Korean J. Math. 28 (2020), No. 1, pp. 123–136 http://dx.doi.org/10.11568/kjm.2020.28.1.123

IHARA ZETA FUNCTION OF FINITE GRAPHS WITH CIRCUIT RANK TWO

Sanghoon Kwon: and Seungmin Lee

Abstract. In this paper, we give an explicit formula as a rational function for the Ihara zeta function of every finite connected graph without degree one vertices whose circuit rank is two.

1. Introduction

Ihara zeta function is a zeta function associated with graphs that re- sembles the Selberg’s dynamical zeta function for a geodesic flow on a Riemannian manifold. In the Selberg’s dynamical zeta function of a Rie- mannian manifold, primitive closed orbits (the primitive periodic points of the geodesic flow) of the geodesic flow play the role of primes in the Riemann’s zeta function ([6]). Ihara investigated the p-adic analogue of the Riemann surfaces and found the similar idea of Selberg’s zeta func- tion can be adapted to the p-adic and positive-characteristic cases. He also showed that one can compute the zeta function effectively, via a product of polynomials and the determinant of a certain matrix ([2]). As Serre remarked, a biregular tree is a p-adic analogue of Riemann surfaces in the sense that an F -points of a rank one semisimple algebraic group over an ultrametric local field F acts on the tree ([7]). Thus, we

Received September 18, 2019. Revised March 3, 2020. Accepted March 16, 2020. 2010 Mathematics Subject Classification: 05C50, 15A24, 37C30. Key words and phrases: Ihara zeta function, circuit rank, dumbbell graphs, matrix analysis. : This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government 2019R1I1A3A01058926 and the grant funded by Catholic Kwandong University 201901200001. c The Kangwon-Kyungki Mathematical Society, 2020. This is an Open Access article distributed under the terms of the Creative com- mons Attribution Non-Commercial License (http://creativecommons.org/licenses/by -nc/3.0/) which permits unrestricted non-commercial use, distribution and reproduc- tion in any medium, provided the original work is properly cited. 124 Sanghoon Kwon and Seungmin Lee may interpret Ihara’s zeta function and the determinant formula purely in terms of graphs. For instance, the zeta function of tree lattices are studied by many authors including [1] and [5]. Recently, a new weighted zeta function for a graph is also introduce in [3]. The explicit form, however, as a rational function of the zeta function of finite graphs are rarely known, although we have the determinant formula. The purpose of this paper is to provide explicit formulas of the Ihara zeta function of finite connected graphs which have circuit rank two, without degree one vertices. The circuit rank of an undirected graph is the minimum number of edges that must be removed from the graph to break all its cycles, making it into a tree. It follows directly from the definition that the circuit rank r of a connected graph pV,Eq is equal to |E|´|V |`1. Up to homeomorphism, there are three types of graphs with circuit rank two which we call a dumbbell graph (Figure 1), a figure eight graph (Figure 2), and a bicyclic graph (Figure 3).

2 3 3 1 2 4

4 n 1 2 ` 1 5

5 7 m 6 6 7

Figure 1. Dumbbell graph, n, m ě 3, ` ě 0

Let us recall the definition of the Ihara’s zeta function of a finite graph following [8]. In order to define the zeta function of graphs, we need to figure out what primes in graphs are. Let G “ pVG,EGq be a finite, connected, and undirected graph with a set VG of vertices and a set EG of edges. If G is any undirected finite connected graph with unorient edges set E and vertex set V , we orient its edges arbitrarily and ´1 obtain 2 |E| oriented edges labelled by e1, e2, ¨ ¨ ¨ , en, en`1 “ e1 , en`2 “ ´1 ´1 e2 , ¨ ¨ ¨ , e2n “ en where n “ |E|. Let C “ pe1, e2, ¨ ¨ ¨ , esq be a primitive ´1 ´1 cycle without backtracking. That is, ei`1 ‰ ei , es ‰ e1 for all i and C ‰ Df for f ą 1 and a closed path D in G. We say two primitive cycles Zeta function of circuit rank two graphs 125

