgraph by Hu.
For a basic introduction to the lexicographic product of two graphs we suggest the
books [7, 12]. One of the main problems in the study of G ◦ H consists of finding exact
values or tight bounds for specific parameters of these graphs and express them in terms
of known invariants of G and H . In particular, we cite the following works on domination
theory of lexicographic product graphs: (total) domination [14, 18, 19, 21], Roman domi-
nation [14], weak Roman domination [20], rainbow domination [15], super domination [6],
doubly connected domination [2], secure domination [13], double domination [3] and total
Roman domination [5].
We assume that the reader is familiar with the basic concepts and terminology of domi-
nation in graph. If this is not the case, we suggest the textbooks [8, 9, 11]. In particular, we
use the standard notation γ(G) and γt(G) for the domination number and the total domi-
nation number of a graph G, respectively. Throughout the paper, N(v) will denote the set
of neighbours or open neighbourhood of v in G. The closed neighbourhood of v, denoted
by N [v], equals N(v) ∪ {v}. A vertex v ∈ V (G) such that N [v] = V (G) is said to be a
universal vertex. For the remainder of the paper, definitions will be introduced whenever a
concept is needed.
2 The case where γ(H) ≥ 2
The next theorem merges two results obtained in [14] and [21].
Theorem 2.1 ([14] and [21]). For any graph G with no isolated vertex and any nontrivial
graph H ,
γ(G ◦H) =
{
γ(G), if γ(H) = 1,
γt(G), if γ(H) ≥ 2.
Below we present two theorems that complete the tools we need to deduce our first
result.
Theorem 2.2 ([1]). For any graph G with no isolated vertex,
2γ(G) ≤ γtR(G) ≤ min{2γt(G), 3γ(G)}.
A. Cabrera Martı́nez and J. A. Rodrı́guez-Velázquez: Closed formulas for the total Roman . . . 235
Theorem 2.3 ([4]). For any graph G with no isolated vertex and any nontrivial graph H ,
γt(G ◦H) = γt(G).
From the results above we deduce the following main theorem.
Theorem 2.4. For any graph G with no isolated vertex and any graph H with γ(H) ≥ 2,
γtR(G ◦H) = 2γt(G).
Proof. The result immediately follows by applying Theorems 2.1, 2.3 and 2.2, in this order,
i.e., 2γt(G) = 2γ(G ◦H) ≤ γtR(G ◦H) ≤ 2γt(G ◦H) = 2γt(G).
Notice that, since the general optimization problem of finding the total domination
number of a graph is NP-hard [16], by Theorem 2.4 we can conclude that the problem of
finding the total Roman domination number is NP-hard. Even so, we would like to point
out that there are several families of graphs for which the total domination number can be
found in polynomial time [10].
3 The case where γ(H) = 1
The following two lemmas are the main tools in this section.
Lemma 3.1. Let G be a graph with no isolated vertex. For any nontrivial graph H with
γ(H) = 1, there exists a γtR(G ◦H)-function f satisfying the following two conditions.
(i) f(V (Hu)) ≤ 2 for every u ∈ V (G).
(ii) If f(V (Hu)) = 2, then f(u, v) = 2 for some universal vertex v of H .
Proof. Given a TRDF f on G ◦H , we define the set Rf = {x ∈ V (G) : f(V (Hx)) ≥ 3}.
Let f be a γtR(G◦H)-function such that |Rf | is minimum among all γtR(G◦H)-functions.
Let v ∈ V (H) be a universal vertex and suppose that there exists u ∈ Rf . We differentiate
the following two cases.
Case 1. There exists u′ ∈ N(u) such that f(V (Hu′)) ≥ 1. Let f ′ be the function defined
by f ′(V (Hu)) = f ′(u, v) = 2 and f ′(x, y) = f(x, y) for every x ∈ V (G) \ {u}. It is
readily seen that f ′ is a γtR(G ◦H)-function with |Rf ′ | < |Rf |, which is a contradiction.
Case 2. f(N(u) × V (H)) = 0. In this case, we choose a vertex u′ ∈ N(u) and define a
function f ′ as f ′(V (Hu′)) = f ′(u′, v) = 1, f ′(V (Hu)) = f ′(u, v) = 2 and f ′(x, y) =
f(x, y) for every x ∈ V (G) \ {u, u′}. As in Case 1, f ′ is a γtR(G ◦ H)-function with
|Rf ′ | < |Rf |, which is a contradiction.
According to the two cases above, (i) follows. Now, for any γtR(G ◦ H)-function
f(V0, V1, V2) satisfying (i), we define R′f = {x ∈ V (G) : f(V (Hx)) = 2 and V (Hx) ∩
V2 = ∅}. Let g(V ′0 , V ′1 , V ′2) be a γtR(G◦H)-function such that |R′g| is minimum among all
γtR(G ◦H)-functions satisfying (i). Suppose that there exists u ∈ R′g . If there exists u′ ∈
N(u) such that, g(V (Hu′)) = 2, then the function g′ defined by g′(V (Hu)) = g′(u, v) =
1 and g′(x, y) = g(x, y) for every x ∈ V (G) \ {u}, is a TRDF on G ◦ H of weight
ω(g′) < ω(g) = γtR(G ◦H), which is a contradiction. Hence, g(N(u)× V (H)) ≤ 1 and
we can differentiate the following two cases.
236 Ars Math. Contemp. 20 (2021) 233–241
Case 1′. There exists u′ ∈ N(u) such that g(V (Hu′)) = 1. In this case, we define a
function g′ by g′(V (Hu)) = g′(u, v) = 2 and g′(x, y) = g(x, y) for every x ∈ V (G) \
{u}. Notice that g′ is a γtR(G ◦ H)-function satisfying (i) and |R′g′ | < |R′g|, which is a
contradiction.
Case 2′. g(N(u) × V (H)) = 0. We fix u′ ∈ N(u). Notice that there exists u′′ ∈
N(u′)\{u}, with V (Hu′′)∩V ′2 ̸= ∅. Hence, we can define a function g′ as g′(V (Hu′)) =
g′(u′, v) = g′(V (Hu)) = g
′(u, v) = 1 and g′(x, y) = g(x, y) for every x ∈ V (G) \
{u, u′}. As in Case 1′, g′ is a γtR(G ◦H)-function satisfying (i) and |R′g′ | < |R′g|, which
is a contradiction.
According to the two cases above, R′g = ∅, and so there exists a γtR(G ◦H)-function
h defined as h(V (Hu)) = h(u, v) = 2 whenever g(V (Hu)) = 2 and h(V (Hu)) =
g(V (Hu)) otherwise. Therefore, h satisfies (i) and (ii).
Lemma 3.2. Let G be a graph with no isolated vertex and H a nontrivial graph with
γ(H) = 1. Let f(V0, V1, V2) be a γtR(G ◦H)-function, A = {x ∈ V (G) : V (Hx)∩V1 ̸=
∅} and B = {x ∈ V (G) : V (Hx) ∩ V2 ̸= ∅}. If f satisfies Lemma 3.1, then B is a
dominating set and A ∪B is a total dominating set of G.
Proof. Let f(V0, V1, V2) be a γtR(G ◦ H)-function which satisfies Lemma 3.1. Let C =
V (G) \ (A∪B). Obviously, if x ∈ C, then V (Hx) ⊆ V0, which implies that x is adjacent
to some vertex in B and, since H is a nontrivial graph and f satisfies Lemma 3.1, if x ∈ A,
then there exists y ∈ V (H) such that (x, y) ∈ V0, and so x is adjacent to some vertex in B.
Hence, B is a dominating set of G. Now, since the subgraph of G ◦H induced by V1 ∪ V2
does not have isolated vertices, the subgraph of G induced by A∪B does not have isolated
vertices, which implies that A ∪B is total dominating set of G.
For any graphG, let D(G) be the set of dominating sets ofG, and Dt(G) the set of total
dominating sets of G. We now proceed to introduce our main tool, which is the following
domination parameter.
ξ(G) = min{|A|+ 2|B| : B ∈ D(G) and A ∪B ∈ Dt(G)}.
We say that an ordered pair (A,B) of subsets of V (G) is a ξ(G)-pair if B ∈ D(G),
A ∪B ∈ Dt(G) and ξ(G) = |A|+ 2|B|.
Theorem 3.3. For any graph G with no isolated vertex and any nontrivial graph H with
γ(H) = 1,
γtR(G ◦H) = ξ(G).
Proof. Let v be a universal vertex ofH . From any ξ(G)-pair (A,B) we define the function
f(V0, V1, V2) as V2 = B×{v}, V1 = A×{v} and V0 = V (G◦H)\(V1∪V2). Since V2 is a
dominating set ofG◦H and V1∪V2 is a total dominating set ofG◦H , we can conclude that
f is a TRDF onG◦H . Therefore, γtR(G◦H) ≤ ω(f) = |V1|+2|V2| = |A|+2|B| = ξ(G).
Now, let f ′(V ′0 , V
′
1 , V
′
2) be a γtR(G ◦ H)-function which satisfies Lemma 3.1. Let
A = {x ∈ V (G) : f ′(V (Hx)) = 1} and B = {x ∈ V (G) : f ′(V (Hx)) = 2}. By
Lemma 3.2, B is a dominating set of G and A∪B is a total dominating set, which implies
that ξ(G) ≤ |A|+ 2|B| = |V ′1 |+ 2|V ′2 | = γtR(G ◦H). Therefore, the result follows.
A. Cabrera Martı́nez and J. A. Rodrı́guez-Velázquez: Closed formulas for the total Roman . . . 237
a
db c e
f
2 1 2 1 2
a
db c e
f
2 2 2 2
Figure 1: The labels correspond to two different γtR(G)-functions f1(V0, V1, V2), on the
left, and f2(W0,W1,W2), on the right. In this case, γtR(G) = 2γt(G) = 8, V2 = {a, d, f}
is a γ(G)-set and W2 = {a, b, e, f} is the only γt(G)-set.
Let G be the graph shown in Figure 1 and H a nontrivial graph with γ(H) = 1. Notice
that γtR(G ◦ H) = ξ(G) = γtR(G) = 8, where f1(V0, V1, V2) and f2(W0,W1,W2)
are γtR(G)-functions for V1 = {b, e}, V2 = {a, d, f}, W1 = ∅, W2 = {a, b, e, f}.
Furthermore, both (V1, V2) and (W1,W2) are ξ(G)-pairs, where V2 is a γ(G)-set and
|V1|+ |V2| > γt(G), while W2 is a γt(G)-set which does not contain any γ(G)-set.
The following bounds were given in [5]. In fact the lower bound was stated for any
connected non-trivial graph G, although it also holds for any graph G with no isolated
vertex.
Theorem 3.4 ([5]). For any graph H and any graph G with no isolated vertex,
γtR(G) ≤ γtR(G ◦H) ≤ 2γt(G).
Furthermore, if γ(H) = 1, then
γtR(G ◦H) ≤ 3γ(G).
In order to improve some of these bounds, we need to introduce some additional termi-
nology. Given a set S ⊆ V (G), we define
ψ(S) = min{|S′| : S′ ⊆ V (G) \ S and S ⊆ N(S′ ∪ S)}.
We also define the following parameter associated to G.
µ(G) = min{ψ(S) : S is a γ(G)-set}.
It is readily seen that 0 ≤ µ(G) ≤ γ(G). Furthermore, µ(G) = 0 if and only if γt(G) =
γ(G), while µ(G) = γ(G) if and only if for every γ(G)-set S and every pair of different
vertices x, y ∈ S we have that N [x] ∩ N [y] = ∅, i.e., if and only if every γ(G)-set is a
2-packing of G.
With the notation above in mind, we state the following theorem.
Theorem 3.5. Let G and H be two graphs with no isolated vertex. If γ(H) = 1, then
max{γtR(G), γt(G) + γ(G)} ≤ γtR(G ◦H) ≤ min{2γ(G) + µ(G), 2γt(G)}.
Proof. Our main tool is Theorem 3.3. For any ξ(G)-pair (A,B) we have that γtR(G◦H) =
ξ(G) = 2|B|+ |A| ≥ |(A ∪B)|+ |B| ≥ γt(G) + γ(G).
Now, let S be a γ(G)-set with µ(G) = ψ(S) and S′ ⊆ V (G) \ S a set of minimum
cardinality among the subsets of V (G)\S satisfying that S ⊆ N(S′∪S). Since S∪S′ is a
total dominating set, γtR(G◦H) = ξ(G) ≤ |S∪S′|+ |S| = 2|S|+ |S′| = 2γ(G)+µ(G).
Finally, by Theorem 3.4, γtR(G) ≤ γtR(G ◦ H) ≤ 2γt(G), which completes the
proof.
238 Ars Math. Contemp. 20 (2021) 233–241
Since µ(G) ≤ γ(G), we can conclude that the bound γtR(G ◦ H) ≤ 2γ(G) + µ(G)
is never worse than the known bound γtR(G ◦ H) ≤ 3γ(G). In order to see that the
upper bounds given by Theorem 3.5 are tight, we take the graph G shown in
Figure 1 and any nontrivial graphH with γ(H) = 1. In this case, γtR(G◦H) = 2γt(G) =
2γ(G) + µ(G) = 8.
We would point out the following result which is a direct consequence of Theorems 2.2
and 3.5.
Theorem 3.6. If G is a graph with γt(G) = γ(G) and H is a nontrivial graph with
γ(H) = 1, then
γtR(G ◦H) = γtR(G) = 2γ(G).
We now proceed to characterize the graphs achieving the lower bounds given by Theo-
rem 3.5.
Theorem 3.7. Let G and H be two graphs with no isolated vertex. If γ(H) = 1, then the
following statements are equivalent.
(i) γtR(G ◦H) = γtR(G).
(ii) There exists a γtR(G)-function f(V0, V1, V2) such that V2 is dominating set of G.
Proof. If there exists a γtR(G)-function f(V0, V1, V2) such that V2 is dominating set of
G, then γtR(G ◦ H) = ξ(G) ≤ |V1 ∪ V2| + |V2| = |V1| + 2|V2| = γtR(G). Since
γtR(G) ≤ γtR(G ◦H), we conclude that γtR(G ◦H) = γtR(G).
Conversely, assume that γtR(G ◦ H) = γtR(G). Let g(V ′0 , V ′1 , V ′2) be a γtR(G ◦ H)-
function satisfying Lemma 3.1. Let A = {x ∈ V (G) : g(V (Hx)) = 1} and B = {x ∈
V (G) : g(V (Hx)) = 2}. By Lemma 3.2, B is a dominating set of G and A ∪ B is a total
dominating set. Hence, we can define a TRDF h(V ′′0 , V
′′
1 , V
′′
2 ) from V
′′
1 = A and V
′′
2 = B.
Since ω(h) = |A|+ 2|B| = |V ′1 |+ 2|V ′2 | = γtR(G ◦H) = γtR(G), we conclude that h is
a γtR(G)-function where V ′′2 is a dominating set, as desired.
The next result gives a characterization for the case γtR(G ◦ H) = γt(G) + γ(G)
whenever γ(H) = 1.
Theorem 3.8. Let G and H be two graphs with no isolated vertex. If γ(H) = 1, then the
following statement are equivalent.
(i) γtR(G ◦H) = γt(G) + γ(G).
(ii) There exists a γt(G)-set that contains some γ(G)-set.
Proof. If there exists a γt(G)-set X which contains a γ(G)-set B, then γtR(G ◦ H) =
ξ(G) ≤ |X \ B| + 2|B| = |X| + |B| = γt(G) + γ(G), and by (i) we conclude that
γtR(G ◦H) = γt(G) + γ(G).
Conversely, assume that γtR(G◦H) = γt(G)+γ(G) and let (A,B) be a ξ(G)-pair. If
the total dominating setA∪B is a γt(G)-set, then we are done, asB is a dominating set and
from γt(G)+ γ(G) = γtR(G ◦H) = ξ(G) = |A|+2|B| = |A∪B|+ |B| = γt(G)+ |B|
we deduce that B is a γ(G)-set. Suppose to the contrary, that |A ∪ B| > γt(G). In such a
case, γt(G) + γ(G) = ξ(G) = |A|+ 2|B| ≥ |A ∪B|+ |B| > γt(G) + γ(G), which is a
contradiction. Therefore, the result follows.
Figure 2 shows a graph G such that γtR(G ◦H) = γt(G) + γ(G) = 7 > 6 = γtR(G)
for every nontrivial graph H with γ(H) = 1.
A. Cabrera Martı́nez and J. A. Rodrı́guez-Velázquez: Closed formulas for the total Roman . . . 239
a
db c e
Figure 2: The γt(G)-set D = {a, b, d, e} contains the γ(G)-set S = {a, b, d}.
4 Small values of γtR(G ◦ H)
In this short section we characterize the graphs G and H for which γtR(G ◦H) ∈ {3, 4}.
Theorem 4.1. For any graph G and H with no isolated vertex, the following statements
are equivalent.
(i) γtR(G ◦H) = 3.
(ii) γ(G) = γ(H) = 1.
Proof. If γtR(G ◦ H) = 3, then by Theorem 2.4 we deduce that γ(H) = 1. Moreover,
by Theorem 3.5 we have that 3 = γtR(G ◦H) ≥ γt(G) + γ(G) ≥ 3. Hence, γ(G) = 1,
as required. Conversely, if γ(G) = γ(H) = 1, then by Theorem 3.8 we deduce that
γtR(G ◦H) = 3.
Theorem 4.2. For any graph G and H with no isolated vertex, γtR(G ◦ H) = 4 if and
only if one of the following conditions are satisfied.
(i) γt(G) = 2 and γ(H) ≥ 2.
(ii) γt(G) = γ(G) = 2 and γ(H) = 1.
Proof. We first notice that if conditions (i) or (ii) holds, then by Theorem 2.4 or by Theo-
rem 3.5, respectively, it follows that γtR(G ◦H) = 4.
