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′ ∞ ΓG(1) gk(G) γavg(G) = = k · . ΓG(1) X ΓG(1) k=0

The study of the average genus of a graph began by Gross and Furst [9], and much further developed by Chen and Gross [1, 2, 4]. Two lower bounds were obtained in [3] for the average genus of two kinds of graphs. In [18], Stahl gave the asymptotic result for average genus of linear graph families. The exact value 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, 11, 14, 16, 19] etc. For general background in topological , we refer the readers to see Gross and Tucker [8] or White [21]. One objective of this 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 Bn.

Figure 1 demonstrates the graphs B1,B2,B3. The bouquets {Bn,n > 1} are very important graphs in theory. First, since any connected graph can be reduced to a bouquet by contracting a spanning to a point, bouquets are fundamental building blocks of topological graph theory. Second, as that demonstrated in [6, 7], Cayley graphs and many other regular graphs are covering spaces of bouquets.

B B B

Figure 1: The bouquets B1,B2, and B3.

For the genus distribution of Bn, Gross, Robbins and Tucker [10] proved that the numbers gm(Bn) of embeddings of the Bn in the oriented surface of genus m satisfy the following recurrence for n> 2,

2 (n + 1)gm(Bn) = 4(2n − 1)(2n − 3)(n − 1) (n − 2)gm−1(Bn−2) (1.1) + 4(2n − 1)(n − 1)gm(Bn−1) with initial conditions

gm(B0)=1 for m = 0 and gm(B0)=1 for m> 0,

gm(B1)=1 for m = 0 and gm(B1)=1 for m> 0.

With an aid of edge-attaching surgery technique, the total embedding polynomials of Bn was computed in [12]. Stahl [17] also did some researches on the average genus of Bn. By [17,

2 Theorem 2.5] and Euler formula, one easily sees that

2n n + 1 1 1 lim γavg(Bn) − − = 0. (1.2) n→∞  2 2 X k  k=1

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 Tay- lor formulas, we derive an explicit expression of γavg(Bn). By this expression, (1.2) follows immediately. Our methods are totally different from that in [17] and we don’t need to make estimates on s(n, k). In Section 2, we will give the computation of γavg(Bn) in detail. Another objective of this paper is to give an explicit expression of the average genus for dipoles Dn (two vertices, n−multiple edges). Figure 2 demonstrates the graphs D1, D2, D3.

D D D

Figure 2: The dipoles D1, D2, and D3.

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 distributions of

Dn are given by [12] and [15]. In this paper, the calculation of γavg(Dn) are similar as that in γavg(Bn). But the processes are more complicated, so we still give their details in Section 3.

It is not difficult to obtain the following recurrence relation for γavg(Bn)

2 n − 1 γavg(Bn) = γavg(Bn 1)+ γavg(Bn 2) + 1 . n + 1 − n + 1 − 

Since the coefficients before γavg(Bn−2) and γavg(Bn−1) depend on n, there are little tools to obtain the explicit expression of γavg(Bn). In this paper, we give a method to get explicit expression of γavg(Bn). Our method is new and deeply depends on the knowledge in ordinary differential equations and series theory.

2 The average genus of Bn

The main objective of this section is to prove the following theorem.

Theorem 2.1. The average genus of Bn is given by

n−1 n + 1 1 + (−1)m 1 + (−1)n γ (B )= − − . avg n 2 X 2(m + 1) 4(n + 1) m=0

3 In particular, we have

n − ln n − c + 1 − ln 2 γavg(Bn)= + o(1), 2 where c ≈ 0.5772 is the Euler constant.