3 3 4 2 2 4

5 1 5

6 n m 6 7 7

Figure 2. Figure eight graph, n, m ě 3

2 1 1 ` 3 2

2 4

3

1 5 4 n m

Figure 3. Bicyclic graph, n, m ě 2, ` ě 0 are equivalent if we can get one from the other by changing the starting vertex. A prime in the graph G is an equivalence class rCs of primitive cycles. The length of the path C is the number s of edges in C, denoted by νpCq. Definition 1.1. The Ihara zeta function of graph G is defined at a complex number u, for which |u| is sufficiently small, by νpP q ´1 ZGpuq “ p1 ´ u q rP s ź where the product is over all primes rP s in G. 126 Sanghoon Kwon and Seungmin Lee

If G “ pVG,EGq, then recall that the circuit rank r of G is equal to |EG| ´ |VG| ` 1. Let VG “ tv1, ¨ ¨ ¨ , vnu. If G is a simple graph (a graph without loops and multiple edges), then the adjacency matrix A of G is the square matrix such that aij “ 1 if vi and vj are adjacent and aij “ 0 otherwise. Let D be the diagonal matrix with dii “ degG vi and Q “ D ´ I. Theorem 1.2 ( [1] (see also [2], Theorem 2)). Let G be a connected graph pVG,EGq, and let r be the circuit rank of G. Then, the zeta function of G is given by 1 ZGpuq “ . p1 ´ u2qr´1 det pI ´ Au ` Qu2q

We are ready to state our main results. We denote by Dn,m,` the dumbbell graph of type pn, m, `q, which consists of two vertex-disjoint cycles Cm,Cn and a path P` pn, m ě 1, ` ě 0q joining those cycles. It has n ` m ` ` number of vertices and n ` m ` ` ` 1 number of edges.

Theorem 1.3. Let D be the dumbbell graph Dn,m,` with n, m ě 3 and ` ě 0. Then, the zeta function of D is given by 1 Z puq “ . D p1 ´ unq p1 ´ umq p1 ´ un ´ um ` un`m ´ 4un`m`2``2q We remark that the formula for the cases n “ 1, m “ 1 and n “ 1, m “ 2 are given in [4]. The figure eight graph of type pn, mq, denoted by En,m, consists of two circles Cn and Cm with the same starting vertex. It has n ` m ´ 1 number of vertices and n ` m number of edges.

Theorem 1.4. Let E be the figure eight graph En,m with n, m ě 3. Then, the zeta function of E is given by 1 Z puq “ . E p1 ´ unq p1 ´ umq p1 ´ un ´ um ´ 3un`mq

The bicyclic graph Bn,m,` of type pn, m, `q consists of two cycles of length n ` ` ` 1 and m ` ` ` 1 which shares ` ` 2 vertices.

Theorem 1.5. Let B be the bicyclic graph Bn,m,` with n, m ě 2 and ` ě 0. Then, the zeta function ZBpuq of B is given by ´1 . p2un`m```1 ´ un`m ´ un```1 ´ um```1 ` 1q p2un`m```1 ` un`m ` un```1 ` um```1 ´ 1q Zeta function of circuit rank two graphs 127

2. Determinants of some tridiagonal matrices

In this section, we define some tridiagonal matrices which are fre- quently used in the proof of the theorems. Let

1 ` u2 ´u ´u 1 ` u2 ´u ¨ ˛ An “ ´u ¨ ¨ ¨ ´u , ˚ ´u 1 ` u2 ´u ‹ ˚ 2 ‹ ˚ ´u 1 ` 2u ‹ ˚ ‹ ˝ ‚ 1 ` 2u2 ´u ´u 1 ` u2 ´u ¨ ˛ Bn “ ´u ¨ ¨ ¨ ´u , ˚ ´u 1 ` u2 ´u ‹ ˚ 2 ‹ ˚ ´u 1 ` u ‹ ˚ ‹ ˝ ‚ and 1 ` u2 ´u ´u 1 ` u2 ´u ¨ ˛ Kn “ ´u ¨ ¨ ¨ ´u ˚ ´u 1 ` u2 ´u ‹ ˚ 2 ‹ ˚ ´u 1 ` u ‹ ˚ ‹ ˝ ‚ be n ˆ n matrices defined for u. The determinant of the above tridiagonal matrices are given by