Conversely, assume that γtR(G ◦ H) = 4. If γ(H) ≥ 2, then Theorem 2.4 leads to
γt(G) = 2. From now on, we assume that γ(H) = 1. By Theorem 3.8, we have that
4 = γtR(G ◦ H) ≥ γt(G) + γ(G). Hence, 1 ≤ γ(G) ≤ 2. If γ(G) = 1, then by
Theorem 4.1 we obtain that γtR(G ◦H) = 3, which is a contradiction. Hence, γ(G) = 2
and so γt(G) = 2. Therefore, the result follows.
5 Open problems
By Theorem 3.3 we learned that, if we want to know the behaviour of γtR(G ◦ H) when
γ(H) = 1, then it is crucial to obtain the exact value or derive tight bounds on ξ(G). In
this sense, the study of ξ(G) is an interesting challenge.
In particular, Theorem 3.5 states that
max{γtR(G), γt(G) + γ(G)} ≤ ξ(G) ≤ min{2γ(G) + µ(G), 2γt(G)}.
The graphs achieving the equalities ξ(G) = γtR(G) and ξ(G) = γt(G)+ γ(G) were char-
acterized in Theorems 3.7 and 3.8, respectively. Therefore, the problems of characterizing
the graphs achieving the equalities ξ(G) = 2γt(G) and ξ(G) = 2γ(G) + µ(G) = 3γ(G)
remain open.
240 Ars Math. Contemp. 20 (2021) 233–241
ORCID iDs
Abel Cabrera Martı́nez https://orcid.org/0000-0003-2806-4842
Juan Alberto Rodrı́guez-Velázquez https://orcid.org/0000-0002-9082-7647
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 243–260
https://doi.org/10.26493/1855-3974.2359.a7b
(Also available at http://amc-journal.eu)
Wiener-type indices of Parikh word
representable graphs*
Nobin Thomas
Research scholar, APJ Abdul Kalam Technological University,
Thiruvananthapuram, Kerala 695 016 India, and
Amal Jyothi College of Engineering, Kanjirappally, Kerala 686 518 India
Lisa Mathew
Amal Jyothi College of Engineering, Kanjirappally, Kerala 686 518 India
Sastha Sriram
Department of Mathematics, School of Arts, Science and Humanities,
SASTRA Deemed University, Tanjore, Tamil Nadu 613 401 India
K. G. Subramanian †
School of Mathematics, Computer Science and Engineering,
Liverpool Hope University, Liverpool L16 9JD, United Kingdom
Received 10 June 2020, accepted 14 February 2021, published online 9 November 2021
Abstract
A new class of graphs G(w), called Parikh word representable graphs (PWRG), corre-
sponding to words w that are finite sequence of symbols, was considered in the recent past.
Several properties of these graphs have been established. In this paper, we consider these
graphs corresponding to binary core words of the form aub over a binary alphabet {a, b}.
We derive formulas for computing the Wiener index of the PWRG of a binary core word.
Sharp bounds are established on the value of this index in terms of different parameters
related to binary words over {a, b} and the corresponding PWRGs. Certain other Wiener-
type indices that are variants of Wiener index are also considered. Formulas for computing
these indices in the case of PWRG of a binary core word are obtained.
Keywords: Graphs, words, Parikh matrix, Parikh word representable graphs.
Math. Subj. Class. (2020): 68R10, 68R15
*The authors would like to thank the reviewers for their very useful comments which enabled them to revise
the paper improving the presentation of the paper.
†Honorary Visiting Professor.
E-mail addresses: nobinvazhayil@gmail.com (Nobin Thomas), lisamathew@amaljyothi.ac.in (Lisa
Mathew), sriram.discrete@gmail.com (Sastha Sriram), kgsmani1948@gmail.com (K. G. Subramanian)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
244 Ars Math. Contemp. 20 (2021) 243–260
1 Introduction
While words that are finite sequences of symbols are the fundamental and central objects in
computing models developed in theoretical studies of computer science, graphs are mathe-
matical models of pairwise relations between objects found useful for analyzing and solv-
ing different kinds of problems. An interesting area of investigation is relating graphs and
words and there are many studies in this direction (see, for example, [7, 10, 15, 18, 19, 24]).
On the other hand, in the study of numerical quantities related to subwords (also called
scattered subwords) of a word, the notion of Parikh matrix of a word over an ordered
alphabet was introduced in [26]. This has opened up a new direction of research in the area
of combinatorics on words [23] and many problems on words and subwords have been
investigated (see, for example, [1, 2, 4, 25, 33, 34, 36, 37, 38] and references therein),
resulting in a number of interesting results. Parikh word representable graph (PWRG) is
one such notion introduced in [3] linking the two areas of study on properties of words and
of graphs. Based on the notions of subwords of a word and the Parikh matrix of a word
[26] with entries of the matrix giving the counts of certain subwords in the word, PWRG
related to a word was introduced in [3]. Relationship of these graphs with corresponding
words and partitions was recently studied in [27].
In the field of chemical graph theory [14], undirected graphs, referred to as molecular
graphs are considered providing graph representations of organic compounds or equiva-
lently their molecular structures with atoms other than hydrogen often represented by ver-
tices and covalent chemical bonds by edges. In fact in chemical graph theory there have
been attempts to capture the molecular structure in terms of the topological index of the
corresponding graph. There has been a great interest in various topological indices associ-
ated with graphs due to their application in the area of chemical graph theory [8]. There are
a number of studies (see, for example, [14]) of various topological indices of graphs estab-
lishing formulae for computing the indices and also providing upper and lower bounds on
the values of such indices. The Wiener index (also called Wiener number) [40] is the first
topological index introduced by Harold Wiener. Knor et al. [22] provide an excellent sum-
mary of results relating to Wiener index besides providing conjectures and problems on this
index. Wiener index and its variants for different classes of graphs are widely investigated
indices (see, for example, [9, 12, 14, 21, 28, 29, 39] and references therein).
In this paper we study the Wiener index of a PWRG of a binary core word and derive
formulas for computing this index besides establishing sharp bounds on their values, given
different parameters related to the graphs. We also obtain formulas for evaluating certain
other indices that are variants of Wiener index, such as multiplicative Wiener index [13],
terminal Wiener index [11], peripheral Wiener index [16], hyper-Wiener index [20, 30] in
the case of a PWRG of a binary core word.
2 Preliminaries
The basic notions and notations relating to words and subwords can be found in [23,
31]. We recall some essential concepts and results. An ordered alphabet Σ which is
a set of symbols {a1, a2, . . . , as} with an ordering < on its symbols is written as Σ =
{a1 < a2 < · · · < as} . A word v is a subword of a word w over Σ if and only if we can
find words x1, x2, . . . , xn, y0, y1, . . . , yn over Σ, some of them possibly empty, such that
w = y0x1y1x2y2 · · · yn−1xnyn and v = x1x2 · · ·xn. The number of occurrences of a
word u as a subword of w is denoted by |w|u. For example, in the word w = aababaaab =
N. Thomas et al.: Wiener-type indices of Parikh word representable graphs 245
a2baba3b over the ordered binary alphabet Σ = {a < b}, the number of distinct occur-
rences of the subword ab is 11 so that |w|ab = 11. The set of all words over an alphabet Σ,
including the empty word λ with no symbols, is denoted by Σ∗.
Definition 2.1 ([6]). A binary word w over an alphabet {a, b} is said to be fair if
|w|ab = |w|ba.
Example 2.2. The binary word abbbaab is a fair word since |w|ab = |w|ba = 6.
Definition 2.3 ([38]). Consider the binary word w ∈ Σ∗ where Σ = {a < b}. The core of
w, denoted by core(w), is the unique word w0 of w with the smallest possible length such
that w ∈ b∗w0a∗. A word w ∈ Σ∗ is said to be a core word if and only if core(w) = w.
Clearly, a non empty word w over Σ = {a < b} is a core word if and only if w starts
with a and ends with b.
We now recall the relationship between binary core words and partitions following the
discussion in [38, pages 62–63].
Lemma 2.4 ([38]). Every nonempty binary core word can be identified with a partition of
a positive integer.
Proof. Suppose w ∈ Σ∗ is a nonempty core word and has the form an1ban2b · · · an|w|b b
where n1 ≥ 1 and nk is nonnegative for each k, 2 ≤ k ≤ |w|b. Thus w can be identified
with the partition
|w|ab = (n1 + n2 + · · ·+ n|w|b) + · · ·+ (n2 + n1) + n1.
Clearly, distinct core words are identified with distinct partitions.
Conversely, suppose that m1 +m2 + · · · +ml is a partition of some positive integer,
where m1 ≥ m2 ≥ · · · ≥ ml ≥ 1. It is clear that the word
w = amlbaml−1−mlbaml−2−ml−1b · · · am1−m2b
can be identified with the given partition.
We shall use the notation p(w) = (n1+n2+ · · ·+nl)+(n1+n2+ · · ·+nl−1)+ · · ·+
(n1 +n2 + · · ·+nl−i+1) + · · ·+ (n1 +n2) + (n1) to indicate the partition corresponding
to the word w = an1ban2b · · · anlb.
We now recall the notion of Parikh word representable graph (PWRG) [3]. For basic
concepts pertaining to graphs we refer to [5].
Definition 2.5 ([3]). For a word w = a1a2 · · · an of length n where for 1 ≤ i ≤ n,
ai ∈ Σ = {a < b}, we associate a simple graph G = G(w) with n vertices {1, 2, . . . , n}.
Each vertex i has the label ai and represents the position of the letter ai, 1 ≤ i ≤ n, in w.
For each occurrence of the subword ab in w, there is an edge in G(w) joining the vertices
corresponding to the positions of a and b in w. We say that the graph G is Parikh binary
word representable by the binary word w. In other words, we say that a graph G is Parikh
binary word representable if there exists a binary word w such that G is Parikh binary word
representable by the binary word w.
246 Ars Math. Contemp. 20 (2021) 243–260
Since every connected Parikh binary word representable graph corresponds to a core
word, we deal with only core words and the corresponding graphs in the rest of this paper.
As an illustration, if the core word is w = aabab, then in the Parikh word representable
graph as shown in Figure 1, the vertices 1, 2 and 4 have label a while the vertices 3 and 5
have the label b. The number of edges in the graph is |w|ab = 5. For example there is a
subword ab in w formed by the symbol a in position 1 and the symbol b in position 3 and
so in the graph there is an edge joining the vertex 1 with the vertex 3.
3
1 42
5
G(aabab)
Figure 1: The Parikh word representable graph of the word aabab.
3 Wiener index of Parikh word representable graphs
Let G = (V,E) be a connected graph with vertex set V (G) and edge set E(G). The
distance between the vertices u and v of G is denoted by d(u, v) and is defined as the
length of a shortest path between u and v in G.
Definition 3.1. The Wiener index W (G) of a connected graph G = (V,E), is the sum of
distances d(u, v) between all the vertices u and v of G. In other words
W (G) =
∑
{u,v}⊆V (G)
d(u, v).
We now obtain a formula for computing the Wiener index of Parikh word representable
graph of a binary word.
Theorem 3.2. The Wiener index of a Parikh word representable graph G(w) for w =
an1ban2b · · · anlb, n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l, is
W (G(w)) =
(
l∑
i=1
ni
)2
+
l∑
i=1
(l + 2i− 3)ni + l(l − 1).
Proof. In the Parikh word representable graph G(w) corresponding to the word w =
an1ban2b · · · anlb, we consider pairs of vertices (u, v), with u, v ∈ {1, 2, . . . , n} where
the label of u appears before the label of v in w. There are now four cases to be considered:
(i) u and v are both labeled a;
(ii) u is labeled a and v is labeled b;
(iii) u is labeled b and v is labeled a;
N. Thomas et al.: Wiener-type indices of Parikh word representable graphs 247
(iv) u and v are both labeled b.
The contribution to the Wiener index of the Parikh word representable graph from each of
these four cases may be calculated as follows:
(i) The distance between any two vertices labeled a is 2 and there are n1+n2+ · · ·+nl
vertices labeled a. Hence the total contribution from these pairs of vertices is
(n1 + n2 + · · ·+ nl)C2 × 2 = (n1 + n2 + · · ·+ nl)2 − (n1 + n2 + · · ·+ nl).
(ii) The distance between u labeled a and v labeled b is 1 and there are n1+(n1+n2)+
· · ·+ (n1 + n2 + · · ·+ nl) such pairs. Hence the total contribution is
n1 + (n1 + n2) + · · ·+ (n1 + n2 + · · ·+ nl) = ln1 + (l − 1)n2 + · · ·+ nl.
(iii) The distance between u labeled b and v labeled a is 3 and there are (n2 +n3 + · · ·+
nl) + (n3 + · · · + nl) + · · · + nl = n2 + 2n3 + · · · + (l − 1)nl such pairs. Hence
the total contribution is
3(n2 + 2n3 + · · ·+ (l − 1)nl).
(iv) The distance between any two vertices labeled b is 2 and there are l vertices labeled
b. Hence the total contribution from these pairs of vertices is
lC2 × 2 = l(l − 1).
Hence the Wiener index of G(w) is given by
W (G(w)) = (n1 + n2 + · · ·+ nl)2 + (l − 1)n1 + (l + 1)n2 + · · ·
+ 3(l − 1)nl + l(l − 1)
=
(
l∑
i=1
ni
)2
+
l∑
i=1
(l + 2i− 3)ni + l(l − 1).
Example 3.3. For the PWRG in Figure 1 corresponding to the word w = a2bab, we have
l = 2, n1 = 2, n2 = 1 and so W (G(w)) = 16 which can also be verified from the formula
in the Definition 3.1 by actually computing the distances d(u, v) for all unordered pairs
(u, v) of vertices.
We now derive an alternate form of the expression for the Wiener index of the Parikh
word representable graph G(w) for w = an1ban2b · · · anlb, n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l.
An interesting aspect of this alternate form is that the expression in the formula is elegant
involving only the parameters related to the word.
Theorem 3.4. The Wiener index of a Parikh word representable graph G(w), for w =
an1ban2b · · · anlb, n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l, is
W (G(w)) = |w|2 − |w|+ |w|a|w|b − 2|w|ab.
248 Ars Math. Contemp. 20 (2021) 243–260
Proof. Since w = an1ban2b · · · anlb, we have
∑l
i=1 ni = |w|a, l = |w|b. Also
|w|a|w|b − |w|ab = l
(
l∑
i=1
ni
)
− [(n1 + n2 + · · ·+ nl) + · · ·+ (n1 + n2) + n1]
=
l∑
i=1
ini − |w|a
so that
l∑
i=1
ini = |w|a|w|b + |w|a − |w|ab.
Hence from Theorem 3.2, the Wiener index
W (G(w)) = |w|2a + (|w|b − 3)|w|a + |w|b(|w|b − 1) + 2
l∑
i=1
ini
= |w|2a + |w|2b + 3|w|a|w|b − |w|a − |w|b − 2|w|ab
= |w|2 − |w|+ |w|a|w|b − 2|w|ab
using |w| = |w|a + |w|b.
Corollary 3.5. The Wiener index of a Parikh word representable graph G(w) for a fair
word w = an1ban2b · · · anlb, n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l, is
W (G(w)) = |w|2 − |w|.
Proof. For a binary word w, we have |w|ab + |w|ba = |w|a|w|b. Since w is a fair word,
|w|ab = |w|ba so that |w|a|w|b − 2|w|ab = |w|ba − |w|ab = 0. Hence from Theorem 3.4,
W (G(w)) = |w|2 − |w|.
Theorem 3.6. The Wiener index W (G(w)) of a Parikh word representable graph G(w) =
(V1 ∪ V2, E) with |V1| = |w|a = p, |V2| = |w|b = q for the word w = an1ban2b · · · anqb,
n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l, is bounded above by p2 + q2 + 3pq − 3p − 3q + 2 and
below by p2 + q2 + pq − p− q. The bounds are attained on G(abq−1ap−1b) and G(apbq)
respectively.
Proof. Since G(w) is connected, |w|ab = |E| ≥ p + q − 1 [5]. Also |w|ab ≤ pq [26].
Hence from Theorem 3.4, the Wiener index of G(w) is
W (G(w)) = p2 + q2 + 3pq − p− q − 2|w|ab
≤ p2 + q2 + 3pq − p− q − 2(p+ q − 1) = p2 + q2 + 3pq − 3p− 3q + 2
which is the Wiener index of the Parikh word representable graph G(abq−1ap−1b) and
W (G(w)) ≥ p2 + q2 + 3pq − p− q − 2pq = p2 + q2 + pq − p− q
which is the Wiener index of the Parikh word representable graph G(apbq).
N. Thomas et al.: Wiener-type indices of Parikh word representable graphs 249
Remark 3.7. One of the conjectures listed in [22, page 333], states that for a graph G
with diameter d and order 2d + 1, the Wiener index W (G) ≤ W (C2d+1) where C2d+1
denotes a cycle of length 2d+ 1. Since the diameter of any PWRG G(w) corresponding to
the binary core word w is 3, this conjecture holds good for G(w), if the order of G(w) is 7,
which also equals |w|. In fact, if the binary core word w with |w| = 7, is over the ordered
alphabet {a < b}, the maximum Wiener index of G(w) equals W (C7) which is 42 and this
is attained, by Theorem 3.6, on G(ab3a2b) or G(ab2a3b).
We shall now find an expression for an upper bound on the Wiener index of Parikh
word representable graph with a fixed number of edges. We use the following lemma.
Lemma 3.8. Given a fixed value of e and of l, the maximum value of W (G(w)) over all
Parikh word representable graphs of the form G(w) for w = an1ban2b · · · anlb, n1 ≥ 1,
nk ≥ 0 for 2 ≤ k ≤ l, with e edges, is attained on G(abl−1ae−lb).