Proof. Multiplying both sides of (1.1) by xm, it holds that

m 2 m (n + 1)g (B )x = 4(2n − 1)(2n − 3)(n − 1) (n − 2)g 1(B 2)x X m n X m− n− m>0 m>0 (2.1) m + 4(2n − 1)(n − 1)g (B 1)x . X m n− m>0

Hence, the genus polynomial ΓBn (x) satisfy the following recurrence

2 (n + 1)ΓBn (x) = 4(2n − 1)(2n − 3)(n − 1) (n − 2) · x · ΓBn−2 (x) (2.2) + 4(2n − 1)(n − 1)ΓBn−1 (x)

with initial conditions ΓB1 (x) = 1, ΓB2 (x)=4+2x. By (2.2), we have

′ 2 ′ (n + 1)ΓBn (1) = 4(2n − 1)(2n − 3)(n − 1) (n − 2) · ΓBn−2 (1) 2 ′ + 4(2n − 1)(2n − 3)(n − 1) (n − 2) · ΓBn−2 (1) + 4(2n − 1)(n − 1)ΓBn−1 (1).

Dividing both sides of the above equality by ΓBn (1) = (2n − 1)!, for n > 3, one arrives at that

(n + 1)γavg(Bn) = 2γavg(Bn 1) + (n − 1) γavg(Bn 2) + 1 . (2.3) − −  1 By direct calculation, one sees γavg(B1) = 0, γavg (B2)= 3 . For n 6 0, we define γavg(Bn) = 0 so that (2.3) holds for any integer n > 1. For simplicity of writing, we use an to denote

γavg(Bn) in the rest of proof. Thus, an satisfies the following recurrence relation

2 n − 1 a = a 1 + (a 2 + 1) (2.4) n n + 1 n− n + 1 n−

1 with initial conditions a1 = 0, a2 = 3 . Using (2.4), we have

(n + 1)an = 2an−1 + (n − 1)(an−2 + 1)

4 and

n n n (n + 1)a t = 2 a 1t + (n − 1)(a 2 + 1)t . (2.5) X n X n− X n− n>1 n>1 n>1

n Let u(t)= 1 ant . Then, by (2.5), we obtain Pn> ′ n n−1 n n n t · a t = 2t · a 1t + (n − 2)a 2t + a 2t + (n − 1)t  X n  X n− X n− X n− X n>1 n>1 n>1 n>1 n>1 3 n−3 2 2 n−1 ′ = 2tu(t)+ t (n − 2)a 2t + t u(t)+ t · ( t ) , X n− X n>1 n>2 that is

′ ′ 3 n−3 2 2 t (tu(t)) = 2tu(t)+ t (n − 2)a 2t + t u(t)+ t X n− 1 − t n>3 ′ 3 n−1 2 2 t = 2tu(t)+ t nant + t u(t)+ t X 1 − t n>1 t ′ = 2tu(t)+ t3u′(t)+ t2u(t)+ t2 , 1 − t which implies that u(t) satisfies the following first order linear differential equation

t2 u′(t)(t − t3)+ u(t)(1 − 2t − t2)= . (2.6) (1 − t)2 with initial condition u(t)|t=0 = 0. We solve this equation using variation of parameters and obtain its solution

− t2 − 1 ln(1 − t)+ t2 − 1 ln(t + 1) + 2t u(t)=   . 4(t − 1)2t

Denote

1 (t + 1) ln(1 − t) (t + 1) ln(t + 1) u1(t)= , u2(t)= − , u3(t)= . 2(t − 1)2 4(t − 1)t 4(t − 1)t

Then, we have u(t)= u1(t)+ u2(t)+ u3(t). By Taylor formula, we obtain

n + 1 n u1(t) = t (2.7) X 2 n>0 and

1 1 ln(1 − t) 1 ℓ 1 m u2(t) = (1 + t) · · = (1 + t) · t · − t 4 1 − t t 4 X X m + 1  ℓ>0 m>0

5 n 1 1 n n = (1 + t) · (− )t = bnt , (2.8) 4 X X m + 1 X n>0 m=0 n>0

1 1 n 1 n−1 1 where b0 = − and bn = =0(− )+ =0(− ) ,n > 1. For u3(t), we have 4 4 h Pm m+1 Pm m+1 i

m 1 1 ln(1 + t) 1 ℓ (−1) m u3(t) = − (1 + t) · · = − (1 + t) · t · t 4 1 − t t 4 X X m + 1 ℓ>0 m>0 n m 1 (−1) n n = − (1 + t) · t = cnt , (2.9) 4 X X m + 1 X n>0 m=0 n>0

1 where c0 = − 4 and

n n−1 1 (−1)m (−1)m c = − + ) , n > 1. n 4h X m + 1 X m + 1 i m=0 m=0

Combining (2.7)-(2.9), it holds that

n + 1 a = + b + c n 2 n n n n−1 n n−1 n + 1 1 1 1 1 (−1)m (−1)m = + (− )+ (− ) − + ) , 2 4h X m + 1 X m + 1 i 4h X m + 1 X m + 1 i m=0 m=0 m=0 m=0 which completes the proof of Theorem 2.1.