2 4 2n det An “ det Bn “ 1 ` 2u ` 2u ` ¨ ¨ ¨ ` 2u and 1 ´ u2n`2 det K “ 1 ` u2 ` u4 ` ¨ ¨ ¨ ` u2n “ . n 1 ´ u2

2 4 2n Let us denote by fnpuq “ 1 ` u ` u ` ¨ ¨ ¨ ` u so that we have det Kn “ fnpuq and det An “ det Bn “ 2fnpuq ´ 1. In the sequel, we write fn for fnpuq for simplicity. For n, m ě 3 and ` ě 0, let Hn,m,`, An,`, and Bm,` be the matrices given by 128 Sanghoon Kwon and Seungmin Lee

n m

1 ` u2 ´u ´u ¨ ¨ ¨ ´u n » fi ´u 1 ` u2 ´u — ffi — ´u 1 ` 2u2´u ffi — K` ffi — 2 ffi — An ´u 1 ` u ´u ffi — ffi Hn,m,` “ — ´u ¨ ¨ ¨ ´u ffi ` — ffi — ´u 1 ` u2 ´u B ffi — m ffi — ´u 1 ` 2u2´u ffi — ffi — 2 ffi — ´u 1 ` u ´u ffi m — ffi — ´u ¨ ¨ ¨ ´u ffi — ffi — ´u 1 ` u2ffi — ffi — ffi – fl `

n `

1 ` u2 ´u » ´u ¨ ¨ ¨ ´u fi n — 2 ffi — ´u 1 ` u ´u ffi — ffi — 2 ffi An,` “ — ´u 1 ` 2u ´u ffi — ffi — 2 ffi — ´u 1 ` u ´u ffi — ffi — ffi — ´u ¨ ¨ ¨ ´u ffi ` — ffi — ffi — ´u 1 ` u2 ffi — ffi — ffi – fl

` m

1 ` u2 ´u

` » ´u ¨ ¨ ¨ ´u fi — 2 ffi — ´u 1 ` u ´u ffi — ffi — 2 ffi Bm,` “ — ´u 1 ` 2u ´u ffi . — ffi — 2 ffi — ´u 1 ` u ´u ffi — ffi m — ffi — ´u ¨ ¨ ¨ ´u ffi — ffi — ffi — ´u 1 ` u2 ffi — ffi — ffi – fl Zeta function of circuit rank two graphs 129

The determinant of the above matrices are given as follows.

Lemma 2.1. Let An,`, Bm,` and Hn,m,` be the above matrices. Then, we have

det An,` “ det Kn´1p1 ´ det K`q ` det An det K`

“ fn´1p1 ´ f`q ` p2fn ´ 1qf`,

deg Bm,` “ det Km´1p1 ´ det K`q ` det Bm det K`

“ fm´1 p1 ´ f`q ` p2fm ´ 1q f`,

det Hn,m,` “ det Bm,` det An ` det Bm,`´1p1 ´ det Knq

“ rfm´1 p1 ´ f`q ` p2fm ´ 1q f`s p2fn ´ 1q

` rfm´1 p1 ´ f`´1q ` p2fm ´ 1q f`´1s p1 ´ f`q .