Proof. Since the number of edges in G(w) is e = |w|ab, |w|b = l and e ≥ |w|a + |w|b − 1
as G(w) is a connected graph with |w|a + |w|b vertices, we have |w|a ≤ e− l + 1. Hence
from Theorem 3.4, the Wiener index of G(w) is
W (G(w)) = |w|a(|w|a − 1) + l2 + 3|w|al − l − 2e
≤ (e− l + 1)(e− l) + l2 + 3|w|al − l − 2e
≤ (e− l + 1)(e− l) + l2 + 3(e− l + 1)l − l − 2e = e2 − l2 + el + l − e
which is the Wiener index of the Parikh word representable graph G(abl−1ae−lb).
Theorem 3.9. An upper bound of the Wiener index W (G(w)) of a Parikh word repre-
sentable graph G(w), w = an1ban2b · · · anlb, n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l, with e edges
is given by
W (G(w)) ≤
{
5m2 − 6m+ 2(m ≥ 1), if e = 2m− 1;
5m2 −m(m ≥ 1), if e = 2m.
The bound is sharp and is attained on G(abm−1am−1b) when e = 2m − 1 and on
G(abm−1amb) when e = 2m.
Proof. From Lemma 3.8, the Wiener index of the Parikh word representable graph G(w),
w = an1ban2b · · · anlb, n1 ≥ 1, has a maximum for G(abl−1ae−lb) and is given by
W (G(w)) = e2 − l2 + el + l − e. We now use the fact that a quadratic expression
ax2 + bx + c, a < 0, has a maximum when x = − b2a . When e = 2m − 1,m ≥ 1,
we have
W (G(w)) = −l2 + 2ml + (4m2 − 6m+ 2).
If e = 2m− 1 has a fixed value, this quadratic expression in l has a maximum when l = m
and the maximum is 5m2 − 6m+ 2. When e = 2m,m ≥ 1, we have
W (G(w)) = −l2 + l(1 + 2m) + (4m2 − 2m).
Again if e = 2m has a fixed value, this quadratic expression has a maximum when l =
[m+ 12 ] = m where [x] is the integral part of x and the maximum is 5m
2 + 2m.
We shall now evaluate the Wiener index of a Parikh word representable graph corre-
sponding to a specific partition of a given integer.
250 Ars Math. Contemp. 20 (2021) 243–260
Theorem 3.10. Suppose m1+m2+ · · ·+ml is a partition of some positive integer, where
m1 ≥ m2 ≥ · · · ≥ ml ≥ 1. Then the Wiener index of the Parikh word representable graph
G corresponding to this partition is given by
W (G) = m21 − 2e+ (3l − 1)m1 + l(l − 1)
where e is the number of edges of G.
Proof. From Lemma 2.4, the word w = amlbaml−1−mlbaml−2−ml−1b · · · am1−m2b corre-
sponds to the given partition. Now |w| = m1 + l, |w|a = m1, |w|b = l, so that using the
formula in Theorem 3.4, we have
W (G) = W (G(w)) = |w|2 − |w|+ |w|a|w|b − 2|w|ab
= (m1 + l)
2 − (m1 + l) + lm1 − 2e = m21 − 2e+ (3l − 1)m1 + l(l − 1)
since |w|ab = e.
4 Multiplicative Wiener index
Definition 4.1 ([32]). The Wiener polynomial of a graph G is
W (G;x) =
∑
{u,v}⊆V (G)
xd(u,v).
Theorem 4.2. The Wiener polynomial of a Parikh word representable graph G(w), for
w = an1ban2b · · · anlb, n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l, is
W (G(w);x) = (|w|a|w|b−|w|ab)x3+
1
2
(
|w|a(|w|a−1)+ |w|b(|w|b−1)
)
x2+(|w|ab)x.
Proof. We consider pairs of vertices (u, v) in the Parikh word representable graph G(w)
corresponding to the word w = an1ban2b · · · anlb, with u, v ∈ {1, 2, . . . , n} where the
label of u appears before the label of v in w. As discussed in the proof of Theorem 3.2, the
vertex pairs are of four types, namely, types 1, 2, 3 and 4. Also, as discussed in the proof
of Theorem 3.4, there are 12 |w|a(|w|a − 1) pairs of vertices of type 1 and
1
2 |w|b(|w|b − 1)
of type 4 and the distance between each such pair is 2. Likewise, there are |w|ab pairs of
vertices of type 2 with distance 1 and |w|a|w|b − |w|ab pairs of vertices of type 3 with
distance 3. Hence the Wiener polynomial is
(|w|a|w|b − |w|ab)x3 +
1
2
(
|w|a(|w|a − 1) + |w|b(|w|b − 1)
)
x2 + (s|w|ab)x.
Definition 4.3 ([40]). The Wiener polarity index of G, denoted by Wp(G), is defined as
Wp(G) = |{(u, v) ⊆ V (G) : d(u, v) = 3}|.
Theorem 4.4. The Wiener polarity index of a Parikh word representable graph G(w), for
w = an1ban2b · · · anlb, n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l, is given by
Wp(G(w)) = |w|a|w|b − |w|ab.
N. Thomas et al.: Wiener-type indices of Parikh word representable graphs 251
Proof. In the Parikh word representable graph G(w), for the word w = an1ban2b · · · anlb
as in the hypothesis, the pairs of vertices (u, v) of type 3 as mentioned in the proof of
Theorem 4.2, are at a distance 3 and these pairs contribute to the Wiener polarity index of
G(w). In fact there are n2 + n3 + · · ·+ nl vertices with label a that are at distance 3 from
the vertex with label, the first b in w. Likewise for other vertices corresponding to the other
b′s in w. Hence
Wp(G(w)) = (n2 + n3 + · · ·+ nl) + (n3 + n4 + · · ·+ nl) + · · ·+ nl
=
l∑
i=1
ini −
l∑
i=1
ni = w|a|w|b − |w|ab.
Definition 4.5 ([13]). The multiplicative version of the Wiener index of a graph G is
π(G) =
∏
{u,v}⊆V (G)
d(u, v).
Theorem 4.6. The multiplicative Wiener index of a Parikh word representable graph G(w),
for w = an1ban2b · · · anlb, n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l, is
π(G(w)) = 2
|w|a(|w|a−1)+|w|b(|w|b−1)
2 3|w|a|w|b−|w|ab .
Proof. Considering pairs of vertices as in Theorem 4.2, we obtain the required result, since
there are |w|a(|w|a−1)+|w|b(|w|b−1)2 pairs of vertices at distance 2 in G(w) while there are
|w|a|w|b − |w|ab pairs of vertices at distance 3. It is to be noted that pairs of vertices at
distance 1 contribute value 1 to the product defining π(G(w)).
Corollary 4.7. The multiplicative Wiener index of a Parikh word representable graph
G(w) for a fair word w = an1ban2b · · · anlb, n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l, is
π(G(w)) = 2
|w|2−|w|
2
(
3
2
)|w|ab .
Proof. For a binary word w, we have |w|ab + |w|ba = |w|a|w|b. Since w is a fair word,
|w|ab = |w|ba so that |w|a|w|b − 2|w|ab = |w|ba − |w|ab = 0. Also |w| = |w|a + |w|b.
Hence from Theorem 4.6, π(G(w)) = 2
|w|2−|w|
2
(
3
2
)|w|ab .
Lemma 4.8. Given a fixed value of e and of l, the maximum value of π(G(w)) over all
Parikh word representable graphs of the form G(w) for w = an1ban2b · · · anlb, n1 ≥ 1,
nk ≥ 0 for 2 ≤ k ≤ l, with e edges, is attained on G(abl−1ae−lb).
Proof. Since |w|b = l and the number of edges in G(w) is e = |w|ab, we have |w|a ≤
e− l+1 as G(w) is a connected graph with |w|a+ |w|b vertices so that e ≥ |w|a+ |w|b−1.
Note that in a connected graph G with n vertices and e edges, e ≥ n− 1 [5]. Hence from
Theorem 4.6, the multiplicative Wiener index of G(w) is
π(G(w)) = 2
|w|a(|w|a−1)+|w|b(|w|b−1)
2 3|w|a|w|b−|w|ab
≤ 2
(e−l+1)(e−l)+l(l−1)
2 3(e−l+1)l−e
which is the multiplicative Wiener index of the Parikh word representable graph
G(abl−1 ae−lb).
252 Ars Math. Contemp. 20 (2021) 243–260
Theorem 4.9. An upper bound of the multiplicative Wiener index π(G(w)) of a Parikh
word representable graph G(w), w = an1ban2b · · · anlb, n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l,
with e edges is given by
π(G(w)) ≤
{
2m
2−m3m
2−2m+1(m ≥ 1), if e = 2m− 1;
2m
2
3m
2−m(m ≥ 1), if e = 2m.
The bound is sharp and is attained on G(abm−1am−1b) when e = 2m − 1 and on
G(abm−1amb) when e = 2m.
Proof. From Lemma 4.8, the multiplicative Wiener index of the Parikh word representable
graph G(w), w = an1ban2b · · · anlb, n1 ≥ 1, has a maximum for G(abl−1ae−lb) and is
given by
π(G(w)) = 2
(e−l+1)(e−l)+l(l−1)
2 3(e−l+1)l−e.
On taking logarithms we get,
ln(π(G(w))) = [(e− l + 1)(e− l) + l2 − l] ln 2
2
+ [(e− l + 1)l − e] ln 3
= l2(ln 2− ln 3)− l((e+ 1)(ln 2− ln 3) + e
(
(e+ 1)
2
ln 2− ln 3
)
.
We now use the fact that a quadratic expression ax2+ bx+ c, a < 0, has a maximum when
x = − b2a . Let F (l) = l
2(ln 2− ln 3)− l((e+1)(ln 2− ln 3)+ e( (e+1)2 ln 2− ln 3). When
e = 2m− 1,m ≥ 1, F (l) has a maximum when l = m and so π(G(w)) has the maximum
2m
2−m3m
2−2m+1. When e = 2m,m ≥ 1, F (l) has a maximum when l = [m + 12 ] = m
where [x] is the integral part of x and π(G(w)) has the maximum 2m
2
3m
2−m.
Theorem 4.10. The multiplicative Wiener index π(G(w)) of a Parikh word representable
graph G(w) = (V1 ∪ V2, E) with |V1| = |w|a = p, |V2| = |w|b = q for the word
w = an1ban2b · · · anqb, n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l, is bounded above by
2
p(p−1)+q(q−1)
2 3pq−p−q+1
and below by
2
p(p−1)+q(q−1)
2 .
The bounds are attained on G(abq−1ap−1b) and G(apbq) respectively.
Proof. Since G(w) is connected, |w|ab = |E| ≥ p + q − 1 [5]. Also |w|ab ≤ pq [26].
Hence from Theorem 4.6, the multiplicative Wiener index of G(w) is
π(G(w)) ≥ 2
p(p−1)+q(q−1)
2
which is the multiplicative Wiener index of the Parikh word representable graph G(apbq)
and
π(G(w)) ≤ 2
p(p−1)+q(q−1)
2 3pq−p−q+1
which is the multiplicative Wiener index of the Parikh word representable graph
G(abq−1 ap−1b).
N. Thomas et al.: Wiener-type indices of Parikh word representable graphs 253
5 Hyper-Wiener index of Parikh word representable graphs
We now derive formuas for computing hyper-Wiener index of Parikh word representable
graphs of binary words.
Definition 5.1 ([20, 30]). The hyper-Wiener index of a connected graph G is given by
WW (G) =
∑
{u,v}⊆V (G)
d(u, v) + d2(u, v)
where d(u, v) is the distance between the vertices u and v of G.
Theorem 5.2. The hyper-Wiener index WW (G(w)) of a Parikh word representable graph
G(w) for w = an1ban2b · · · anlb, n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l, is
WW (G(w)) = 3
(
l∑
i=1
ni
)2
+
l∑
i=1
(2l + 10i− 13)ni + 3l(l − 1).
Proof. As in the proof of Theorem 3.2, there are four cases for the vertex pairs (u, v). The
contribution to the hyper-Wiener index of the Parikh word representable graph from each
of these cases may be calculated as follows:
(i) The contribution from pairs of vertices labeled a is
(n1+n2+ · · ·+nl)C2× (2+4) = 3(n1+n2+ . . .+nl)2−3(n1+n2+ · · ·+nl).
(ii) The contribution from pairs of vertices (u, v) where u is labeled a and v is labeled
b is
2(n1 + (n1 + n2) + . . .+ (n1 + n2 + . . .+ nl)) = 2(ln1 + (l − 1)n2 + · · ·+ nl).
(iii) The contribution from pairs of vertices (u, v) where u is labeled b and v is labeled
a is
12(n2 + 2n3 + · · ·+ (l − 1)nl)).
(iv) The contribution from pairs of vertices labeled b is
lC2 × 6 = 3l(l − 1).
Hence the hyper-Wiener index of G(w) is given by
WW (G(w) = 3
(
l∑
i=1
ni
)2
+
l∑
i=1
(2l + 10i− 13)ni + 3l(l − 1).
We now derive an alternate form of the expression for the hyper-Wiener index of the
Parikh word representable graph G(w) for w = an1ban2b · · · anlb, n1 ≥ 1, nk ≥ 0 for
2 ≤ k ≤ l.
254 Ars Math. Contemp. 20 (2021) 243–260
Theorem 5.3. The hyper-Wiener index of a Parikh word representable graph G(w) for
w = an1ban2b · · · anlb, n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l, is
WW (G(w)) = 3|w|2a + 3|w|2b + 12|w|a|w|b − 3|w|a − 3|w|b − 10|w|ab
= 3|w|2 − 3|w|+ 6|w|a|w|b − 10|w|ab.
Proof. Since w = an1ban2b · · · anlb, we have
∑l
i=1 ni = |w|a, l = |w|b. Also, as in the
proof of Theorem 3.4,
l∑
i=1
ini = |w|a|w|b + |w|a − |w|ab.
Hence from Theorem 5.2, the hyper-Wiener index
WW (G(w)) = 3|w|2a + (2|w|b − 13)|w|a + 3|w|b(|w|b − 1) + 10
l∑
i=1
ini
= 3|w|2a + 3|w|2b + 12|w|a|w|b − 3|w|a − 3|w|b − 10|w|ab
= 3|w|2 − 3|w|+ 6|w|a|w|b − 10|w|ab
using |w| = |w|a + |w|b.
Theorem 5.4. The hyper-Wiener index of a Parikh word representable graph G(w) =
(V1 ∪ V2, E) with |V1| = |w|a = p, |V2| = |w|b = q for the word w = an1ban2b · · · anqb,
n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l, is bounded above and below by
3(p2 + q2) + 12pq − 13p− 13q + 10
and
3(p2 + q2) + 2pq − 3p− 3q.
The bounds are attained on G(abq−1ap−1b) and G(apbq) respectively.
Proof. As done in the proof of Theorem 3.6, using the inequalities |w|ab ≥ p+ q − 1 [5],
|w|ab ≤ pq [26], we have from Theorem 5.3, the hyper-Wiener index of G(w) is
WW (G(w)) = 3p2 + 3q2 + 12pq − 3p− 3q − 10|w|ab
≤ 3(p2 + q2) + 12pq − 13p− 13q + 10
which is the hyper-Wiener index of the Parikh word representable graph G(abq−1ap−1b)
and
W (G(w)) ≥ 3(p2 + q2) + 2pq − 3p− 3q
which is the Wiener index of the Parikh word representable graph G(apbq).
The hyper-Wiener index of a Parikh word representable graph corresponding to a spe-
cific partition of a given integer can be evaluated proceeding as in the proof of Theorem 3.10
and is given in the following theorem.
N. Thomas et al.: Wiener-type indices of Parikh word representable graphs 255
Theorem 5.5. Suppose m1 +m2 + · · ·+ml is a partition of some positive integer, where
m1 ≥ m2 ≥ · · · ≥ ml ≥ 1. Then the hyper-Wiener index of the Parikh word representable
graph G corresponding to this partition is given by
WW (G) = 3(m21 + (4l − 1)m1 + l(l − 1))− 10e
where e is the number of edges of G.
We shall now find an expression for an upper bound on the hyper-Wiener index of
Parikh word representable graph with a fixed number of edges. We use the following
lemma.
Lemma 5.6. Given a fixed value of e and of l, the maximum value of WW (G(w)) over all
Parikh word representable graphs of the form G(w) for w = an1ban2b · · · anlb, n1 ≥ 1,
nk ≥ 0 for 2 ≤ k ≤ l, with e edges, is attained on G(abl−1ae−lb).
Proof. Proceeding as done in the proof of Lemma 3.8, the number of edges in G(w) is
e = |w|ab, |w|b = l and e ≥ |w|a+ |w|b−1 as G(w) is a connected graph with |w|a+ |w|b
vertices, we have |w|a ≤ e − l + 1. Hence from Theorem 5.3, the hyper-Wiener index of
G(w) is
WW (G(w)) = 3|w|a(|w|a − 1) + 3l2 + 12l|w|a − 3l − 10e
≤ 3(e− l + 1)(e− l) + 3l2 + 12l(e− l + 1)− 3l − 10e
= 3e2 − 6l2 + 6el + 6l − 7e
which is the hyper-Wiener index of the Parikh word representable graph G(abl−1ae−lb).
Theorem 5.7. An upper bound of the Wiener index W (G(w)) of a Parikh word repre-
sentable graph G(w) for w = an1ban2b · · · anlb, n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l, with e
edges is given by
WW (G) ≤
{
18m2 − 26m+ 10(m ≥ 1), if e = 2m− 1;
18m2 − 8m(m ≥ 1), if e = 2m.
The bound is sharp and is attained on G(abm−1am−1b) when e = 2m − 1 and on
G(abm−1amb) when e = 2m.
