Rongquan Feng and Jin Ho Kwak

Transcription

1 J. Korean Math. Soc ), No. 6, pp ZETA FUNCTIONS OF GRAPH BUNDLES Rongquan Feng and Jin Ho Kwak Abstract. As a continuation of computing the zeta function of a regular covering graph by Mizuno and Sato in [9], we derive in this paper computational formulae for the zeta functions of a graph bundle and of any regular or irregular) covering of a graph. If the voltages to derive them lie in an abelian or dihedral group and its fibre is a regular graph, those formulae can be simplified. As a by-product, the zeta function of the cartesian product of a graph and a regular graph is obtained. The same work is also done for a discrete torus and for a discrete Klein bottle. 1. Introduction In this paper we consider an undirected finite simple graph. Let G be a graph with vertex set V G) and edge set EG). The degree deg G v) of a vertex v in G is the number of edges of G incident with v. An automorphism of G is a permutation of the vertex set V G) that preserves adjacency. The set of automorphisms forms a permutation group, called the automorphism group AutG) of G. A v 0, v n )-path P of length n in G is a sequence P = v 0, v 1,..., v n 1, v n ) of n + 1 vertices and n edges such that consecutive vertices share an edge we do not require that all vertices are distinct). Sometimes, the path P is also considered as a subgraph of G. We say that a path has a backtracking if v i 1 = v i+1 for some i, 1 i n 1. A v 0, v n )- path is called a cycle if v 0 = v n. The inverse cycle of a cycle C = v 0, v 1,..., v n 1, v 0 ) is the cycle C 1 = v 0, v n 1,..., v 1, v 0 ). A subpath v 1,..., v m 1, v m ) of a cycle C = v 1,..., v m,..., v 1 ) is called a tail if deg C v 1 ) = 1, deg C v i ) =, i m 1, and Received July 15, Mathematics Subject Classification: 05C50, 05C5, 15A15, 15A18. Key words and phrases: zeta function, graph bundle, voltage assignment, discrete torus or Klein bottle. The first author is supported by NSF of China No ) and the second author is supported by Com MaC-KOSEF in Korea.

2 170 Rongquan Feng and Jin Ho Kwak deg C v m ) 3, where deg C v) is the degree of v in the subgraph C. Each cycle C without backtracking determines a unique tail-less, backtracking-less cycle C by removing all tails of C. Note that any backtrackingless tail-less cycle C is just a cycle such that both C and C have no backtracking see [4], [11]). Two backtracking-less, tail-less cycles C 1 = v 1,..., v m ) and C = w 1,..., w m ) are called equivalent if there is k such that w j = v j+k for all j, where the subscripts are module m. Let [C] denote the equivalence class which contains a cycle C. A backtracking-less, tail-less cycle C is primitive if C is not obtained by going r times around some other cycle B, i.e., C B r for r. Note that each equivalence class of primitive, backtracking-less and tail-less cycles of a graph G corresponds to a conjugacy class of the fundamental group π 1 G, v) of G for a vertex v V G). The Ihara) zeta function [13] of a graph G is defined to be the function of u C with u sufficiently small, given by ZG, u) = Z G u) = [C]1 u νc) ) 1, where [C] runs over all equivalence classes of primitive, backtracking-less and tail-less cycles of G and νc) denotes the length of C. Clearly, the zeta function of a disconnected graph is the product of the zeta functions of its connected components. Zeta functions of graphs were originated from zeta functions of regular graphs by Ihara [5], where their reciprocals are expressed as explicit polynomials. A zeta function of a regular graph G associated to a unitary representation of the fundamental group of G was developed by Sunada [14]. Hashimoto [4] treated multivariable zeta functions of bipartite graphs. Northshield [11] proved that the number of spanning trees in a graph G can be expressed in terms of the zeta function Z G u). Let G be a connected graph with V G) = {v 1,..., v m }. The adjacency matrix AG) = a ij ) is an m m matrix with a ij = 1 if v i and v j are adjacent and a ij = 0 otherwise. Let D G denote the diagonal matrix with diagonal entries d G i = deg G v i ), 1 i m, and let Q G = D G I. The Ihara s result on zeta functions of regular graphs is generalized as follows. Theorem 1. Bass [1]) Let G be a connected graph with m vertices and s edges. Then the reciprocal of the zeta function of G is given by Z G u) 1 = 1 u ) s m det I AG)u + Q G u ).