Proof. Using row and column operations, we have det Bm,``2 “ p1 ` 2 2 u q det Bm,``1 ´ u det Bm,`. It follows from the characteristic equation of linear recurrence relations that we have 2` det Bm,` “ c1 ` c2u for some constants c1 and c2. Since 2 2 2m 2 2 2m´2 det Bm,1 “ p1 ` u qp1 ` 2u ` ¨ ¨ ¨ ` 2u q ´ u p1 ` u ` ¨ ¨ ¨ ` u q and

2 4 2 2m 2 4 2 2m´2 det Bm,2 “ p1`u `u qp1`2u `¨ ¨ ¨`2u q´pu `u qp1`u `¨ ¨ ¨`u q, we have 1 ` u2 ` ¨ ¨ ¨ ` u2m c “ ´ 1 u2 ´ 1 and u4p1 ` u2 ` u4 ` ¨ ¨ ¨ ` u2m´4 ` 2u2m´2q c “ . 2 u2 ´ 1 This yields ´p1 ` u2 ` ¨ ¨ ¨ ` u2mq ` u2``4p1 ` u2 ` ¨ ¨ ¨ ` u2m´4 ` 2u2m´2q det B “ m,` u2 ´ 1 “fm´1 p1 ´ f`q ` p2fm ´ 1q f`.

The result for An,` can be obtained analogously. 130 Sanghoon Kwon and Seungmin Lee

2 For the case of Hn,m,`, we have det Hn`2,m,` “ p1 ` 2u q det Hn`1,m,` ´ 2 2n u det Hn,m,` and hence det Hn,m,` “ c3 `c4u for some constants c3 and c4. Since

det H1,m,` ´p1 ` u2 ` ¨ ¨ ¨ ` u2mq ` u2``4p1 ` u2 ` ¨ ¨ ¨ ` u2m´4 ` 2u2m´2q “p1 ` 2u2q u2 ´ 1 ˆ ˙ ´p1 ` u2 ` ¨ ¨ ¨ ` u2mq ` u2``2p1 ` u2 ` ¨ ¨ ¨ ` u2m´4 ` 2u2m´2q ´ u2 u2 ´ 1 ˆ ˙ and det H2,m,` ´p1 ` u2 ` ¨ ¨ ¨ ` u2mq ` u2``4p1 ` u2 ` ¨ ¨ ¨ ` u2m´4 ` 2u2m´2q “p1 ` 2u2 ` 2u4q u2 ´ 1 ˆ ˙ ´p1 ` u2 ` ¨ ¨ ¨ ` u2mq ` u2``2p1 ` u2 ` ¨ ¨ ¨ ` u2m´4 ` 2u2m´2q ´ pu2 ` u4q , u2 ´ 1 ˆ ˙ we have ´u2p1 ` u2 ` ¨ ¨ ¨ ` u2mq ` u2``4p´1 ` 2u2qp1 ` u2 ` ¨ ¨ ¨ ` u2m´4 ` 2u2m´2q c4 “ pu2 ´ 1q2 and

2 2m 2``6 2 2m´4 2m´2 2 p1 ` u ` ¨ ¨ ¨ ` u q ´ u p1 ` u ` ¨ ¨ ¨ ` u ` 2u q c3 “ b1 ´ c4u “ . pu2 ´ 1q2

It follows that the determinant of the matrix Hn,m,` is equal to

p1 ´ u2n`2qp1 ` u2 ` ¨ ¨ ¨ ` u2mq ` u2``4p2u2n`2 ´ u2n ´ u2qp1 ` u2 ` ¨ ¨ ¨ ` u2m´4 ` 2u2m´2q pu2 ´ 1q2

“ rfm´1 p1 ´ f`q ` p2fm ´ 1q f`s p2fn ´ 1q ` rfm´1 p1 ´ f`´1q ` p2fm ´ 1q f`´1s p1 ´ f`q . This completes the proof of lemma.

3. Proof of theorem 1.3

Let D “ Dn,m,` be a dumbbell graph with n, m ě 3 and ` ě 0. We use the Ihara’s determinant formula (Theorem 1.2). Let us denote by Zeta function of circuit rank two graphs 131