Proof. From Lemma 5.6, the hyper-Wiener index of the Parikh word representable graph
G(w), w = an1ban2b · · · anlb, n1 ≥ 1, has a maximum for G(w) for w = abl−1ae−lb and
is given by WW (G(w)) = 3e2 − 6l2 + 6el + 6l − 7e. We use the fact that a quadratic
expression ax2+bx+c, a < 0, has a maximum when x = − b2a . When e = 2m−1,m ≥ 1,
we have WW (G(w)) = −6l2 + 12ml + (12m2 − 26m+ 10). If e = 2m− 1 has a fixed
value, this quadratic expression in l has a maximum when l = m and the maximum is
18m2 − 26m + 10. When e = 2m,m ≥ 1, we have WW (G(w)) = −6l2 + l(6 +
12m) + (12m2 − 14m). Again if e = 2m has a fixed value, this quadratic expression has
a maximum when l = [m+ 12 ] = m where [x] is the integral part of x and the maximum is
18m2 − 8m.
256 Ars Math. Contemp. 20 (2021) 243–260
6 Terminal and peripheral Wiener indices
By considering only pendant vertices in a graph, a special kind of Wiener index, called
terminal Wiener index, has been introduced and studied [11]. Here we consider this notion
in the context of Parikh word representable graphs.
Definition 6.1 ([11]). The terminal Wiener index TW (G) of a connected graph G is the
sum of distances between all pairs of pendant vertices of G.
Theorem 6.2. The terminal Wiener index of a Parikh word representable graph G =
G(an1ban2b · · · anlb), n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ l, is given by
TW (G) =
{
nl(nl − 1), if n1 ̸= 1;
nl(nl − 1) + k(k − 1) + 3knl, if n1 = 1, for 2 ≤ j ≤ k ≤ l, nj = 0.
Proof. If n1 ̸= 1, the only possible pendant vertices are the a’s in the last block and the
distance between any two of them is 2 so that
TW (G) = 2× nlC2 = nl(nl − 1).
Note that if nl = 0, then TW (G) = 0.
If n1 = 1, n2 = n3 = · · · = nk = 0 and nk+1 ̸= 0, 2 ≤ k < l, l > 1, then the first
k b’s are also a pendant vertices and the distance between any one of these b’s and any one
of the a’s in the last block is 3 while that between any two b’s is 2 as well as the distance
between any two a′s in the last block is 2. Hence
TW (G) = nl(nl − 1) + k(k − 1) + 3knl.
When k = l, then n1 = 1 and ni = 0, 2 ≤ i ≤ l so that all the b′s are the only pendant
vertices and hence
TW (G) = l(l − 1).
If n1 = 1, l = 1, clearly TW (G) = 1 as the corresponding graph G(ab) has only one edge
with the end labels a and b.
Another special kind of Wiener index, called peripheral Wiener index, has also been
studied [16].
Definition 6.3 ([16]). The peripheral Wiener index of a connected graph G is given by
Wp(G) =
∑
{u,v}⊆P (G) d(u, v) where P (G) is the set of all peripheral vertices which are
the vertices of G with their eccentricities equal to diameter of G.
Theorem 6.4. The peripheral Wiener index of the Parikh word representable graph G(w)
for w = an1ban2ban3b · · · anrbl−r+1, n1 ≥ 1, nk ≥ 0 for 2 ≤ k ≤ r − 1, and nr > 0
with r at least 2 and at the most l, is
PW (G(w)) =
(
r∑
i=2
ni
)2
+
r∑
i=2
(r + 2i− 4)ni + (r − 1)(r − 2).
Furthermore PW (G(w)) = W (G(w)) if w = anbl, where W (G(w)) is the Wiener index
of G(w).
N. Thomas et al.: Wiener-type indices of Parikh word representable graphs 257
Proof. From the definition of G(w), the Parikh word representable graph corresponding
to the word w = an1ban2ban3b · · · anrbl−r+1, it is clear that G(w) is a bipartite graph of
diameter 3 and radius 2. Also,the vertices corresponding to all a′s in ani , 2 ≤ i ≤ r and to
all b′s except the last l − r + 1 b′s are of eccentricity 3 and hence they are the peripheral
vertices of G(w). We consider pairs of peripheral vertices (u, v), where the label of u
appears before the label of v in w. As in Theorem 3.2, the vertex pairs of (u, v) can be
divided into four cases as given below:
1. u and v are both labeled a;
2. u is labeled a and v is labeled b;
3. u is labeled b and v is labeled a;
4. u and v are both labeled b.
The contribution to the peripheral Wiener index of the Parikh word representable graph
from each of these cases may be calculated as follows:
1. The contribution from the pairs of peripheral vertices labeled a is
(n2+n3+n4+· · ·+nr)C2×2 = (n2+n3+n4+· · ·+nr)2−(n2+n3+n4+· · ·+nr).
2. The contribution from the pairs of peripheral vertices (u, v) where u is labeled a and
v is labeled b is
n2 + (n2 + n3) + (n2 + n3 + n4) + · · ·+ (n2 + n3 + · · ·+ nr−1)
= (r − 2)n2 + (r − 3)n3 + · · ·+ nr−1.
3. The contribution from the pairs of peripheral vertices (u, v) where u is labeled b and
v is labeled a is
3[(n2 + n3 + · · ·+ nr) + (n3 + n4 + · · ·+ nr) + · · ·+ nr]
= 3[(r − 1)nr + (r − 2)nr−1 + · · ·+ n2].
4. The contribution from the pairs of peripheral vertices labeled b is
(r − 1)C2 × 2 = (r − 1)(r − 2).
Hence the peripheral Wiener index of G(w) is given by
PW (G(w)) =
(
r∑
i=2
ni
)2
+
r∑
i=2
(r + 2i− 4)ni + (r − 1)(r − 2).
Furthermore, if w = anbl then all the vertices of G(w) labeled a as well as b are peripheral
vertices of G(w). Thus PW (G(w)) = W (G(w)) where W (G(w)) is the Wiener index
of G(w).
258 Ars Math. Contemp. 20 (2021) 243–260
7 Conclusion
We have derived formulas for evaluating the Wiener index and certain other variants of
Wiener topological indices for Parikh word representable graphs [3] of binary core words.
There are problems that remain to be investigated. For example, the lower bound of Wiener
index of a Parikh word representable graph G(w) of a binary core word w when the number
of edges of G(w) is a given fixed value, needs to be examined. Bipartite graphs have been
utilized in studies of structural features in the areas of molecular biology and chemistry
(see, for example, [17, 35]). Parikh word representable graphs (PWRG) corresponding to
binary core words are bipartite graphs. It will be of interest to examine the role of PWRG
in such studies of structural features and relationships. Also, it will also be of interest to
study other kinds of topological indices for this class of graphs.
ORCID iDs
Nobin Thomas https://orcid.org/0000-0002-4057-3220
Lisa Mathew https://orcid.org/0000-0002-3722-3326
Sastha Sriram https://orcid.org/0000-0003-0604-2159
K. G. Subramanian https://orcid.org/0000-0001-8726-5850
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 261–274
https://doi.org/10.26493/1855-3974.2248.d3f
(Also available at http://amc-journal.eu)
A double Sylvester determinant*
Darij Grinberg †
Drexel University, Korman Center, 15 S 33rd Street, Philadelphia PA, 19104, USA
Received 7 February 2020, accepted 22 August 2020, published online 18 November 2021
Abstract
Given two (n+ 1) × (n+ 1)-matrices A and B over a commutative ring, and some
k ∈ {0, 1, . . . , n}, we consider the
(
n
k
)
×
(
n
k
)
-matrix W whose entries are (k + 1)×(k + 1)-
minors of A multiplied by corresponding (k + 1)× (k + 1)-minors of B. Here we require
the minors to use the last row and the last column (which is why we obtain an
(
n
k
)
×
(
n
k
)
-
matrix, not a
(
n+1
k+1
)
×
(
n+1
k+1
)
-matrix). We prove that the determinant detW is a multiple of
detA if the (n+ 1, n+ 1)-th entry of B is 0. Furthermore, if the (n+ 1, n+ 1)-th entries
of both A and B are 0, then detW is a multiple of (detA) (detB). This extends a previous
result of Olver and the author.
Keywords: Determinant, compound matrix, Sylvester’s determinant, polynomials.
Math. Subj. Class. (2020): 15A15, 11C20
1 Introduction
Let n and k be nonnegative integers, and let A = (ai,j)1≤i≤n+1, 1≤j≤n+1 be an (n+ 1)×
(n+ 1)-matrix over some commutative ring. Let Pk be the set of all k-element subsets
of {1, 2, . . . , n}. For any such subset K ∈ Pk, let K+ denote the subset K ∪ {n+ 1}
of {1, 2, . . . , n+ 1}. If U and V are two subsets of {1, 2, . . . , n+ 1}, then subVU A shall
denote the |U |×|V |-submatrix of A containing only the entries au,v with u ∈ U and v ∈ V.
Let WA be the Pk × Pk-matrix1 whose (I, J)-th entry (for all I ∈ Pk and J ∈ Pk) is
det
(
subJ+I+ A
)
.
*The author would like to thank Christian Krattenthaler, Peter Olver and Victor Reiner for enlightening dis-
cussions, and Peter Olver for the joint work that led to this paper. The SageMath computer algebra system [14]
has been used for experimentation leading up to some of the results below.
†Homepage: http://www.cip.ifi.lmu.de/˜grinberg/
E-mail address: darijgrinberg@gmail.com (Darij Grinberg)
1This means a matrix whose rows and columns are indexed by the k-element subsets of {1, 2, . . . , n}. If you
pick a total order on the set Pk , then you can view such a matrix as an
(n
k
)
×
(n
k
)
-matrix.
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
262 Ars Math. Contemp. 20 (2021) 261–274
(Thus, the entries of WA are all (k + 1) × (k + 1)-minors of A that use the last row and
the last column.) A particular case of a celebrated result going back to Sylvester [15] (see
[12, §2.7] or [13, Teorema 2.9.1] or [10] for modern proofs) then says that
det (WA) = a
p
n+1,n+1 · (detA)
q
, where p =
(
n− 1
k
)
and q =
(
n− 1
k − 1
)
.
Now, consider a second (n+ 1) × (n+ 1)-matrix B = (bi,j)1≤i≤n+1, 1≤j≤n+1 over
the same ring. Let WA,B (later to be just called W ) be the Pk ×Pk-matrix whose (I, J)-th
entry (for all I ∈ Pk and J ∈ Pk) is
det
(
subJ+I+ A
)
det
(
subJ+I+ B
)
.
What can be said about det (WA,B)? In general, very little2. However, under some as-
sumptions, it splits off factors. Namely, we shall show (Theorem 2.1) that det (WA,B) is
a multiple of detA if bn+1,n+1 = 0. We shall then conclude (Theorem 2.2) that if both
an+1,n+1 and bn+1,n+1 are 0, then det (WA,B) is a multiple of (detA) (detB). In either
case, the quotient (usually a much more complicated polynomial3) remains mysterious; our
proofs are indirect and reveal little about it. Our second result generalizes a curious prop-
erty of
(
n
2
)
×
(
n
2
)
-determinants [6, Theorem 10] that arose from the study of the n-body
problem (see Example 2.4 for details).
2 The theorems
Let us first introduce the standing notations.
Let N = {0, 1, 2, . . .}. Let K be a commutative ring. If a and b are two elements of K,
then we write a | b when b is a multiple of a (that is, b ∈ Ka).
If m ∈ N, then [m] shall mean the set {1, 2, . . . ,m}.
Fix an n ∈ N. If K is any subset of [n], then K+ shall mean the subset K ∪ {n+ 1}
of [n+ 1].
Fix k ∈ {0, 1, . . . , n}. Let Pk be the set of all k-element subsets of [n]. This is a finite
set; thus, any Pk × Pk-matrix (i.e., any matrix whose rows and columns are indexed by
k-element subsets of [n]) has a well-defined determinant4. Such matrices appear frequently
in classical determinant theory (see, e.g., the “k-th compound determinants” in [11] and in
[12, §2.6], as well as the related “Generalized Sylvester’s identity” in [12, §2.7] and [13,
Teorema 2.9.1] and [10]).
If A ∈ Ku×v is a u× v-matrix, and if I ⊆ [u] and J ⊆ [v], then subJI A shall mean the
submatrix of A obtained by removing all rows whose indices are not in I and removing all
columns whose indices are not in J . (Rigorously speaking, if A = (ai,j)1≤i≤u, 1≤j≤v and
I = {i1 < i2 < · · · < ip} and J = {j1 < j2 < · · · < jq}, then subJI A is defined to be the
matrix (aix,jy )1≤x≤p, 1≤y≤q .) When |I| = |J |, then the submatrix sub
J
I A is square; its
determinant det(subJI A) is called a minor of A.
2For example, if n = 3 and k = 2, then det
(
WA,B
)
is an irreducible polynomial in the (altogether
2 (n+ 1)2 = 32) variables ai,j and bi,j with 110268 monomials.
3Again, irreducible in the case when n = 3 and k = 2.
4Here, we are using the concepts of P × P -matrices (where P is a finite set) and their determinants. Both of
these concepts are folklore; a brief introduction can be found in [5, §1].
D. Grinberg: A double Sylvester determinant 263
Our main two results are the following:
Theorem 2.1. Let
A = (ai,j)1≤i≤n+1, 1≤j≤n+1 ∈ K
(n+1)×(n+1) and
B = (bi,j)1≤i≤n+1, 1≤j≤n+1 ∈ K
(n+1)×(n+1)
be such that bn+1,n+1 = 0. Let W be the Pk × Pk-matrix whose (I, J)-th entry (for all
I ∈ Pk and J ∈ Pk) is
det
(
subJ+I+ A
)
det
(
subJ+I+ B
)
.
Then detA | detW .
Theorem 2.2. Let
A = (ai,j)1≤i≤n+1, 1≤j≤n+1 ∈ K
(n+1)×(n+1) and
B = (bi,j)1≤i≤n+1, 1≤j≤n+1 ∈ K
(n+1)×(n+1)
be such that an+1,n+1 = 0 and bn+1,n+1 = 0. Define the Pk × Pk-matrix W as in
Theorem 2.1. Then (detA) (detB) | detW .
Example 2.3. For this example, set k = 1. Then Pk = P1 = {{1} , {2} , . . . , {n}}. Thus,
the map
[n] → Pk, i 7→ {i}
is a bijection. Use this bijection to identify the elements 1, 2, . . . , n of [n] with the elements
{1} , {2} , . . . , {n} of Pk. Thus, the Pk×Pk-matrix W in Theorem 2.1 becomes the n×n-
matrix(
det
(
sub
{j}+
{i}+ A
)︸ ︷︷ ︸
=ai,jan+1,n+1
−ai,n+1an+1,j
det
(
sub
{j}+
{i}+ B
)︸ ︷︷ ︸
=bi,jbn+1,n+1
−bi,n+1bn+1,j
)
1≤i≤n, 1≤j≤n
=
(
(ai,jan+1,n+1 − ai,n+1an+1,j) (bi,j bn+1,n+1︸ ︷︷ ︸
=0
−bi,n+1bn+1,j)
)
1≤i≤n, 1≤j≤n
= ((ai,jan+1,n+1 − ai,n+1an+1,j)(−bi,n+1bn+1,j))1≤i≤n, 1≤j≤n.
This is the matrix obtained from (ai,jan+1,n+1 − ai,n+1an+1,j)1≤i≤n, 1≤j≤n by multiply-
ing the i-th row with −bi,n+1 for all i ∈ [n] and multiplying the j-th column with bn+1,j
for all j ∈ [n]. Thus, the claim of Theorem 2.1 follows from the classical fact that
det
(
(ai,jan+1,n+1 − ai,n+1an+1,j)1≤i≤n, 1≤j≤n
)
= an−1n+1,n+1 · detA.
This fact is known as Chio pivotal condensation (see, e.g., [7, Theorem 0.1]), and is a
particular case of Sylvester’s identity ([12, §2.7]).
264 Ars Math. Contemp. 20 (2021) 261–274
Example 2.4. For this example, set k = 2, and consider the situation of Theorem 2.1
again. Then Pk = P2 = {{i, j} | 1 ≤ i < j ≤ n}. If {i, j} ∈ P2 and {k, l} ∈ P2 satisfy
i < j and k < l, then the ({i, j} , {k, l})-th entry of W is
det
(
sub
{k,l}+
{i,j}+ A
)
det
(
sub
{k,l}+
{i,j}+ B
)
= det
 ai,k ai,l ai,n+1aj,k aj,l aj,n+1
an+1,k an+1,l an+1,n+1
 det
 bi,k bi,l bi,n+1bj,k bj,l bj,n+1
bn+1,k bn+1,l 0
 .
Note that bn+1,n+1 = 0. If we furthermore assume that
an+1,n+1 = 0, and
an+1,i = ai,n+1 = 1 for all i ∈ [n] , and
bn+1,i = bi,n+1 = 1 for all i ∈ [n] ,
then this entry rewrites as
det
ai,k ai,l 1aj,k aj,l 1
1 1 0
 det
bi,k bi,l 1bj,k bj,l 1
1 1 0

= (aj,k + ai,l − ai,k − aj,l) (bj,k + bi,l − bi,k − bj,l) .
Hence, [6, Theorem 10] can be obtained from Theorem 2.2 by setting k = 2 and A = CS
and B = CT (and observing that the matrix W then equals to WS,T ).
3 The proofs
Our proofs of Theorem 2.1 and Theorem 2.2 will rely on some basic commutative algebra:
the notion of a unique factorization domain (“UFD”); the concepts of coprime, prime and
irreducible elements; the localization of a commutative ring at a multiplicative subset. This
all appears in most textbooks on abstract algebra; for example, [8, Sections VIII.4 and
VIII.10] is a good reference5.