3 The average genus of Dn

The main objective of this section is to prove the following theorem.

Theorem 3.1. γavg(D1)= γavg(D2) = 0 and for n > 3, we have

n+1 m 2 2 m m 4(−1) m + m − 12(−1) m − 5m + 6(−1) + 6 · (n − m + 2) γavg(Dn)= X 2(m − 3)(m − 2)(m − 1)m m=4 (3.1)

In particular, we have

n − ln n − c γavg(Dn)= + o(1), (3.2) 2 where c ≈ 0.5772 is the Euler constant.

6 Proof. Once we have obtained (3.1), by the soft Mathematica or the series theory, we have

n+1 4(−1)mm2 + m2 − 12(−1)mm − 5m + 6(−1)m + 6 1 1 = 1 − + o( ) (3.3) X (m − 3)(m − 2)(m − 1)m n n m=4 and

n+1 m 2 2 m m 4(−1) m + m − 12(−1) m − 5m + 6(−1) + 6 · (m − 2) X (m − 3)(m − 2)(m − 1)m m=4 n+1 m 2 m m 4(−1) m − 12(−1) m − 5m + 6(−1) + 6 · (m − 2) = X (m − 3)(m − 2)(m − 1)m m=4 n+1 m2 · (m − 2) + X (m − 3)(m − 2)(m − 1)m m=4 n+1 7 m = − + o(1) + 4 X (m − 3)(m − 1) m=4 n+1 n+1 7 1 1 = − + o(1) + + 4 X (m − 3) X (m − 3)(m − 1) m=4 m=4 n+1 7 1 3 = − + o(1) + + = −1+ o(1) + ln n + c, (3.4) 4 X (m − 3) 4 m=4 where c is the Euler constant. Combining (3.3)(3.4), we complete the proof of (3.2). Now we give a proof of (3.1). By [15, Theorem 5.2], we have

3 2 2 (n + 2)gk(Dn+1)= n(2n + 1)gk(Dn)+ n (n − 1) gk−1(Dn−1) − n(n − 1) gk(Dn−1) and

k k 3 2 k (n + 2)g (D +1)x = n(2n + 1) g (D )x + n (n − 1) g 1(D 1)x X k n X k n X k− n− k>0 k>0 k>0 2 k −n(n − 1) g (D 1)x , X k n− k>0 where gk(Dn)=0 for k = −1. Therefore, ΓDn (x) satisfies the following recurrence relation

2 2 (n + 2)ΓDn+1 (x) − n(2n + 1)ΓDn (x)= n(n − 1) (n x − 1) · ΓDn−1 (x) (3.5)

with initial conditions ΓD1 (x)=ΓD2 (x) = 1. Taking derivations on both sides of (3.5) and setting x = 1, one sees that

′ ′ 2 2 ′ 3 2 2 (n + 2)ΓDn+1 (1) = n(2n + 1)ΓDn (1) + n(n − 1) (n − 1) · ΓDn−1 (1) + n (n − 1) (n − 2)! .

7 Dividing both sides of the above equality by n!2, we obtain

2n + 1 (n2 − 1) (n + 2)γ (D +1)= γ (D )+ · γ (D 1)+ n avg n n avg n n avg n− and

2 2 n(n + 2)γavg(Dn+1) = (2n + 1)γavg(Dn) + (n − 1) · γavg(Dn−1)+ n .