2 RD the matrix I ´ Au ` Qu for D given as follows. n m

1 ` u2 ´u ´u » ´u ¨ ¨ ¨ ´u fi n — ´u 1 ` u2 ´u ffi — ffi — 2 ffi — ´u ´u 1 ` 2u ´u ffi — ffi — u 1 u2 u ffi — ´ ` ´ ffi — ffi RD “ — ´u ¨ ¨ ¨ ´u ffi ` — ffi — 2 ffi — ´u 1 ` u ´u ffi — ffi — ´u 1 ` 2u2´u ´u ffi — ffi — 2 ffi — ´u 1 ` u ´u ffi m — ffi — ffi — ´u ¨ ¨ ¨ ´u ffi — ffi — ´u ´u 1 ` u2ffi — ffi — ffi – fl ` To show the Theorem 1.3, it is enough to calculate the determinant of RD. Using the row and column operation we have det RD 4 2 2 2 2 “ u det Hn´2,m´2,` ´ u p1 ` u q det Hn´2,m´1,` ´ u p1 ` u q det Hn´1,m´2,` 2 m`n n`2 ` p1 ` u q det Hn´1,m´1,` ` 4u det K` ` 2u det K` det Km´2 m`2 4 m`2 ` 2u det K` det Kn´2 ` u det K` det Kn´2 det Km´2 ` 2u det An´2,` 2 2 ´ u p1 ` u q det Km´2 det An´2,` 4 m 2 n`2 ` u det Km´2 det An´2,` ´ 2u p1 ` u q det An´2,` ` 2u det Bm´2,` 4 ` u det Kn´2 det Bm´2,` n 2 2 2 ´ 2u p1 ` u q det Bm´1,` ´ u p1 ` u q det Kn´2 det Bm´1,`.

Replacing the determinants of Hn,m,`, Kn, An,`, Bn,` by fn in the above expression, we have

4 det RD “ u rpfm´3 p1 ´ f`q ` p2fm´2 ´ 1q f`q p2fn´2 ´ 1q

` pfm´3 p1 ´ f`´1q ` p2fm´2 ´ 1q f`´1q p1 ´ fn´2qs 2 2 ´ u p1 ` u q rpfm´2 p1 ´ f`q ` p2fm´1 ´ 1q f`q p2fn´2 ´ 1q

` pfm´2 p1 ´ f`´1q ` p2fm´1 ´ 1q f`´1q p1 ´ fn´2q s 2 2 ´ u p1 ` u q rpfm´3 p1 ´ f`q ` p2fm´2 ´ 1q f`q p2fn´1 ´ 1q 132 Sanghoon Kwon and Seungmin Lee

` pfm´3 p1 ´ fL´1q ` p2fm´2 ´ 1q f`´1q p1 ´ fn´1qs 2 ` p1 ` u q rpfm´2 p1 ´ f`q ` p2fm´1 ´ 1q f`q p2fn´1 ´ 1q