The content of a polynomial p over a UFD is defined to be the greatest common divisor
of the coefficients of p. For example, the polynomial 4x2 + 6y2 ∈ Z [x, y] has content
gcd (4, 6) = 2. (Of course, in a general UFD, the greatest common divisor is defined only
up to multiplication by a unit.) The following known facts are crucial to us:
Proposition 3.1. A polynomial ring over Z in finitely many indeterminates is always
a UFD. □
Proposition 3.1 appears, e.g., in [8, Remark after Corollary 8.21]. For a constructive
proof of Proposition 3.1, we refer to [9, Chapter IV, Theorems 4.8 and 4.9] or to [2, Es-
say 1.4, Corollary of Theorem 1 and Corollary 1 of Theorem 2].
Proposition 3.2. Let p be an irreducible element of a UFD K. Then the quotient ring
K/ (p) is an integral domain.
5We call “multiplicative subset” what Knapp (in [8, Section VIII.10]) calls a “multiplicative system”.
D. Grinberg: A double Sylvester determinant 265
Proof of Proposition 3.2. First of all, we recall that any irreducible element of a UFD is
prime (indeed, this follows from [8, Proposition 8.13]). Thus, the element p of K is prime.
Hence, [8, Proposition 8.14] shows that the ideal (p) of K is prime. Therefore, the quotient
ring K/ (p) is an integral domain. This proves Proposition 3.2.
We shall furthermore use the following properties of contents (whose proofs are easy):
Proposition 3.3. Let U be a UFD. Let p ∈ U [x1, x2, . . . , xm] be a polynomial over U.
Assume that the content of p is 1. Also assume that p is irreducible when considered as
a polynomial in F [x1, x2, . . . , xm], where F is the field of fractions of U. Then p is also
irreducible when considered as a polynomial in U [x1, x2, . . . , xm].
Proposition 3.4. Let U be a UFD. Let p, q ∈ U [x1, x2, . . . , xm] be two polynomials over
U. Assume that both p and q have content 1, and assume furthermore that p and q don’t
have any indeterminates in common (i.e., there is no i ∈ [m] such that degxi p > 0 and
degxi q > 0). Then p and q are coprime.
The next simple fact states that for any positive integer n, the determinant of the
“generic n × n-matrix” (i.e., of the n × n-matrix whose n2 entries are distinct indeter-
minates in a polynomial ring over Z) is irreducible as a polynomial over Z:
Corollary 3.5. Let n be a positive integer. Let G be the multivariate polynomial ring
Z
[
ai,j | (i, j) ∈ [n]2
]
. Let A ∈ Gn×n be the n × n-matrix (ai,j)1≤i≤n, 1≤j≤n. Then the
element detA of G is irreducible.
Proof of Corollary 3.5. A well-known fact (e.g., [1, Lemma 5.12]) shows that detA is
irreducible as an element of Q
[
ai,j | (i, j) ∈ [n]2
]
. This yields (using Proposition 3.3) that
detA is irreducible as an element of Z
[
ai,j | (i, j) ∈ [n]2
]
as well, since the polynomial
detA has content 1. This proves Corollary 3.5.
An element a of a commutative ring A is said to be regular if every b ∈ A satisfying
ab = 0 must satisfy b = 0. (Regular elements are also known as non-zero-divisors.) In a
polynomial ring, each indeterminate is regular; hence, each monomial (without coefficient)
is regular (since any product of two regular elements is regular).
We recall a few standard concepts from commutative algebra. Let K be a commutative
ring. A multiplicative subset of K means a subset S of K that contains the unity 1K of K
and has the property that every a, b ∈ S satisfy ab ∈ S.
If S is a multiplicative subset of K, then the localization of K at S is defined as follows:
Let ∼ be the binary relation on the set K× S defined by
((r, s) ∼ (r′, s′)) ⇐⇒ (t (rs′ − sr′) = 0 for some t ∈ S) .
Then it is easy to see that ∼ is an equivalence relation. The set L of its equivalence
classes [(r, s)] can be equipped with a ring structure via the rules [(r, s)] + [(r′, s′)] =
[(rs′ + sr′, ss′)] and [(r, s)] · [(r′, s′)] = [(rr′, ss′)] (with zero element [(0, 1)] and unity
[(1, 1)]). The resulting ring L is commutative, and is known as the localization of K at S.
(This generalizes the construction of Q from Z known from high school.)
The element [(r, s)] of L is denoted by rs . There is a canonical ring homomorphism
from K to L that sends each r ∈ K to [(r, 1)] = r1 ∈ L.
266 Ars Math. Contemp. 20 (2021) 261–274
When all elements of the multiplicative subset S are regular, the statement “t(rs′ −
sr′) = 0 for some t ∈ S” in the definition of the relation ∼ can be rewritten in the
equivalent (but much simpler) form “rs′ = sr′” (which is even more reminiscent of the
construction of Q).
The following fact is easy to see:
Proposition 3.6. Let K be a commutative ring. Let S be a multiplicative subset of K such
that all elements of S are regular. Let L be the localization of the ring K at S. Then:
(a) The canonical ring homomorphism from K to L is injective. We shall thus consider
it as an embedding.
(b) If K is an integral domain, then L is an integral domain.
(c) Let a and b be two elements of K. Then we have the following logical equivalence:
(a | b in L) ⇐⇒ (a | sb in K for some s ∈ S) .
Matrices over arbitrary commutative rings can behave a lot less predictably than matri-
ces over fields. However, matrices over integral domains still show a lot of the latter good
behavior, such as the following:
Proposition 3.7. Let P be a finite set. Let M be an integral domain. Let W ∈ MP×P be
a P × P -matrix over M. Let u ∈ MP be a vector such that u ̸= 0 and Wu = 0. Here, u
is considered as a “column vector”, so that Wu is defined by
Wu =
∑
q∈P
wp,quq

p∈P
, where W = (wp,q)(p,q)∈P×P and u = (up)p∈P .
Then detW = 0.
Proof of Proposition 3.7. Let m = |P |. Then we can view the P × P -matrix W as an
m × m-matrix (by “numerical reindexing”, as explained in [5, §1]), and we can view the
vector u as a column vector of size m. Let us do this from here on.
Let F be the quotient field of the integral domain M. Thus, there is a canonical embed-
ding of M into F. Hence, we can view the matrix W ∈ Mm×m as a matrix over F, and we
can view the vector u ∈ Mm as a vector over F. Let us do so from here on. We are now
in the realm of classical linear algebra over fields: The vector u ∈ Fm is nonzero (since
u ̸= 0) and belongs to the kernel of the m × m-matrix W ∈ Fm×m (since Wu = 0).
Hence, the kernel of the matrix W is nontrivial. In other words, this matrix W is singular.
Thus, detW = 0 by a classical fact of linear algebra. This proves Proposition 3.7.
Let us next recall an identity for determinants (a version of the Cauchy–Binet formula):
Lemma 3.8. Let n ∈ N, m ∈ N and p ∈ N. Let A ∈ Kn×p be an n × p-matrix. Let
B ∈ Kp×m be a p×m-matrix. Let k ∈ N. Let P be a subset of [n] such that |P | = k. Let
Q be a subset of [m] such that |Q| = k. Then
det
(
subQP (AB)
)
=
∑
R⊆[p];
|R|=k
det
(
subRP A
)
· det
(
subQR B
)
.
D. Grinberg: A double Sylvester determinant 267
Lemma 3.8 is [4, Corollary 7.251] (except that we are using the notation subJI C for
what is called subw(J)w(I) C in [4]). It also appears in [3, Chapter I, (19)] (where it is stated
using p-tuples instead of subsets).
The next lemma is just a particular case of Theorem 2.1, but it is a helpful stepping
stone on the way to proving the latter theorem:
Lemma 3.9. Let
A = (ai,j)1≤i≤n+1, 1≤j≤n+1 ∈ K
(n+1)×(n+1) and
B = (bi,j)1≤i≤n+1, 1≤j≤n+1 ∈ K
(n+1)×(n+1)
be such that bn+1,n+1 = 0. Assume further that
an+1,j = 0 for all j ∈ [n] . (3.1)
Define the Pk × Pk-matrix W as in Theorem 2.1. Then detA | detW .
The following proof is inspired by [6, proof of Theorem 10].
Proof of Lemma 3.9. We WLOG assume that K is the polynomial ring over Z in n2 +
(n+ 1) + ((n+ 1)
2 − 1) indeterminates
ai,j for all i ∈ [n] and j ∈ [n] ;
ai,n+1 for all i ∈ [n+ 1] ;
bi,j for all i ∈ [n+ 1] and j ∈ [n+ 1] except for bn+1,n+1.
And, of course, we assume that the entries of A and B that are not zero by assumption are
these indeterminates.6
The ring K is a UFD (by Proposition 3.1).
We WLOG assume that n > 0 (otherwise, the result follows from detW =
det
(
0
)
= 0).
The set Pk is nonempty (since k ∈ {0, 1, . . . , n}); thus, |Pk| ≥ 1.
Let A be the n × n-matrix (ai,j)1≤i≤n, 1≤j≤n ∈ K
n×n. Then, because of (3.1), we
have
detA = an+1,n+1 · detA (3.2)
(by [4, Theorem 6.43], applied to n+ 1 instead of n).
The matrix A is a completely generic n× n-matrix (i.e., its entries are distinct indeter-
minates); thus, its determinant detA is an irreducible polynomial in the polynomial ring
Z
[
ai,j | (i, j) ∈ [n]2
]
(by Corollary 3.5). Hence, detA also is an irreducible polynomial
in the ring K (since K differs from Z
[
ai,j | (i, j) ∈ [n]2
]
only in having more variables,
which clearly cannot contribute any factors to detA). Thus, Proposition 3.2 (applied to
p = detA) shows that the quotient ring K/(detA) is an integral domain.
Let M be the quotient ring K/(detA). Then M is an integral domain (since K/(detA)
is an integral domain). All monomials in the variables bi,j (with (i, j) ̸= (n+ 1, n+ 1))
are nonzero in M. Likewise, an+1,n+1 ̸= 0 in M.
6These assumptions are legitimate, because if we can prove Lemma 3.9 under these assumptions, then the
universal property of polynomial rings shows that Lemma 3.9 holds in the general case.
268 Ars Math. Contemp. 20 (2021) 261–274
Let w be the element
∏
j∈[n] bn+1,j ∈ M. (Strictly speaking, we mean the canonical
projection of
∏
j∈[n] bn+1,j ∈ K onto the quotient ring M.) Then, w is a nonzero element
of the integral domain M (since bn+1,j ̸= 0 in M for all j ∈ [n]).
For each i ∈ [n], we define zi ∈ M by zi =
∏
j∈[n]; j ̸=i bn+1,j (projected onto M).
This is a nonzero element of M. In M, we have
bn+1,izi = bn+1
∏
j∈[n];
j ̸=i
bn+1,j =
∏
j∈[n]
bn+1,j = w (3.3)
for all i ∈ [n].
We need another piece of notation: If M is a p × q-matrix, and if u ∈ [p] and v ∈ [q],
then M∼u,∼v denotes the (p− 1)× (q − 1)-matrix obtained from M by removing the u-th
row and the v-th column.
The matrix A∼1,∼(n+1) has determinant 0 (because (3.1) shows that its last row consists
of zeroes). In other words, det
(
A∼1,∼(n+1)
)
= 0.
Also, due to (3.1), we see that each i ∈ [n] satisfies
det (A∼1,∼i) = an+1,n+1 · det
(
A∼1,∼i
)
(3.4)
(by [4, Theorem 6.43], applied to A∼1,∼i instead of A), because the last row of the matrix
A∼1,∼i is (0, 0, . . . , 0, an+1,n+1).
For each i ∈ [n+ 1], we define an element ui ∈ M by
ui =
{
zi (−1)i det (A∼1,∼i) , if i ∈ [n] ;
1, if i = n+ 1.
Claim 1. All these n+ 1 elements u1, u2, . . . , un+1 of M are nonzero.
Proof of Claim 1. Let i ∈ [n]. Then, det
(
A∼1,∼i
)
̸= 0 in M because det
(
A∼1,∼i
)
is a
polynomial of smaller degree than detA, and thus is not a multiple of detA. Now,
ui = zi (−1)i
=an+1,n+1·det(A∼1,∼i) (by (3.4))︷ ︸︸ ︷
det (A∼1,∼i)
= zi︸︷︷︸
̸=0 in M
̸=0 in M︷ ︸︸ ︷
(−1)i an+1,n+1︸ ︷︷ ︸
̸=0 in M
·
̸=0 in M︷ ︸︸ ︷
det
(
A∼1,∼i
)
̸= 0 in M (since M is an integral domain).
Thus, u1, u2, . . . , un are nonzero. Moreover, un+1 is nonzero (since un+1 = 1). Thus, we
are done.
Let u = (uJ)J∈Pk ∈ M
Pk be the vector defined by
uJ =
∏
j∈J
uj .
D. Grinberg: A double Sylvester determinant 269
Then the entries of the vector u are nonzero (because they are products of the nonzero
elements u1, u2, . . . , un+1 of the integral domain M). Since the vector u has at least one
entry (because |Pk| ≥ 1), we thus conclude that u ̸= 0.
Let ∆ be the diagonal matrix ∆ = diag (u1, u2, . . . , un+1) ∈ M(n+1)×(n+1).
Let x ∈ Mn+1 be the column vector defined by
x =
(
(−1)1 det (A∼1,∼1) , (−1)2 det (A∼1,∼2) , . . . , (−1)n+1 det(A∼1,∼(n+1))
)T
.
Let (e1, e2, . . . , en+1) be the standard basis of the free M-module Mn+1. Thus, for any
(n+ 1)× (n+ 1)-matrix C ∈ M(n+1)×(n+1) and any j ∈ {1, 2, . . . , n+ 1}, we have
(the j-th column of the matrix C) = Cej . (3.5)
Now, using Laplace expansion, it is easy to see that
Ax = − detA · e1. (3.6)
To prove Equation (3.6), consider the adjugate adjA of the matrix A. A standard fact ([4,
Theorem 6.100]) says that A ·adjA = detA ·In+1. But the definition of adjA reveals that
the first column of the matrix adjA is −x. Hence, the first column of the matrix A · adjA
is A · (−x) = −Ax. On the other hand, the first column of the matrix A ·adjA is detA ·e1
(since A · adjA = detA · In+1). Comparing the preceding two sentences, we conclude
that −Ax = detA · e1, so that Ax = −detA · e1. This proves Equation (3.6).
Also, Equation (3.5) (applied to C = BT and j = n+ 1) yields
BT en+1 =
(
the (n+ 1)-st column of the matrix BT
)
= (bn+1,1, bn+1,2, . . . , bn+1,n+1)
T
.
Hence,
∆BT en+1 = ∆(bn+1,1, bn+1,2, . . . , bn+1,n+1)
T
= (u1bn+1,1, u2bn+1,2, . . . , un+1bn+1,n+1)
T
(3.7)
(since ∆ = diag (u1, u2, . . . , un+1)).
Claim 2. We have
uibn+1,i = w · (−1)i det (A∼1,∼i) for each i ∈ [n+ 1] . (3.8)
Proof of Claim 2. Let i ∈ [n+ 1]. If i = n+ 1, then both sides of (3.8) are zero (because
bn+1,n+1 = 0 and det
(
A∼1,∼(n+1)
)
= 0). If i ̸= n+1, then i ∈ [n] and thus the definition
of ui yields ui = zi(−1)i det(A∼1,∼i). Hence,
uibn+1,i = zi (−1)i det (A∼1,∼i) bn+1,i = bn+1,izi︸ ︷︷ ︸
=w (by (3.3))
(−1)i det (A∼1,∼i)
= w · (−1)i det (A∼1,∼i) .
Hence, Equation (3.8) is proven in both cases.
270 Ars Math. Contemp. 20 (2021) 261–274
Now, (3.7) becomes
∆BT en+1
= (u1bn+1,1, u2bn+1,2, . . . , un+1bn+1,n+1)
T
=
(
w · (−1)1 det (A∼1,∼1) , w · (−1)2 det (A∼1,∼2) , . . . ,
w · (−1)n+1 det
(
A∼1,∼(n+1)
) )T
(by (3.8))
= w ·
(
(−1)1 det (A∼1,∼1) , (−1)2 det (A∼1,∼2) , . . . , (−1)n+1 det
(
A∼1,∼(n+1)
))T
︸ ︷︷ ︸
=x (by the definition of x)
= wx.
Hence,
A∆BT en+1 = Awx = w ·
=− detA·e1 (by (3.6))︷︸︸︷
Ax = −w · detA︸ ︷︷ ︸
=an+1,n+1·detA (by (3.2))
· e1
= −w · an+1,n+1 · detA︸ ︷︷ ︸
=0 (since we are in M)
· e1 = 0.
In other words, the (n+ 1)-st column of the matrix A∆BT is 0 (since the (n+ 1)-st col-
umn of the matrix A∆BT is A∆BT en+1 (by (3.5), applied to C = A∆BT and j = n+1)).
Now, fix I ∈ Pk. Then, the last column of the matrix subI+I+(A∆BT ) is 0 (because
this column is a piece of the (n+ 1)-st column of the matrix A∆BT , but as we have just
shown the latter column is 0). Thus, det
(
subI+I+(A∆B
T )
)
= 0.
But Lemma 3.8 (applied to M, n + 1, n + 1, n + 1, ∆BT , k + 1, I+ and I+ instead
of K, n, m, p, B, k, P and Q) yields
det
(
subI+I+(A∆B
T )
)
=
∑
R⊆[n+1];
|R|=k+1
det
(
subRI+ A
)
det
(
subI+R (∆B
T )
)
.
Comparing this with det
(
subI+I+(A∆B
T )
)
= 0, we obtain
0 =
∑
R⊆[n+1];
|R|=k+1
det
(
subRI+ A
)
det
(
subI+R (∆B
T )
)
.
In the sum on the right hand side, all addends for which n + 1 /∈ R are zero (because if
R ⊆ [n+ 1] satisfies |R| = k + 1 and n+ 1 /∈ R, then the last row of the matrix subRI+ A
consists of zeroes (by (3.1), since n + 1 /∈ R but n + 1 ∈ I+), and therefore we have
det
(
subRI+ A
)
= 0), and thus can be discarded. Hence, we are left with
0 =
∑
R⊆[n+1];
|R|=k+1;
n+1∈R
det
(
subRI+ A
)
det
(
subI+R (∆B
T )
)
.