For simplicity of writing, we use an to denote γavg(Dn) in the rest of proof. Then, for n > 2, we have the following recurrence relation for an

2 2 n(n + 2)an+1 = (2n + 1)an + (n − 1)an−1 + n (3.6) with initial conditions a1 = a2 = 0. Let n−3 n−2 u(t)= a t = a +1t . X n X n n>1 n>2 Then, by (3.6) and calculation, we have

n−2 n−2 2 n−2 2 n−2 n(n + 2)a +1t = (2n + 1)a t + (n −1)a 1t + n t (3.7) X n X n X n− X n>2 n>2 n>2 n>2 and

′ n−4 n−3 n−3 u (t) = a (n − 3)t = a +1(n − 2)t = a +1(n − 2)t , X n X n X n n>4 n>3 n>2 ′′ n−5 n−4 n−4 u (t) = a (n − 3)(n − 4)t = a +1(n − 2)(n − 3)t = a +1(n − 2)(n − 3)t . X n X n X n n>5 n>4 n>2

Also, by calculation and the above two equalities, we obtain

n−2 2 n−2 n(n + 2)an+1t = n − 5n + 6 + 7(n − 2) + 8 an+1t X X   n>2 n>2 = t2u′′(t) + 7tu′(t) + 8u(t), n−2 n−1 n−1 (2n + 1)ant = (2n + 3)an+1t = 2(n − 2) + 7 an+1t X X X  n>2 n>2 n>2 = 2t2u′(t) + 7tu(t), 2 n−2 2 n−2 2 n (n − 1)a 1t = (n − 1)a 1t = (n + 4n + 3)a +1t X n− X n− X n n>2 n>4 n>2 2 n = n − 5n + 6 + 9(n − 2) + 15 a +1t X h i n n>2 = t4u′′(t) + 9t3u′(t) + 15t2u(t),

8 n2tn−2 = n(n − 1)tn−2 + ntn−2 = v′′(t)+ ntn−2 − t−1 X X X X n>2 n>2 n>2 n>0 v′(t) 3t − 4 − t2 = v′′(t)+ − t−1 = , t (t − 1)3

n ′ n−1 ′′ n−2 where v(t)= 0 t , v (t)= 0 nt , v (t)= 0 n(n−1)t . Substituting the above Pn> Pn> Pn> equalities into (3.7), by the definition of u(t), one arrives at that u(t) satisfies the following second order linear differential equation

3t − 4 − t2 (t2 − t4)u′′(t) + (7t − 2t2 − 9t3)u′(t) + (8 − 7t − 15t2)u(t)= (t − 1)3

1 ′ 5 with initial conditions u(0) = a3 = γavg(D3)= 2 , u (0) = a4 = γavg(D4)= 6 . With the help of a computer, the solution to the above equation is

1 v(t) u(t)= + , 4(t − 1)t2 4(t − 1)2t4 where

v(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 formula, we have

1 1 = (− )tm, 4(t − 1)t2 X 4 m>−2 m 2 2 m m 24(−1) m + m − 12(−1) m − 5m + 6(−1) + 6 v(t)= t2 − t3 + tm, X (m − 3)(m − 2)(m − 1)m m>4 1 m + 5 = tm. 4(t − 1)2t4 X 4 m>−4

Therefore,

1 n + 3 n + 2 a +3 = − + − n 4 4 4 n+4 m 2 2 m m 24(−1) m + m − 12(−1) m − 5m + 6(−1) + 6 n − m + 5 + · X (m − 3)(m − 2)(m − 1)m 4 m=4 n+4 4(−1)mm2 + m2 − 12(−1)mm − 5m + 6(−1)m + 6 n − m + 5 = · X (m − 3)(m − 2)(m − 1)m 2 m=4 n+4 n 4(−1)mm2 + m2 − 12(−1)mm − 5m + 6(−1)m + 6 = 2 X (m − 3)(m − 2)(m − 1)m m=4

9 n+4 4(−1)mm2 + m2 − 12(−1)mm − 5m + 6(−1)m + 6 m − 5 − · , X (m − 3)(m − 2)(m − 1)m 2 m=4 which yields the desired result (3.1).