` pfm´2 p1 ´ f`´1q ` p2fm´1 ´ 1q f`´1q p1 ´ fn´1qs m`n n`2 m`2 4 ` 4u f` ` 2u f`fm´2 ` 2u f`fn´2 ` u f`fn´2fm´2 m`2 ` 2u rfn´3 p1 ´ f`q ` p2fn´2 ´ 1q f`s 2 2 ´ u p1 ` u qfm´2 rfn´2 p1 ´ f`q ` p2fn´1 ´ 1q f`s 4 ` u fm´2 rfn´3 p1 ´ f`q ` p2fn´2 ´ 1q f`s m 2 ´ 2u p1 ` u q rfn´2 p1 ´ f`q ` p2fn´1 ´ 1q f`s n`2 ` 2u pfm´3 p1 ´ f`q ` p2fm´2 ´ 1q f`q 4 ` u fn´2 rfm´3 p1 ´ f`q ` p2fm´2 ´ 1q f`s n 2 ´ 2u p1 ` u q rfm´2 p1 ´ f`q ` p2fm´1 ´ 1q f`s 2 2 ´ u p1 ` u qfn´2 rfm´2 p1 ´ f`q ` p2fm´1 ´ 1q f`s . Expanding the whole equation, we get n`2 n n`2 m`2 4 det RD “ 2u fm´3 ´ 2u fm´2 ´ 2u fm´2 ` 2u fn´3 ` u fm´2fn´3 4 m m`2 2 4 ` 2u fn´2 ´ 2u fn´2 ´ 2u fn´2 ´ 3u fm´2fn´2 ´ 3u fm´2fn´2 2 4 2 4 ´ u fn´1 ´ u fn´1 ` fm´2fn´1 ` 2u fm´2fn´1 ` u fm´2fn´1 2 2 2 2 ´ f`´1 1 ´ u ` 1 ` 3u fm´2 ´ 2 1 ` u fm´1 1 ` u fn´2 2 2 m n n`2 m`n ´ 1 ``u fn´1 ` f` 1 ´˘u ` 2u `` 2u ´˘ 2u ˘ ` 4u m`2 2 4 m m`2 ´ `2u f˘n´3 `˘3u fn´“2 ´ 3u fn´2 ` 2u fn´2 ` 8u fn´2 4 m m`2 ´ 2fn´1 ` 2u fn´1 ´ 4u fn´1 ´ 4u fn´1 2 n 2 2 ´ 2 1 ` u fm´1 1 ` 2u ` 3u fn´2 ´ 2 1 ` u fn´1 4 2 2 ` fm`´2t´u˘fn´3 `` u 4 ` 13u fn´2 ` ˘ ˘ 2 n 2 ` 1 ` 4u 1 ` 2u ´` 2 1 ` u˘ fn´1 u . ` ˘ ` ` ˘ ˘ ‰ This can by factored as m n n`m`2``2 m`n n m 2 det RD “ pu ´ 1q pu ´ 1q 4u ´ u ` u ` u ´ 1 { u ´ 1 which yields ´ ¯ ` ˘ 1 ZDpuq “ . p1 ´ unq p1 ´ umq p1 ´ un ´ um ` un`m ´ 4un`m`2``2q Zeta function of circuit rank two graphs 133

by the Ihara’s determinant formula.

4. Proof of theorem 1.4

Let En,m be the figure eight graph with n, m ě 3. Let us define the pn ` m ´ 1q ˆ pn ` m ´ 1q square matrix Hn,m by

n

1 ` u2 ´u » ´u ¨ ¨ ¨ ´u fi n 2 — ´u 1 ` u ´u ffi — ffi H “ — 2 ffi n,m — ´u 1 ` 3u ´u ffi — 2 ffi — ´u 1 ` u ´u ffi — ffi m — ffi — ´u ¨ ¨ ¨ ´u ffi — ffi — ´u 1 ` u2 ffi — ffi — ffi – fl m

Using the row and column expansion, the determinant of Hn,m is given by

2 det Hn,m “ 1 ` 3u det Kn´1 det Km´1 2 2 ` ` u˘det Kn´2 det Km´1 ` u det Kn´1 det Km´2 2 2 2 “ 1 ` 3u fn´1fm´1 ` u fn´2fm´1 ` u fn´1fm´2

2 Let RE be the` matrix˘I ´ Au ` Qu for E given as follows.

n

1 ` u2 ´u ´u » ´u ¨ ¨ ¨ ´u fi n 2 — ´u 1 ` u ´u ffi — ffi R “ — 2 ffi E — ´u ´u 1 ` 3u ´u ´u ffi — 2 ffi — ´u 1 ` u ´u ffi — ffi m — ffi — ´u ¨ ¨ ¨ ´u ffi — ffi — ´u ´u 1 ` u2 ffi — ffi — ffi – fl m 134 Sanghoon Kwon and Seungmin Lee

The determinant of RE is given by

2 2 2 2 det RE “ p1 ` u q det Hn´1,m´1 ´ p1 ` u qu det Hn´1,m´2 2 m 2 m ´ p1 ` u qu det Kn´2 ´ p1 ` u qu det Kn´2 2 2 2 2 ´ p1 ` u qu det Kn´2 det Km´2 ´ p1 ` u qu det Hn´2,m´1 4 m`2 ` u det Hn´2,m´2 ` u det Kn´3 m`2 4 ` u det Kn´3 ` u det Kn´3 det Km´2 2 n n`2 ´ p1 ` u qu det Km´2 ` u det Km´3 2 n`2 n`2 2 ´ p1 ` u q det Km´2u ` u det Km´3 ´ u det Kn´2 det Km´1.