D. Grinberg: A double Sylvester determinant 271
But the subsets R of [n+ 1] satisfying |R| = k + 1 and n+ 1 ∈ R can be parametrized as
J+ with J ranging over Pk. Hence, this rewrites further as
0 =
∑
J∈Pk
det
(
subJ+I+ A
)
det
(
subI+J+(∆B
T )
)
.
It is easily seen that det
(
subI+J+(∆B
T )
)
= det
(
subJ+I+ B
)
uJ for each J ∈ Pk (indeed,
recall the definition of ∆ and the fact that un+1 = 1 and that det
(
CT
)
= detC for each
square matrix C). Thus, the above equality simplifies to
0 =
∑
J∈Pk
det
(
subJ+I+ A
)
det
(
subJ+I+ B
)
uJ .
Now, forget that we fixed I . We thus have proven that
0 =
∑
J∈Pk
det
(
subJ+I+ A
)
det
(
subJ+I+ B
)
uJ (3.9)
for each I ∈ Pk. This rewrites as Wu = 0 (indeed, the left hand side of (3.9) is the I-th
entry of the zero vector 0, whereas the right hand side of (3.9) is the I-th entry of Wu).
Now, consider the matrix W as a matrix in MPk×Pk . Then, Proposition 3.7 (applied to
P = Pk) yields detW = 0 in M (since u ̸= 0 and Wu = 0). In view of the definition of
M, this rewrites as detA | detW in K.
Let us consider the matrix W again as a matrix over K. Each entry of W has the form
det
(
subJ+I+ A
)
det
(
subJ+I+ B
)
for some I, J ∈ Pk.
Claim 3. det
(
subJ+I+ A
)
is a multiple of an+1,n+1 for all I, J ∈ Pk.
Proof of Claim 3. Let I, J ∈ Pk. Then, the equality (3.1) shows that the last row of the
matrix subJ+I+ A is (0, 0, . . . , 0, an+1,n+1). Hence, an application of [4, Theorem 6.43]
shows that det
(
subJ+I+ A
)
= an+1,n+1 det
(
subJI A
)
. Thus, det
(
subJ+I+ A
)
is a multiple
of an+1,n+1.
By Claim 3, all entries of W are multiples of an+1,n+1. Hence, the determinant of W
is a multiple of (an+1,n+1)
|Pk|, thus a multiple of an+1,n+1 (since |Pk| ≥ 1). In other
words, an+1,n+1 | detW in K.
Recall that K is a UFD. Also, the two polynomials an+1,n+1 and detA in K both
have content 1, and don’t have any indeterminates in common; thus, these two polynomials
are coprime (by Proposition 3.4). Hence, any polynomial in K that is divisible by both
an+1,n+1 and detA must be divisible by the product an+1,n+1 · detA as well. Thus, from
an+1,n+1 | detW and detA | detW , we obtain an+1,n+1 · detA | detW . In view of
(3.2), this rewrites as detA | detW . This proves Lemma 3.9.
We shall now derive Theorem 2.2 from Lemma 3.9, following the same idea as in [12,
§2.7] and [13, Teorema 2.9.1] and [10]:
Proof of Theorem 2.1. We WLOG assume that n > 0 (otherwise, the result follows from
detW = det
(
0
)
= 0).
272 Ars Math. Contemp. 20 (2021) 261–274
We WLOG assume that K is the polynomial ring over Z in (n+ 1)2 + ((n+ 1)2 − 1)
indeterminates
ai,j for all i ∈ [n+ 1] and j ∈ [n+ 1] ;
bi,j for all i ∈ [n+ 1] and j ∈ [n+ 1] except for bn+1,n+1.
And, of course, we assume that the entries of A and B that are not zero by assumption
are these indeterminates. Proposition 3.1 shows that the ring K is a UFD (since it is a
polynomial ring over Z).
Let S be the multiplicative subset
{
apn+1,n+1 | p ∈ N
}
of K. Then, all elements of S
are regular (since they are monomials in a polynomial ring).
Let L be the localization of the commutative ring K at the multiplicative subset S. Then,
Proposition 3.6(a) shows that the canonical ring homomorphism from K to L is injective;
we shall thus consider it as an embedding. Also, Proposition 3.6(b) shows that L is an
integral domain.
Claim 1. We claim that
detA | detW in L. (3.10)
Proof of Claim 1. Consider A, B and W as matrices over L. The entry an+1,n+1 of A
is invertible in L (by the construction of L). Hence, we can subtract appropriate scalar
multiples7 of the (n+ 1)-st column of A from each other column of A to ensure that
all entries of the last row of A become 0, except for an+1,n+1. (Specifically, for each
j ∈ [n], we have to subtract aj,n+1/an+1,n+1 times the (n+ 1)-st column of A from the j-
th column of A.) All these column transformations preserve the determinant detA, and also
preserve the minors det
(
subJ+I+ A
)
for all I, J ∈ Pk (because when the (n+ 1)-st column
of A is subtracted from another column of A, the matrix subJ+I+ A either stays the same or
undergoes an analogous column transformation8, which preserves its determinant); thus,
they preserve the matrix W . Hence, we can replace A by the result of these transformations.
This new matrix A satisfies (3.1). Hence, Lemma 3.9 (applied to L instead of K) yields
that detA | detW in L. This proves (3.10).
But we must prove that detA | detW in K. Fortunately, this is easy: Since K embeds
into L, we can translate our result “detA | detW in L” as “detA | apn+1,n+1 detW in K
for an appropriate p ∈ N” (by Proposition 3.6(c), applied to a = detA and b = detW ).
Consider this p.
Claim 2. The polynomial an+1,n+1 ∈ K is coprime to detA.
Proof of Claim 2. The polynomial detA contains the monomial a1,n+1a2,n · · · an+1,1 =∏
i∈[n+1] ai,n+2−i, and thus is not a multiple of an+1,n+1. Hence, it is coprime to an+1,n+1
(since the only non-unit divisor of an+1,n+1 is an+1,n+1 itself, up to scaling by units).
So we know that an+1,n+1 is coprime to detA. Hence, its power a
p
n+1,n+1 is co-
prime to detA as well. Hence, we can cancel the apn+1,n+1 from the divisibility detA |
apn+1,n+1 detW , and conclude that detA | detW in K. This proves Theorem 2.1.
7The scalars, of course, come from L here.
8Here we are using the fact that n + 1 ∈ J+ (so that the matrix subJ+I+ A contains part of the (n+ 1)-st
column of A).
D. Grinberg: A double Sylvester determinant 273
Proof of Theorem 2.2. We WLOG assume that K is the polynomial ring over Z in the
((n+ 1)
2 − 1) + ((n+ 1)2 − 1) indeterminates
ai,j for all i ∈ [n+ 1] and j ∈ [n+ 1] except for an+1,n+1;
bi,j for all i ∈ [n+ 1] and j ∈ [n+ 1] except for bn+1,n+1.
And, of course, we assume that the entries of A and B that are not zero by assumption are
these indeterminates. The ring K is a UFD (by Proposition 3.1).
WLOG assume that n > 0 (otherwise, the result follows from detW = det
(
0
)
= 0).
Thus, the monomial a1,n+1a2,n · · · an+1,1 =
∏
i∈[n+1] ai,n+2−i occurs in the polynomial
detA with coefficient ±1. Hence, the polynomial detA has content 1. Similarly, the
polynomial detB has content 1.
Theorem 2.1 yields detA | detW . The same argument yields detB | detW (since
the matrices A and B play symmetric roles in the construction of W ). But Proposition 3.4
shows that the polynomials detA and detB in K are coprime (because they have content 1,
and don’t have any indeterminates in common). Thus, any polynomial in K that is divisible
by both detA and detB must be divisible by the product (detA) (detB) as well. Thus,
from detA | detW and detB | detW , we obtain (detA) (detB) | detW . This proves
Theorem 2.2.
4 Further questions
While Theorems 2.1 and 2.2 are now proven, the field appears far from fully harvested.
Three questions readily emerge:
Question 4.1. What can be said about detWdetA (in Theorem 2.1) and
detW
(detA)(detB) (in Theo-
rem 2.2)? Are there formulas?
Question 4.2. Are there more direct proofs of Theorems 2.1 and 2.2, avoiding the use of
polynomial rings and their properties and instead “staying inside K”? Such proofs might
help answer the previous question.
Question 4.3. The entries of our matrix W were products of minors of two (n+ 1) ×
(n+ 1)-matrices that each use the last row and the last column. What can be said about
products of minors of two (n+m)× (n+m)-matrices that each use the last m rows and
the last m columns, where m is an arbitrary positive integer? The “Generalized Sylvester’s
identity” in [12, §2.7] answers this for the case of one matrix. It is not quite obvious what
the right analogues of the conditions an+1,n+1 = 0 and bn+1,n+1 = 0 are; furthermore,
nontrivial examples become even more computationally challenging.
ORCID iD
Darij Grinberg https://orcid.org/0000-0002-9661-8432
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 275–288
https://doi.org/10.26493/1855-3974.2229.5f1
(Also available at http://amc-journal.eu)
Boundary-type sets of strong product
of directed graphs*
Bijo S. Anand
Department of Mathematics, Sree Narayana College, Punalur, Kollam, India-691305
Manoj Changat
Department of Futures Studies, University of Kerala, Trivandrum, India-695581
Prasanth G. Narasimha-Shenoi † , Mary Shalet Thottungal Joseph ‡
Department of Mathematics, Government College Chittur, Palakkad, India-678104
Received 26 January 2020, accepted 8 March 2021, published online 18 November 2021
Abstract
Let D = (V,E) be a strongly connected digraph and let u and v be two vertices in
D. The maximum distance md(u, v) is defined as md(u, v) = max{d⃗(u, v), d⃗(v, u)},
where d⃗(u, v) denotes the length of a shortest directed u-v path in D. This is a metric. The
boundary, contour, eccentricity and periphery sets of a strongly connected digraph D with
respect to this metric have been defined. The boundary-type sets of the strong product of
two digraphs is investigated in this article.
Keywords: Maximum distance, boundary-type sets, strongly connected digraph, strong product.
Math. Subj. Class. (2020): 05C12, 05C20, 05C76
*The authors thank the anonymous referees for their valuable suggestions, which helped improve this article.
†Supported by Science and Engineering Research Board, a statutory body of Government of India under
their Extra Mural Research Funding No. EMR/2015/002183 and MATRICS Scheme No. MTR/2018/000012.
Supported by Kerala State Council for Science Technology and Environment of Government of Kerala under
their SARD project grant Council(P) No. 436/2014/KSCSTE.
‡Corresponding author. Supported by Science and Engineering Research Board, a statutory body of Gov-
ernment of India under their Extra Mural Research Funding No. EMR/2015/002183. Supported by Kerala State
Council for Science Technology and Environment of Government of Kerala under their SARD project grant Coun-
cil(P) No. 436/2014/KSCSTE.
E-mail addresses: bijos_anand@yahoo.com (Bijo S. Anand), mchangat@keralauniversity.ac.in (Manoj
Changat), prasanthgns@gmail.com (Prasanth G. Narasimha-Shenoi), mary_shallet@yahoo.co.in (Mary Shalet
Thottungal Joseph)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
276 Ars Math. Contemp. 20 (2021) 275–288
1 Introduction
Directed graphs or in short digraphs have immense applications in almost all areas of sci-
ence and even in sociology. A directed network is a network in which each edge has a
direction, pointing from one vertex to another. They can be represented as directed graphs.
Road traffic networks are the most frequently met examples of one-way networks. A
two-way street is one in which vehicles are allowed to travel in both directions. The advan-
tages of a one-way street network over a two-way street pattern are discussed in [14]. But
when one-way traffic is introduced in a two-way network, the distance between places in
one of the directions may increase. So the problem of designing a network is to minimize
the distance between places and the cost of construction.
The one-way problem was first studied by Robbins [13]. It finds applications in various
fields like computer science, biology, etc. In [2], directed graphs are used to analyze the
local properties of internet connectivity. Neurons are connected in intricate communication
networks established during development to convey sensory information from peripheral
receptors of sensory neurons to the central nervous system and to convey commands from
the central nervous system to effector organs [12].
The boundary-type sets of a graph, the boundary, contour, eccentricity, and periph-
ery sets of a graph were studied in [5] and [7]. It is very difficult to identify the various
boundary-type sets in large networks. So we try to decompose the network into smaller
networks and identify the boundary-type sets. The four standard graph products, namely
Cartesian, direct, strong, and lexicographic products can be extended to digraphs as well.
Marc Hellmuth and Tilen Marc developed a polynomial-time algorithm for determining the
prime factor decomposition of strong product of digraphs [11].
The directed distance defined in digraphs is generally not a metric. As we are concerned
with the problem of designing the network to minimize the distance between places at a
minimum cost, we consider the distance maximum distance or in short, m-distance which
is a metric that was introduced by Chartrand and Tian in [8]. It gives the maximum of the
directed distances in either direction and is denoted by md(u, v). So minimizing md(u, v)
results in minimizing the distance between the nodes in both directions. An undirected
graph G can be identified as a symmetric digraph, that is, one for which (u, v) ∈ E(G) if
and only if (v, u) ∈ E(G), and the metric md is the usual distance metric in undirected
graphs.
The boundary-type sets of the Cartesian product of two digraphs were studied in [6]. In
this paper, a similar study is conducted for the strong product of digraphs.
2 Preliminaries
A directed graph or a digraph D consists of a non-empty finite set V (D) of elements called
vertices and a finite set E(D) of ordered pairs of distinct vertices called arcs or edges [1].
We call V (D) the vertex set and E(D) the edge set of D. We write D = (V,E) to denote
the digraph D with vertex set V and edge set E. For an edge (u, v), the first vertex u of
the ordered pair is the tail of the edge and the second vertex v is the head; together they
are the endpoints. This definition of a digraph does not allow loops (edges whose head and
tail coincide) or parallel edges (pairs of edges with the same tail and the same head). The
underlying graph UD of a digraph D is the simple graph with the vertex set V (D) and the
unordered pair (x, y) ∈ E(UD) if and only if either (x, y) ∈ E(D) or (y, x) ∈ E(D).
The following concepts are taken from [1].
B. S. Anand et al.: Boundary-type sets of strong product of directed graphs 277
For a vertex v in a digraph D = (V,E), the neighborhoods are defined as follows:
N+D (v) = {w ∈ V : (v, w) ∈ E}, N
−
D (v) = {u ∈ V : (u, v) ∈ E}.
The sets N+D (v), N
−
D (v), and ND(v) = N
+
D (v)∪N
−
D (v) are called the out-neighborhood,
in-neighborhood, and neighborhood of v. These neighborhoods are called open neighbor-
hoods of v. Similarly, we can define closed neighborhoods of v (neighbors including v).
The closed neighborhood of v is denoted by ND[v]. That is, ND[v] = ND(v) ∪ {v}.
A directed path is a directed graph with V (P ) ̸= ∅ with distinct vertices u1, u2, . . . , uk
and edges e1, e2, . . . , ek−1 such that ei is an edge directed from ui to ui+1 for 1 ≤ i ≤
k − 1. In this article, a path will always mean a ‘directed path’. A digraph is strongly
connected or strong if, for each ordered pair (u, v) of vertices, there is a path from u to v.
A digraph is weakly connected if its underlying graph is connected. A strong component of
a digraph D is a maximal induced subdigraph of D which is strong. If D1, D2, . . . , Dt are
the strong components of D, then V (D1) ∪ V (D2) ∪ · · · ∪ V (Dt) = V (D) and V (Di) ∩
V (Dj) = ∅ for every i ̸= j.
The length of a path is the number of edges in the path. The directed distance d⃗(u, v)
between two vertices u, v ∈ V (D) is the length of the shortest directed path from u to
v, or infinity if no such path exists. Note that this distance is not a metric, as generally
d⃗(u, v) ̸= d⃗(v, u).
So in [8], Chartrand and Tian introduced two other distances between the vertices u and
v in a strong digraph, namely the maximum distance md(u, v) = max{d⃗(u, v), d⃗(v, u)}
and the sum distance sd(u, v) = d⃗(u, v)+d⃗(v, u), both of which are metrics. In this article,
we deal with the maximum distance, md .
Remark 2.1. md(u, v) is denoted by d(u, v) hereafter.
The m-eccentricity of a vertex v, the m-radius and the m-diameter of a digraph D
are also defined in [8]. Consistent with our notation d(u, v) for maximum distance be-
tween the vertices u and v, we denote them respectively as ecc(v), rad(D), and diam(D).
Thus, ecc(v) = maxu∈V (D) d(v, u), rad(D) = minv∈V (D) ecc(v), and diam(D) =
maxv∈V (D) ecc(v), where ecc(v) denotes m-eccentricity of v.
If a digraph D is strongly connected, then the maximum distance between every pair
of vertices is finite, and hence the m-eccentricity of every vertex in D is finite. Otherwise,
D has more than one strong component, and the maximum distance between two vertices
lying in different strong components of D is infinity. So if D is not strongly connected,
then the m-eccentricity of every vertex in D is infinity.
2.1 Definitions of boundary-type sets
We define the boundary-type sets of a digraph D with respect to the metric maximum
distance. Most of the following definitions are analogous to the definitions in [7]. Let D
be a strong digraph and u, v ∈ V (D). The vertex v is said to be a boundary vertex of u if
no neighbor of v is further away from u than v. Hereafter, we denote ND(v) and ND[v] by
N(v) and N [v], respectively.
A vertex v is called a boundary vertex of D if it is the boundary vertex of some vertex
u ∈ V (D).