3 Zeta functions of graph bundles 171 Note that Theorem 1 is still true for a disconnected graph G. Later, Stark and Terras [13] gave an elementary proof of Theorem 1 and discussed three different zeta functions of a graph. Mizuno and Sato [9] gave a decomposition formula for the zeta function of a regular covering of a graph. In this paper, we compute the zeta function of a graph bundle. In section, a formula for the zeta function of a graph bundle is derived by Theorem 1 and a decomposition formula for the zeta function of any regular or irregular) covering graph is given. In Sections 3 and 4, we compute the zeta function of a graph bundle when its voltages lie in an abelian group or in a dihedral group and whose fibre is a regular graph. The zeta function of a cartesian product of a graph with a regular graph is given in Section 3 too. As an application, in Section 5, the zeta functions of a discrete torus and of a discrete Klein bottle are computed.. Computing zeta functions of graph bundles Let G be a connected graph and let G be the digraph obtained from G by replacing each edge of G with a pair of oppositely directed edges. The set of directed edges of G is denoted by E G). By e 1, we mean the reverse edge to an edge e E G). We denote the directed edge e of G by uv if the initial and terminal vertices of e are u and v, respectively. For a finite group Γ, a Γ-voltage assignment on G is a function φ : E G) Γ such that φe 1 ) = φe) 1 for all e E G). We denote the set of all Γ-voltage assignments on G by C 1 G; Γ). Let F be another graph and let φ C 1 G; AutF )). Now, we construct a new graph G φ F with the vertex set V G φ F ) = V G) V F ), and two vertices u 1, v 1 ) and u, v ) are adjacent in G φ F if either u 1 u E G) and v = v φu 1u ) 1 or u 1 = u and v 1 v EF ). We call G φ F the F -bundle over G associated with φ or, simply a graph bundle) and the first coordinate projection induces the bundle projection p φ : G φ F G. The graphs G and F are called the base and the fibre of the graph bundle G φ F, respectively. Note that the map p φ maps vertices to vertices, but the image of an edge can be either an edge or a vertex. If F = K n, the complement of the complete graph K n of n vertices, then an F -bundle over G is just an n-fold graph covering of G. If φe) is the identity of AutF ) for all e E G), then G φ F is just the cartesian product of G and F. See [6]).

4 17 Rongquan Feng and Jin Ho Kwak Let φ be an AutF )-voltage assignment on G. For each γ AutF ), let G φ,γ) denote the spanning subgraph of the digraph G whose directed edge set is φ 1 γ). Thus the digraph G is the edge-disjoint union of spanning subgraphs G φ,γ), γ AutF ). Let V G) = {u 1, u,..., u m } and V F ) = {v 1, v,..., v n }. We define an order relation on V G φ F ) as follows: for u i, v k ), u j, v l ) V G φ F ), u i, v k ) u j, v l ) if and only if either k < l or k = l and i j. Let P γ) denote the n n permutation matrix associated with γ AutF ) corresponding to the action of AutF ) on V F ), i.e., its i, j)-entry P γ) ij = 1 if v γ i = v j and P γ) ij = 0 otherwise. Then for any γ, δ AutF ), P δγ) = P δ)p γ). The tensor product A B of the matrices A and B is considered as the matrix B having the element b ij replaced by the matrix Ab ij. Kwak and Lee formulated the adjacency matrix AG φ F ) of a graph bundle G φ F as follows. Theorem. [7]) AG φ F ) = γ AutF ) A G φ,γ) ) P γ) + I m AF ), where P γ) is the n n permutation matrix associated with γ AutF ) corresponding to the action of AutF ) on V F ), and I m is the identity matrix of order m = V G). For any vertex u i, v k ) V G φ F ), its degree is d G i +df k, where dg i = deg G u i ) and d F k = deg F v k ). Therefore, D G φ F = D G I n + I m D F and then Q G φ F = D G φ F I mn = D G I m ) I n + I m D F = Q G I n + I m D F. Furthermore, if we set EG) = s and EF ) = t, then EG φ F ) = 1 m n d G i + d F k ) k=1 ) = 1 m n n d G i + m d F k = ns + mt. k=1 The following theorem follows immediately from Theorem 1. Theorem 3. Let G be a connected graph with m vertices and s edges, F a graph with n vertices and t edges and let φ be an AutF )- voltage assignment on G. Then the reciprocal of the zeta function of

5 G φ F is Z G φ F u) 1 = 1 u ) ns+mt) mn det I mn Zeta functions of graph bundles 173 γ AutF ) + Q G I n + I m D F )u ). A G φ,γ) ) P γ) + I m AF ) ) u In the following, we consider three particular cases of Theorem 3: i) F = K n, ii) φ = 1 is trivial, or more generally all voltages lie in an abelian group, and iii) all voltages lie in a dihedral group. The last two cases will be treated in Sections 3 and 4, respectively. As the first case, let F = K n be n isolated vertices. Then any AutK n )-voltage assignment is just a permutation voltage assignment defined in [3], and G φ K n = G φ is just an n-fold covering graph of G. In this case, it may not be a regular covering. Hence, Z G φu) 1 = 1 u ) s m)n det I mn A G φ,γ) ) P γ) u + Q G I n )u. γ Γ A representation ρ of a group Γ over the complex field C is a group homomorphism from Γ to the general linear group GLr, C) of all r r invertible matrices over C. The number r is called the degree of the representation ρ see [15]). Let φ C 1 G; AutK n )) be a permutation voltage assignment on G, and let Γ = φe) e E G) be the subgroup of S n = AutK n ) generated by the voltages φe). Then, it is clear that the homomorphism P : Γ GLn, C) defined by γ P γ) is a representation of Γ, which is called the permutation representation of Γ. Let ρ 1 = 1, ρ,..., ρ l be the nonequivalent irreducible representations of Γ and let f i be the degree of ρ i for 1 i l, so that l f i = Γ. It is well-known [15] that the permutation representation P can be decomposed as the direct sum of irreducible representations: Say P = l m iρ i with multiplicities m i. Then, there exists an invertible matrix M such that l M 1 P γ)m = ρ i γ) I mi )

