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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