Definition 2.2. The boundary ∂(D) of D is the set of all of its boundary vertices
∂(D) = {v ∈ V (D) : ∃u ∈ V (D) such that ∀w ∈ N(v), d(u,w) ≤ d(u, v)}.
278 Ars Math. Contemp. 20 (2021) 275–288
Given u, v ∈ V (D), the vertex v is called an eccentric vertex of u if no vertex in V (D)
is further away from u than v; that is, if d(u, v) = ecc(u). A vertex v is called an eccentric
vertex of digraph D if it is the eccentric vertex of some vertex u ∈ V (D).
Definition 2.3. The eccentricity Ecc(D) of a digraph D is the set of all of its eccentric
vertices
Ecc(D) = {v ∈ V (D) : ∃u ∈ V (D) such that ecc(u) = d(u, v)}.
In a similar way, we can define the eccentricity of any proper subset W of the vertex
set V (D):
Ecc(W ) = {v ∈ V (D) : ∃u ∈ W such that ecc(u) = d(u, v)}.
A vertex v ∈ V (D) is called a peripheral vertex of D if no vertex in V (D) has an
eccentricity greater than ecc(v); that is, if the eccentricity of v is equal to the diameter
diam(D) of D.
Definition 2.4. The periphery Per(D) of a digraph D is the set of all of its peripheral
vertices
Per(D) = {v ∈ V (D) : ecc(u) ≤ ecc(v),∀u ∈ V (D)}
= {v ∈ V (D) : ecc(v) = diam(D)}.
A vertex v ∈ V (D) is called a contour vertex of digraph D if no neighbor vertex of v
has an eccentricity greater than ecc(v). The following definition is from [5].
Definition 2.5. The contour Ct(D) of a digraph D is the set of all of its contour vertices
Ct(D) = {v ∈ V (D) : ecc(u) ≤ ecc(v),∀u ∈ N(v)}.
As in the case of undirected graphs [3] we have,
1. Per(D) ⊆ Ct(D) ∩ Ecc(D),
2. Ecc(D) ∪ Ct(D) ⊆ ∂(D).
This is because a peripheral vertex is a vertex having the maximum eccentricity in the
digraph D and so every peripheral vertex in D is a contour vertex in D as well as the
eccentric vertex of a diametrical vertex in D.
If v is an eccentric vertex of a vertex u, then v is a boundary vertex of u. Also if v
is a contour vertex, then ecc(u) ≤ ecc(v) for all u ∈ N(v). So there exists some vertex
w ∈ V (D) such that d(w, u) ≤ d(w, v) for all u ∈ N(v), and hence v is a boundary vertex
of w.
The open neighborhood N(v) can be replaced by the closed neighborhood N [v] in the
definitions of the boundary and the contour sets. This does not affect the definitions and
is necessary for proving the relationship between the boundary and the contour sets of the
strong product of two digraphs and its factors.
B. S. Anand et al.: Boundary-type sets of strong product of directed graphs 279
3 Strong product of directed graphs
The strong product D1 ⊠ D2 of two digraphs D1 and D2 with vertex sets V (D1) =
{u1, u2, . . . , um} and V (D2) = {v1, v2, . . . , vn} is the digraph having the vertex set
V (D1) × V (D2) and with arc set E(D1 ⊠ D2) defined as follows. A vertex (ui, vr) is
adjacent to (uj , vs) in D1 ⊠D2 if either
1. (ui, uj) ∈ E(D1), vr = vs, or
2. ui = uj , (vr, vs) ∈ E(D2), or
3. (ui, uj) ∈ E(D1), (vr, vs) ∈ E(D2).
The strong product of digraphs is commutative [10]. The distance between two vertices
(g, h) and (g′, h′) in the strong product G ⊠ H of two graphs G and H is given in [9] as
follows:
dG⊠H((g, h), (g
′, h′)) = max{dG(g, g′), dH(h, h′)}.
So in the case of two digraphs D1 and D2, it follows that the directed distance
d⃗D1⊠D2((ui, vr), (uj , vs)) = max{d⃗D1(ui, uj), d⃗D2(vr, vs)}.
Lemma 3.1. Let D1 and D2 be two strongly connected digraphs. Then
dD1⊠D2((ui, vr), (uj , vs)) = max{dD1(ui, uj), dD2(vr, vs)},
eccD1⊠D2(ui, vr) = max{eccD1(ui), eccD2(vr)}.
Proof.
dD1⊠D2((ui, vr), (uj , vs))
= max{d⃗D1⊠D2((ui, vr), (uj , vs)), d⃗D1⊠D2((uj , vs), (ui, vr))}
= max{max{d⃗D1(ui, uj), d⃗D2(vr, vs)},max{d⃗D1(uj , ui), d⃗D2(vs, vr)}}
= max{max{d⃗D1(ui, uj), d⃗D2(vr, vs), d⃗D1(uj , ui), d⃗D2(vs, vr)}}
= max{max{d⃗D1(ui, uj), d⃗D1(uj , ui)},max{d⃗D2(vr, vs), d⃗D2(vs, vr)}}
= max{dD1(ui, uj), dD2(vr, vs)}.
Hence it follows that
eccD1⊠D2(ui, vr)
= max{dD1⊠D2((ui, vr), (uj , vs)) : (uj , vs) ∈ V (D1 ⊠D2)}
= max{max{dD1(ui, uj), dD2(vr, vs)} : uj ∈ V (D1), vs ∈ V (D2)}
= max{max{dD1(ui, uj) : uj ∈ V (D1)},max{dD2(vr, vs) : vs ∈ V (D2)}}
= max{eccD1(ui), eccD2(vr)}.
Corollary 3.2. Let D1 and D2 be two strongly connected digraphs. Then
rad(D1 ⊠D2) = max{rad(D1), rad(D2)},
diam(D1 ⊠D2) = max{diam(D1),diam(D2)}.
280 Ars Math. Contemp. 20 (2021) 275–288
Proof.
rad(D1 ⊠D2) = min
(ui,vr)∈V (D1⊠D2)
ecc(ui, vr)
= min
ui∈V (D1)
vr∈V (D2)
max{eccD1(ui), eccD2(vr)}
= max{ min
ui∈V (D1)
ecc(ui), min
vr∈V (D2)
ecc(vr)}
= max{rad(D1), rad(D2)},
diam(D1 ⊠D2) = max
(ui,vr)∈V (D1⊠D2)
ecc(ui, vr)
= max
ui∈V (D1)
vr∈V (D2)
max{eccD1(ui), eccD2(vr)}
= max{ max
ui∈V (D1)
ecc(ui), max
vr∈V (D2)
ecc(vr)}
= max{diam(D1),diam(D2)}.
The strong product of two directed graphs is strongly connected if and only if both
the digraphs are strongly connected [9]. Also if G and H are two undirected graphs,
NG⊠H [(g, h)] = NG[g]×NH [h] [9]. Since the neigbors of a vertex in a directed graph are
exactly its neighbors in the underlying graph, it follows that
ND1⊠D2 [(ui, vr)] = NG⊠H [(ui, vr)] = NG[ui]×NH [vr] = ND1 [ui]×ND2 [vr],
where G and H are the underlying graphs of D1 and D2, respectively. In [4], Cáceres
et al. presented a description of the boundary-type sets of two undirected graphs and the
description of the boundary is as follows.
For two graphs G and H , ∂(G ⊠H) = (∂(G) × V (H)) ∪ (V (G) × ∂(H)). But this
result does not hold in the case of directed graphs.
Consider the strong product, D1 ⊠ D2 of the digraphs D1 and D2 in Figure 1. The
eccentricity of each vertex is displayed near the vertex in red color.
Per(D1) = Ecc(D1) = Ct(D1) = {u1, u4},
Per(D2) = Ecc(D2) = Ct(D2) = {v1, v2}, and
Per(D1 ⊠D2) = Ecc(D1 ⊠D2) = Ct(D1 ⊠D2)
= {(u1, v1), (u4, v1), (u1, v2), (u4, v2)}.
∂(D1) = {u1, u4},
∂(D2) = {v1, v2}, and
∂(D1 ⊠D2) = {(u1, v1), (u4, v1), (u1, v2), (u4, v2)}.
The reason for (u2, v1), (u2, v2), (u3, v1), (u3, v2) /∈ ∂(D1 ⊠ D2) is explained after the
proof of Theorem 3.3.
Now we present the results concerning the boundary-type sets of the strong product of
two strongly connected digraphs. In all these results, D1 and D2 can be interchanged due
to the commutativity of strong product of digraphs.
B. S. Anand et al.: Boundary-type sets of strong product of directed graphs 281
v2
1
v1
1
D2
(u1, v1)
3
(u2, v1)
2
(u3, v1)
2
(u4, v1)
3
(u1, v2)
3
(u2, v2)
2
(u3, v2)
2
(u4, v2)
3
u1
3
u2
2
u3
2
u4
3
D1
Figure 1: D1 ⊠D2.
We have, ∂(D1 ⊠D2) ⊆ [∂(D1)× V (D2)] ∪ [V (D1)× ∂(D2)].
To this end, let (ui, vr) ∈ ∂(D1 ⊠ D2). Then there exists a vertex (uj , vs) ∈
V (D1 ⊠ D2) such that d((uj , vs), (ui, vr)) ≥ d((uj , vs), (uk, vq)) for every (uk, vq) ∈
N [(ui, vr)]. This implies, max{d(uj , ui), d(vs, vr)} ≥ max{d(uj , uk), d(vs, vq)} for
every uk ∈ N [ui] and for every vq ∈ N [vr]. Hence d(uj , ui) ≥ d(uj , uk) for ev-
ery uk ∈ N [ui], or d(vs, vr) ≥ d(vs, vq) for every vq ∈ N [vr]. Thus, ui ∈ ∂(D1) or
vr ∈ ∂(D2) or both. That is, if (ui, vr) ∈ ∂(D1 ⊠D2), then at least one of the vertices ui
and vr must be a boundary vertex in the corresponding factor graph.
Theorem 3.3. Let D1 and D2 be two strongly connected digraphs. Then ∂(D1 ⊠D2) =
A1 ∪A2 ∪A3, where
A1 = ∂(D1)× ∂(D2),
A2 = {(ui, vr) ∈ V (D1 ⊠D2) : ui ∈ ∂(D1), vr /∈ ∂(D2), and
∃vt ∈ V (D2) such that d(vt, vq) ≤ ecc(ui),∀vq ∈ N [vr]},
A3 = {(ui, vr) ∈ V (D1 ⊠D2) : ui /∈ ∂(D1), vr ∈ ∂(D2), and
∃uℓ ∈ V (D1) such that d(uℓ, uk) ≤ ecc(vr),∀uk ∈ N [ui]}.
Proof. Suppose that (ui, vr) ∈ ∂(D1 ⊠D2).
282 Ars Math. Contemp. 20 (2021) 275–288
Then there exists a vertex (uj , vs) ∈ V (D1 ⊠ D2) such that d((uj , vs), (ui, vr)) ≥
d((uj , vs), (uk, vq)) for all vertices (uk, vq) ∈ N [(ui, vr)]. Since d((uj , vs), (ui, vr)) =
max{d(uj , ui), d(vs, vr)} and d((uj , vs), (uk, vq)) = max{d(uj , uk), d(vs, vq)}, we get
max{d(uj , ui), d(vs, vr)} ≥ max{d(uj , uk), d(vs, vq)} for all uk ∈ N [ui], vq ∈ N [vr].
We distinguish four cases:
1. max{d(uj , ui), d(vs, vr)} = d(uj , ui) and d(vs, vr) ≥ d(vs, vq) for all vq ∈ N [vr];
2. max{d(uj , ui), d(vs, vr)} = d(uj , ui) and d(vs, vr) ≥ d(vs, vq) does not hold for all
vq ∈ N [vr];
3. max{d(uj , ui), d(vs, vr)} = d(vs, vr) and d(uj , ui) ≥ d(uj , uk) for all uk ∈ N [ui];
4. max{d(uj , ui), d(vs, vr)} = d(vs, vr) and d(uj , ui) ≥ d(uj , uk) does not hold for
all uk ∈ N [ui].
In cases 1 and 3, d(uj , ui) ≥ d(uj , uk) for all uk ∈ N [ui] and d(vs, vr) ≥ d(vs, vq)
for all vq ∈ N [vr]. So ui ∈ ∂(D1), vr ∈ ∂(D2), and hence (ui, vr) ∈ A1.
In case 2, ui ∈ ∂(D1) and vr is not a boundary vertex of vs in D2. If there exists any
vertex vt such that vr is a boundary vertex of vt, then we get (ui, vr) ∈ A1. Otherwise,
since vr /∈ ∂(D2), for every vertex vt ∈ V (D2), there exists some vertex vq ∈ N [vr] such
that d(vt, vr) < d(vt, vq). Hence if (ui, vr) is a boundary vertex of a vertex (uℓ, vt) in
D1 ⊠D2, then d((uℓ, vt), (ui, vr)) = max{d(uℓ, ui), d(vt, vr)} = d(uℓ, ui) > d(vt, vr),
for otherwise d(uℓ, ui) ≤ d(vt, vr) and so we get d((uℓ, vt), (ui, vr)) = d(vt, vr) <
d(vt, vq) = d((uℓ, vt), (ui, vq)), where (ui, vq) ∈ N [(ui, vr)].
Let (uk, vq) ∈ N [(ui, vr)]. Then d((uℓ, vt), (uk, vq)) = max{d(uℓ, uk), d(vt, vq)}. If
(ui, vr) is a boundary vertex of (uℓ, vt), then max{d(uℓ, ui), d(vt, vr)} ≥ max{d(uℓ, uk),
d(vt, vq)}. So the necessary condition for the vertex (ui, vr) such that ui ∈ ∂(D1) and vr /∈
∂(D2) to be a boundary vertex of the vertex (uℓ, vt) in D1 ⊠D2 is d(uℓ, ui) ≥ d(vt, vq)
for all vq ∈ N [vr]. Since ecc(ui) ≥ d(uℓ, ui) for all uℓ ∈ V (D1), the necessary condition
becomes ecc(ui) ≥ d(vt, vq) for all vq ∈ N [vr]. Thus, (ui, vr) ∈ A2.
Thus in case 2, (ui, vr) ∈ A1 ∪A2.
In case 4, vr ∈ ∂(D2) and ui is not a boundary vertex of uj in D1. As in case 2, it
follows that (ui, vr) ∈ A1 ∪A3.
Thus in all cases, we get ∂(D1 ⊠D2) ⊆ A1 ∪A2 ∪A3.
Conversely, suppose that (ui, vr) ∈ A1 ∪ A2 ∪ A3. First let (ui, vr) ∈ A1. Then
ui ∈ ∂(D1) and vr ∈ ∂(D2). So there exists vertices uj ∈ V (D1), vs ∈ V (D2) such
that d(uj , ui) ≥ d(uj , uk) for every uk ∈ N [ui], and d(vs, vr) ≥ d(vs, vq) for every
vq ∈ N [vr]. Hence in D1 ⊠ D2, d((uj , vs), (ui, vr)) = max{d(uj , ui), d(vs, vr)} ≥
max{d(uj , uk), d(vs, vq)} = d((uj , vs), (uk, vq)) for all vertices (uk, vq) ∈ N [(ui, vr)].
Thus, A1 ⊆ ∂(D1 ⊠D2).
Now let (ui, vr) ∈ A2. Then ui ∈ ∂(D1), vr /∈ ∂(D2) and there exists some vertex
vt ∈ V (D2) such that d(vt, vq) ≤ ecc(ui), for all vq ∈ N [vr]. Since ui ∈ ∂(D1), there
exists at least one vertex uj ∈ V (D1) such that d(uj , ui) ≥ d(uj , uk) for every uk ∈
N [ui]. Of these vertices, let ub be a vertex such that d(ub, ui) = ecc(ui). Hence in
D1 ⊠D2,
d((ub, vt), (ui, vr)) = max{d(ub, ui), d(vt, vr)}
≥ max{d(ub, uk), d(vt, vq)} = d((ub, vt), (uk, vq))
B. S. Anand et al.: Boundary-type sets of strong product of directed graphs 283
for all (uk, vq) ∈ N [(ui, vr)], since d(vt, vq) ≤ ecc(ui) = d(ub, ui) for all vq ∈ N [vr].
Thus, (ui, vr) is a boundary vertex of (ub, vt) in D1 ⊠D2 and hence A2 ⊆ ∂(D1 ⊠D2).
By analogous arguments and since the strong product of digraphs is commutative, it
follows that A3 ⊆ ∂(D1 ⊠D2).
Hence A1 ∪A2 ∪A3 ⊆ ∂(D1 ⊠D2).
Now consider Figure 1. ecc(v1) = ecc(v2) = 1. N [u2] = N [u3] = {u1, u2, u3, u4},
d(u1, u4) = 3, d(u1, u2) = d(u1, u3) = d(u2, u4) = d(u3, u4) = 2, and d(u2, u3) = 1.
u2 /∈ ∂(D1) and hence (u2, v1), (u2, v2) /∈ ∂(D1 ⊠ D2), since there is no vertex uℓ ∈
V (D1) such that d(uℓ, uk)) ≤ 1 for all uk ∈ N [u2]. For similar reasons, (u3, v1),
(u3, v2) /∈ ∂(D1 ⊠D2).
Consider the strong product of two connected undirected graphs. In the case of an
undirected graph, the maximum distance between two vertices is the usual distance between
the vertices. Also, since we deal with the distance between any two distinct vertices, it
doesn’t matter whether the undirected graphs are simple or not; that is, whether they contain
loops or parallel edges. So we state the result for any two connected nontrivial (not equal
to K1) undirected graphs.
Remark 3.4. The description for the boundary set of the strong product of two graphs
(undirected graphs) G and H presented in [4] holds only for the product of two nontrivial
graphs G and H . To this end, let H = K1 = ({v}, ∅). We have, ∂(K1) = {v} (since
all vertices of a complete graph are boundary vertices of the graph), and hence ∂(G) =̂
∂(G⊠K1) = (∂(G)× {v}) ∪ (V (G)× {v}) =̂ V (G), which is not true in general.