As in the case of dumbbell graphs, we replace all the determinants by the functions of u and use the Ihara’s determinant formula to get 1 Z puq “ . E p1 ´ umqp1 ´ unqp1 ´ um ´ un ` 3um`nq This completes the proof of Theorem 1.4.

5. Proof of Theorem 1.5

Let Bn,m,` be the bicyclic graph with n, m ě 2 and ` ě 0. Similarly, 2 let us denote by RB the matrix I ´ Au ` Qu for B.

n m

1 ` u2 ´u ´u » ´u ¨ ¨ ¨ ´u fi n 2 — ´u 1 ` u ´u ffi — ffi — ´u 1 ` 2u2´u ´u ffi — ffi — 2 ffi — ´u 1 ` u ´u ffi — ffi RB “ — ´u ¨ ¨ ¨ ´u ffi ` — ffi — u 1 u2 u ffi — ´ ` ´ ffi — 2 ffi — ´u ´u 1 ` 2u ´u ffi — ffi — u 1 u2 u ffi — ´ ` ´ ffi m — ffi — ´u ¨ ¨ ¨ ´u ffi — ffi — ´u ´u 1 ` u2ffi — ffi — ffi – fl ` Zeta function of circuit rank two graphs 135

It follows that 2 2 2 2 det RB “ p1 ` u q det Hn´1,m´1,` ´ p1 ` u qu det Hn´1,m´2,` 2 m```1 2 m```1 ` p1 ` u qu det Kn´2 ´ p1 ` u qu det Kn´2 2 2 ´ p1 ` u qu det Kn´2 det Bm´1,` 2 2 4 m```3 ´ p1 ` u qu det Hn´2,m´1,` ` u Hn´2,m´2,` ` u det Kn´3 m```3 4 2 n```1 ` u det Kn´3 ` u det Kn´3 det Bm´1,` ´ p1 ` u qu det Km´2 n```3 n`m 2 n```1 ` u det Km´3 ´ u det K` ´ p1 ` u qu det Km´2 n```3 n`m 2 2 ` u det Km´3 ´ u det K` ´ p1 ` u qu det An´1,` det Km´2 4 4 ` u det An´1,` det Km´3 ` u det Kn´2 det K` det Km´2. Expanding the whole polynomial and applying the Ihara’s determinant formula, we obtain that ZBpuq is equal to

1 ´ . p2un`m```1 ´ un`m ´ un```1 ´ um```1 ` 1q p2un`m```1 ` un`m ` un```1 ` um```1 ´ 1q

Acknowledgements

We would like to thank the anonymous referee for carefully reading our manuscript and for giving constructive comments.

References [1] H. Bass, The Ihara-Selberg zeta function of a tree lattice, Int. J. Math. 3 (1992) 717–797 [2] Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields, J. Math. Soc. Japan 18 (1966), 219–235. [3] N. Konno, H. Mitsuhashi, H. Morita and I. Sato, A new weighted Ihara zeta function for a graph, Linear Algebra and its Applications. 571 (2019) 154–179. [4] S. Kwon and J. Park, Ihara zeta function of dumbbell graphs, Korean J. Math. 26 (2018) 741–746. [5] M. Kotani and T. Sunada, Zeta functions of finite graphs, J. Math. Sci. Univ. Tokyo, 7 (2000), 7–25 [6] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87 [7] J.-P. Serre, Trees, Springer Monographs in Mathematics (2003) 136 Sanghoon Kwon and Seungmin Lee

[8] A. Terras, Zeta function of graphs: A Stroll through the Garden, Cambridge Studies in Advanced Mathematics. 128 (2010)

Sanghoon Kwon Department of Mathematical Education Catholic Kwandong University, Gangneung, Republic of Korea, 25601 E-mail: [email protected] Seungmin Lee Department of Mathematical Education Catholic Kwandong University, Gangneung, Republic of Korea, 25601 E-mail: [email protected]