4 Some remarks

Bouquets and dipoles are two important classes of graphs in topological graph theory. Their average genera are of independent interests. In this paper, we obtain explicit formulas for

γavg(Bn) and γavg(Dn). By Theorems 2.1, 3.1, we have the following relation between γavg(Bn) and γavg(Dn),

1 − ln 2 γ (B )= γ (D )+ + o(1). avg n avg n 2

1−ln 2 It follows that the difference of γavg(Bn) and γavg(Dn) tends to a constant 2 when n tends to infinity. n Since both Bn and Dn are upper-embeddable, the maximum genera of Bn and Dn are 2 n−1   and , respectively. Recall that the minimum genera of Bn and Dn equal 0. Therefore,  2  also by Theorems 2.1, 3.1, we have

γ (B ) γ (D ) lim avg n = 1 and lim avg n = 1. n→∞ n n→∞ n−1 ⌊ 2 ⌋ ⌊ 2 ⌋

This imply that the average genus of Bn (Dn) is more closer to the maximum genus than to the minimum genus.

References

[1] J. Chen, J.L. Gross, Limit points for average genus(I): 3-connected and 2-connected simplicial graphs, J. Combin. Theory (B) 55(1992), 83-103.

[2] J. Chen, J.L. Gross, Kuratowski-type theorems for average genus, J. Combin. The- ory (B) 57(1993), 100-121.

[3] J. Chen, J.L. Gross, R.G. Rieper, Lower bounds for the average genus. J. Graph Theory 19(1995), 281-296.

[4] J. Chen, A linear-time algorithm for isomorphism of graphes of bounded average genus, SIAM J. Discrete Math 7(1994), 614-631.

[5] Y. Chen, Lower Bounds for the Average Genus of a CF-graph, Electronic J. Combin. 17(2010),#R150.

10 [6] J.L. Gross, Every connected regular graph of even degree is a schreier coset graph, J. Combin. Theory (B) 22(3)(1977), 227-232.

[7] J.L. Gross, T.W. Tucker, Generating all graph coverings by permutation voltage assignments, Discrete Mathematics 18(1977), 273-283.

[8] J.L. Gross, T.W. Tucker, Topological Graph Theory, Wiley-Interscience, New York, 1987.

[9] J.L. Gross, M.L. Furst, Hierarchy for imbedding-distribution invariants of a graph, J. Graph Theory 11(2)(1987), 205-220.

[10] J.L. Gross, D.P. Robbins, T.W. Tucker, Genus distributions for bouquets of circles, J. Combin. Theory (B) 47(3)(1989), 292-306.

[11] J.L. Gross, E.W. Klein, R.G. Rieper, On the average genus of a graph, Graphs and Combinatorics 9(2-4)(1993), 153-162.

[12] J.H. Kwak, S.H. Shim, Total embedding distributions for bouquets of circles, Dis- crete Mathematics 248(1)(2002), 93-108.

[13] J.H. Kwak, J. Lee, Genus polynomials of dipoles, Kyungpook Mathematical Journal 33(1995), 115-125.

[14] K.J. Mcgown, A. Tucker, Statistics of genus numbers of cubic fields, arXiv:1611.07088v2 (2016).

[15] R.G. Riper, The enumeraion of graph embeddings, Ph.D. Thesis, Western Michigan University, 1987.

[16] S. Stahl, The average genus of classes of graph embeddings, Congr. Numer. 40(1983), 375-388.

[17] S. Stahl, Region distributions of and stirling numbers, Discrete Mathematics 82(1990), 57-76.

[18] S. Stahl, Permutation-partition pairs (III): embedding distributions of linear fami- lies of graphs, J. Combin. Theory (B) 52(2)(1991), 191-218.

[19] S. Stahl, Bounds for the average genus of the vertex-amalgamation of graphs, Dis- crete Mathematics 142(1995), 235-245.

[20] R.J. Swift, S.A. Wirkus, A Course in Ordinary Differential Equations, CRC Press, Second Edition, 2014.

[21] A.T. White, Graphs Groups and Surfaces, 2nd, North-Holland, Amsterdam, 1984.

11 [22] A.T. White, An introduction to random topological graph theory, Combinatorics, Probability and Computing 3(04)(1994), 545-555.

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