Corollary 3.5. Let D1 and D2 be two nontrivial connected undirected graphs. Then
∂(D1 ⊠D2) = [∂(D1)× V (D2)] ∪ [V (D1)× ∂(D2)].
Proof. By Theorem 3.3, if D1 and D2 are two strongly connected digraphs, ∂(D1⊠D2) =
A1∪A2∪A3. Since D1 and D2 are given to be two nontrivial undirected graphs, ecc(ui) ≥
1 for all ui ∈ V (D1), ecc(vr) ≥ 1 for all vr ∈ V (D2), d(ui, uk) = 1 for all uk ∈ N(ui),
and d(vr, vq) = 1 for all vq ∈ N(vr). Thus,
A1 = ∂(D1)× ∂(D2),
A2 = {(ui, vr) ∈ V (D1 ⊠D2) : ui ∈ ∂(D1), vr /∈ ∂(D2), and
∃vt ∈ V (D2) such that d(vt, vq) ≤ ecc(ui),∀vq ∈ N(vr)}
= ∂(D1)× V (D2),
A3 = {(ui, vr) ∈ V (D1 ⊠D2) : ui /∈ ∂(D1), vr ∈ ∂(D2), and
∃uℓ ∈ V (D1) such that d(uℓ, uk) ≤ ecc(vr),∀uk ∈ N(ui)}
= V (D1)× ∂(D2).
Therefore, ∂(D1⊠D2) = A1∪A2∪A3 = [∂(D1)×V (D2)]∪ [V (D1)×∂(D2)].
Theorem 3.6. Let D1 and D2 be two strongly connected digraphs.
1. If diam(D1) < diam(D2), then Per(D1 ⊠D2) = V (D1)× Per(D2).
2. If diam(D1) = diam(D2), then Per(D1 ⊠D2) = [Per(D1)× V (D2)] ∪
[V (D1)× Per(D2)].
284 Ars Math. Contemp. 20 (2021) 275–288
Proof.
1. Let diam(D2) = n. Let vr ∈ Per(D2).
Then for all ui ∈ V (D1), ecc(ui, vr) = max{ecc(ui), ecc(vr)} = n. Hence
(ui, vr) ∈ Per(D1 ⊠ D2). Also if vr /∈ Per(D2), then since ecc(ui, vr) < n,
(ui, vr) /∈ Per(D1⊠D2). Hence it follows that Per(D1⊠D2) = V (D1)×Per(D2).
2. Let diam(D1) = diam(D2) = n. If ui ∈ Per(D1), then for all vr ∈ V (D2),
(ui, vr) ∈ Per(D1 ⊠D2), since ecc(ui, vr) = max{ecc(ui), ecc(vr)} = n. Hence
(ui, vr) ∈ Per(D1 ⊠ D2). Similarly, if vr ∈ Per(D2), then for all ui ∈ V (D1),
(ui, vr) ∈ Per(D1 ⊠ D2). Hence it follows that [Per(D1) × V (D2)] ∪ [V (D1) ×
Per(D2)] ⊆ Per(D1 ⊠D2).
Conversely, if (ui, vr) ∈ Per(D1 ⊠ D2), then ecc(ui, vr) = max{diam(D1),
diam(D2)} = n. Thus, at least one of ecc(ui) and ecc(vr) must be necessarily
equal to n. Hence ui ∈ Per(D1) or vr ∈ Per(D2), and therefore, Per(D1 ⊠D2) ⊆
[Per(D1)× V (D2)] ∪ [V (D1)× Per(D2)].
Theorem 3.7. Let D1 and D2 be two strongly connected digraphs.
1. If rad(D1) = rad(D2), then
Ecc(D1 ⊠D2) = [Ecc(D1)× V (D2)] ∪ [V (D1)× Ecc(D2)].
2. If rad(D1) < rad(D2), then
Ecc(D1 ⊠D2) =
[ ⋃
ecc(ui)≥rad(D2)
Ecc(ui)× V (D2)
]
∪ [V (D1)× Ecc(D2)] .
Proof.
1. First we will prove that Ecc(D1 ⊠ D2) ⊆ [Ecc(D1) × V (D2)] ∪ [V (D1) ×
Ecc(D2)]. Let (ui, vr) ∈ Ecc(D1 ⊠ D2). Then there exists a vertex (uj , vs) such
that ecc(uj , vs) = d((uj , vs), (ui, vr)) = max{d(uj , ui), d(vs, vr)}. Since ecc(uj , vs) =
max{ecc(uj), ecc(vs)}, and ecc(uj) ≥ d(uj , ui) and ecc(vs) ≥ d(vs, vr), at least one of
ecc(uj) = d(uj , ui) and ecc(vs) = d(vs, vr) must hold. So necessarily ui is an eccentric
vertex of uj , or vr is an eccentric vertex of vs.
Hence (ui, vr) ∈ [Ecc(D1)× V (D2)] ∪ [V (D1)× Ecc(D2)].
Let rad(D1) = rad(D2) = n. Let ui ∈ Ecc(D1). Then there exists a vertex uj ∈
V (D1) such that ecc(uj) = d(uj , ui). Consider the vertex (ui, vr) ∈ V (D1 ⊠D2), where
vr is an arbitrary vertex in D2. Since rad(D2) = n, there exists a vertex vs ∈ V (D2) such
that ecc(vs) = n. Hence d(vs, vr) ≤ n and so ecc(uj , vs) = max{ecc(uj), ecc(vs)} =
max{ecc(uj), n} = ecc(uj). Thus, d((uj , vs), (ui, vr)) = max{d(uj , ui), d(vs, vr)} =
ecc(uj) = ecc(uj , vs). So (ui, vr) is an eccentric vertex of (uj , vs). Thus if ui ∈ Ecc(D1),
then (ui, vr) ∈ Ecc(D1 ⊠ D2) for all vr ∈ V (D2). Similarly, we can prove that if vq ∈
Ecc(D2), then (uk, vq) ∈ Ecc(D1 ⊠D2) for all uk ∈ V (D1).
Hence [Ecc(D1)× V (D2)]∪ [V (D1)×Ecc(D2)] ⊆ Ecc(D1 ⊠D2), and so the result
holds.
B. S. Anand et al.: Boundary-type sets of strong product of directed graphs 285
2. Let rad(D1) < rad(D2) = n. Let ui ∈ V (D1), vr ∈ V (D2). Here two cases arise:
Case 1. vr ∈ Ecc(D2).
Then there exists a vertex vs ∈ V (D2) such that ecc(vs) = d(vs, vr). Let up ∈
V (D1) be such that ecc(up) = rad(D1). Then since rad(D2) > ecc(up), ecc(up, vs) =
max{ecc(up), ecc(vs)} = ecc(vs). Also, d((up, vs), (ui, vr)) = max{d(up, ui),
d(vs, vr)} = ecc(vs). Thus, (ui, vr) is an eccentric vertex of (up, vs). So in this case,
V (D1)× Ecc(D2) ⊆ Ecc(D1 ⊠D2).
Case 2. vr /∈ Ecc(D2).
Let vq ∈ V (D2) be such that ecc(vq) = rad(D2). Let
⋃
ecc(ui)≥rad(D2) Ecc(ui) = A.
Let uk ∈ A. Then there exists a vertex up ∈ V (D1) such that ecc(up) ≥ rad(D2)
and ecc(up) = d(up, uk). Then d((up, vq), (uk, vr)) = max{d(up, uk), d(vq, vr)} =
d(up, uk) = ecc(up) = ecc(up, vq) and hence (uk, vr) is an eccentric vertex of (up, vq).
Hence in this case,
⋃
ecc(ui)≥rad(D2) Ecc(ui)× V (D2) ⊆ Ecc(D1 ⊠D2).
Thus,
[⋃
ecc(ui)≥rad(D2) Ecc(ui)× V (D2)
]
∪ [V (D1)× Ecc(D2)] ⊆ Ecc(D1 ⊠D2).
Conversely, let (uk, vr) ∈ Ecc(D1 ⊠ D2). Then there exists a vertex (uj , vs) ∈
V (D1⊠D2) such that ecc(uj , vs) = d((uj , vs), (uk, vr)) = max{d(uj , uk), d(vs, vr)} =
max{ecc(uj), ecc(vs)}. If vr ∈ Ecc(D2), we get (uk, vr) ∈ V (D1)× Ecc(D2).
Hence suppose that (uk, vr) ∈ Ecc(D1 ⊠ D2) and vr /∈ Ecc(D2). Then for all vs ∈
V (D2), ecc(vs) > d(vs, vr). Thus, ecc(uj , vs) = ecc(uj) = d(uj , uk). If possible,
suppose that uk /∈ A =
⋃
ecc(ui)≥rad(D2) Ecc(ui). Thus, there is no vertex uj in D1 such
that ecc(uj) = d(uj , uk) and ecc(uj) ≥ rad(D2). Hence if uk is an eccentric vertex of uj
in D1, then d(uj , uk) < rad(D2). We have, rad(D1⊠D2) = max{rad(D1), rad(D2)} =
rad(D2). Thus, (uk, vr) cannot be the eccentric vertex of any vertex (uj , vs) ∈ D1 ⊠D2,
since d((uj , vs), (uk, vr)) = max{d(uj , uk), (vs, vr)} ≠ ecc(uj , vs) in this case. This is
a contradiction, and hence uk ∈ A. Hence (uk, vr) ∈
⋃
ecc(ui)≥rad(D2) Ecc(ui)× V (D2).
Hence Ecc(D1⊠D2) ⊆
[⋃
ecc(ui)≥rad(D2) Ecc(ui)×V (D2)
]
∪ [V (D1)×Ecc(D2)].
Theorem 3.8. Let D1 and D2 be two strongly connected digraphs. Then Ct(D1 ⊠D2) =
A1 ∪A2 ∪A3, where
A1 = [Ct(D1)× Ct(D2)],
A2 = {(ui, vr) ∈ V (D1 ⊠D2) : ui ∈ Ct(D1), vr /∈ Ct(D2), and
ecc(vq) ≤ ecc(ui) for all vq ∈ N [vr]},
A3 = {(ui, vr) ∈ V (D1 ⊠D2) : ui /∈ Ct(D1), vr ∈ Ct(D2), and
ecc(uk) ≤ ecc(vr) for all uk ∈ N [ui]}.
Proof. (ui, vr) ∈ Ct(D1 ⊠D2) if and only if ecc(ui, vr) ≥ ecc(uk, vq) for all (uk, vq) ∈
N [(ui, vr)]; if and only if max{ecc(ui), ecc(vr)} ≥ max{ecc(uk), ecc(vq)} for all uk ∈
N [ui] and vq ∈ N [vr]; if and only if one of the following three cases holds.
1. max{ecc(ui), ecc(vr)} = ecc(ui) = ecc(vr). Then, ecc(ui) ≥ ecc(uk) and
ecc(vr) ≥ ecc(vq) for all uk ∈ N [ui] and vq ∈ N [vr].
2. max{ecc(ui), ecc(vr)} = ecc(ui) > ecc(vr). Then, ecc(ui) ≥ ecc(uk) for all
uk ∈ N [ui] and ecc(vr) < ecc(ui), ecc(vq) ≤ ecc(ui) for all vq ∈ N(vr).
286 Ars Math. Contemp. 20 (2021) 275–288
3. max{ecc(ui), ecc(vr)} = ecc(vr) > ecc(ui). Then, ecc(vr) ≥ ecc(vq) for all
vq ∈ N [vr] and ecc(ui) < ecc(vr), ecc(uk) ≤ ecc(vr) for all uk ∈ N(ui).
In case 1, (ui, vr) ∈ Ct(D1 ⊠D2).
In case 2, (ui, vr) ∈ {(ui, vr) ∈ V (D1 ⊠ D2) : ui ∈ Ct(D1), vr /∈ Ct(D2), and
ecc(vq) ≤ ecc(ui) for all vq ∈ N [vr]}.
In case 3, (ui, vr) ∈ {(ui, vr) ∈ V (D1 ⊠ D2) : ui /∈ Ct(D1), vr ∈ Ct(D2), and
ecc(uk) ≤ ecc(vr) for all uk ∈ N [ui]}.
Thus we get, Ct(D1 ⊠D2) = A1 ∪A2 ∪A3.
Consider the contour set of the strong product of two connected undirected graphs. As
in the case of the boundary set, the result holds even when the undirected graphs are not
simple.
Corollary 3.9. Let D1 and D2 be two connected undirected graphs. Then
Ct(D1 ⊠D2) = {(ui, vr) ∈ V (D1 ⊠D2) : ui ∈ Ct(D1), vr /∈ Ct(D2), and
ecc(vr) < ecc(ui)}
∪ {(ui, vr) ∈ V (D1 ⊠D2) : ui /∈ Ct(D1), vr ∈ Ct(D2), and
ecc(ui) < ecc(vr)}
∪ [Ct(D1)× Ct(D2)].
Proof. By Theorem 3.8, when D1 and D2 are two strongly connected digraphs, Ct(D1 ⊠
D2) = A1 ∪A2 ∪A3. Since D1 and D2 are given to be undirected graphs, eccentricity of
two adjacent vertices differ by atmost one. Hence
A1 = Ct(D1)× Ct(D2),
A2 = {(ui, vr) ∈ V (D1 ⊠D2) : ui ∈ Ct(D1), vr /∈ Ct(D2), and
ecc(vq) ≤ ecc(ui) for all vq ∈ N [vr]}
= {(ui, vr) ∈ V (D1 ⊠D2) : ui ∈ Ct(D1), vr /∈ Ct(D2), and
ecc(vr) + 1 ≤ ecc(ui)}
= {(ui, vr) ∈ V (D1 ⊠D2) : ui ∈ Ct(D1), vr /∈ Ct(D2), and ecc(vr) < ecc(ui)},
A3 = {(ui, vr) ∈ V (D1 ⊠D2) : ui /∈ Ct(D1), vr ∈ Ct(D2), and ecc(ui) < ecc(vr)},
since maxuk∈N [ui] ecc(uk) = ecc(ui) + 1. Hence it follows that
Ct(D1 ⊠D2) = {(ui, vr) ∈ V (D1 ⊠D2) : ui ∈ Ct(D1), vr /∈ Ct(D2), and
ecc(vr) < ecc(ui)}
∪ {(ui, vr) ∈ V (D1 ⊠D2) : ui /∈ Ct(D1), vr ∈ Ct(D2), and
ecc(ui) < ecc(vr)}
∪ [Ct(D1)× Ct(D2)].
We have examined the boundary-type sets of the strong product of two strongly con-
nected digraphs D1 and D2. Now suppose that at least one of D1 and D2, say D1, is not
strongly connected. Then the eccentricity of every vertex in D1 is infinity, and hence the
B. S. Anand et al.: Boundary-type sets of strong product of directed graphs 287
eccentricity of every vertex in D1 ⊠ D2 is infinity. Thus, we have ∂(D1) = Per(D1) =
Ecc(D1) = Ct(D1) = V (D1), and ∂(D1 ⊠D2) = Per(D1 ⊠D2) = Ecc(D1 ⊠D2) =
Ct(D1 ⊠ D2) = V (D1) × V (D2). Since rad(D1) = diam(D1) = ∞, the expression
for Per(D1 ⊠D2) in Theorem 3.6, and the expression for Ecc(D1 ⊠D2) in Theorem 3.7
gives V (D1) × V (D2). Since ecc(ui) = ∞ for all ui ∈ V (D1), the expression for
∂(D1 ⊠ D2) in Theorem 3.3, and the expression for Ct(D1 ⊠ D2) in Theorem 3.8 also
gives V (D1)× V (D2). Similar is the case when D2 and both D1 and D2 are not strongly
connected.
Thus, the results derived for the boundary-type sets of the strong product of two strongly
connected digraphs D1 and D2 hold also when the digraphs D1 and D2 are not even weakly
connected. So the results for the boundary-type sets of the strong product of two connected
undirected graphs hold for any two arbitrary undirected graphs.
4 Conclusion
In this article, the relationship between the boundary-type sets of the strong product of two
digraphs, and that of its factors is derived. As ‘maximum distance’ is the generalization
of the usual distance in an undirected graph, these results hold for undirected graphs also.
The results for the periphery and eccentricity sets of the strong product of two undirected
graphs turn out to be the same as the results in [4]. The results for the boundary and contour
sets in the undirected case, as described in [4], are also derived as special cases.
ORCID iDs
Bijo S. Anand https://orcid.org/0000-0002-7221-904X
Manoj Changat https://orcid.org/0000-0001-7257-6031
Prasanth G. Narasimha-Shenoi https://orcid.org/0000-0002-5850-5410
Mary Shalet Thottungal Joseph https://orcid.org/0000-0001-6350-7106
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47–50.
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SIGMAP 2022 Workshop – Announcement
and Call for Papers
The Symmetries in Graphs, Maps, and Polytopes Workshop (SIGMAP) has been held
every four years since Steve Wilson organized the first one in Flagstaff, Arizona in 1998.
The workshop is devoted to the exploration of the symmetries of discrete objects, and has
been an opportunity to share recent advances, discuss open problems, and start new col-
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speakers for SIGMAP 2022 are:
• Gabriel Cunningham (University of Massachusetts Boston, USA)
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• Gareth Jones (University of Southampton, United Kingdom)
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• Pablo Spiga (Università degli Studi di Milano-Bicocca, Italy)
• Klara Stokes (Umeå University, Sweden)
• Gabriel Verret (University of Auckland, New Zealand)
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xvii
The workshop will be held at the University of Alaska Fairbanks campus. UAF is Alaska’s
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situated on 2 250 acres amidst the boreal forest, and features 41.6 km of trails, the Museum
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More information about the workshop is available at
https://www.alaska.edu/sigmap
This is also a call for papers for a special issue of the journal The Art of Discrete and
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