6 174 Rongquan Feng and Jin Ho Kwak for any γ Γ, where A 1 A k denotes the block diagonal matrix with diagonal blocks A 1,..., A k consecutively. Noting that AG) = γ Γ A G φ,γ) ) and l m if i = n, we have I m M) 1 I mn A G φ,γ) ) P γ) u + Q G I n )u γ Γ I m M) l = I mn A G φ,γ) ) ρ i γ)) I mi u + Q G I n )u γ Γ l = I mfi A G φ,γ) ) ρ i γ) u + Q G I fi )u I mi. γ Γ Furthermore, it is known [1] that m 1 is the number of orbits under the action of the group Γ, so that m 1 1. Therefore, the zeta function of a regular or irregular) covering G φ is Z G φu) 1 = 1 u ) s m deti m AG)u + Q G u )) m 1 l u i=1 ) s m)fi deti mfi A G φ,γ) ) ρ i γ))u γ Γ + Q G I fi )u )) m i. Since Z G u) 1 = 1 u ) s m deti m AG)u + Q G u ), we have the following theorem. Theorem 4. Let G be a connected graph with m vertices and s edges, φ C 1 G; AutK n )) a permutation voltage assignment on G, and let Γ = φe) e E G) be the subgroup generated by the voltages φe). Let ρ 1 = 1, ρ,..., ρ l be the irreducible representations of Γ with degrees f 1, f,..., f l, respectively. Then the reciprocal of the zeta function of the n-fold covering G φ of G derived from the voltage assignment φ is Z G φu) 1 = Z G u) 1) m 1 l i= 1 u ) s m)f i T i u)) mi,

7 Zeta functions of graph bundles 175 where 1) T i u) = det I mfi A G φ,γ) ) ρ i γ) u + Q G I fi )u γ Γ and m i is the multiplicity of ρ i in the permutation representation P of Γ. It gives another proof of Corollary 1 in Section in [13]. Corollary 5. Stark and Terras [13]) For every connected covering graph G φ of the graph G, the inverse zeta function Z G u) 1 divides Z G φu) 1. If Γ acts on itself by right multiplication, then Γ can be identified as a regular subgroup of S Γ and the covering G φ is a regular covering of G. In this case, the multiplicity m i is equal to the degree f i of the irreducible representation ρ i. Therefore, we have Theorem in [9] as a corollary. Corollary 6. Mizuno and Sato [9]) The reciprocal of the zeta function of the connected regular covering G φ of G derived from an ordinary voltage assignment φ : E G) Γ is Z G φu) 1 = Z G u) 1 where T i u) is given in Eq 1). l i= 1 u ) s m)f i T i u)) fi, For a voltage assignment φ : E G) Γ and an irreducible representation ρ of Γ, the L-function of G associated to ρ and φ is defined by Z G u, ρ, φ) = det I f ρφc))u νc)) 1, [C] where f is the degree of ρ, φc) is the net voltage on C and [C] runs over all equivalence classes of primitive, backtracking-less and tail-less cycles of G see [4], [5] and [14]). It was proved in [9] that, for the irreducible representations ρ i of Γ, Z G u, ρ i, φ) 1 = 1 u ) s m)f i T i u), where T i u) is given in Eq 1). Therefore we have the following corollary.

8 176 Rongquan Feng and Jin Ho Kwak Corollary 7. Under the same assumptions as in Theorem 4, we have Z G φu) = l Z G u, ρ i, φ) m i, where m i is the multiplicity of ρ i in the permutation representation P of Γ. Example 1. Let n = 3 and Γ = S 3, so that G φ is a 3-fold covering of G. The symmetric group S 3 has three irreducible representations ρ 1 = 1, ρ the sign representation, with degrees f 1 = f = 1 and ρ 3 with degree f 3 = defined as follows: [ ] [ ] µ 0 µ 0 ρ 3 1) = I, ρ 3 13)) = 0 µ, ρ 3 13)) =, 0 µ [ ] [ ] [ 0 µ µ ρ 3 1)) = µ, ρ )) =, ρ )) = µ 0 ], where µ = expπi/3). The permutation representation can be decomposed as P = ρ 1 ρ 3. That is m 1 = 1, m = 0 and m 3 = 1. By Theorem 4, one can get Z G φu) 1 = Z G u) 1 1 u ) s m) det I m A G φ,γ) ) ρ 3 γ) u + Q G I )u, γ S3 where m is the number of vertices of G. Note that S 3 is the dihedral group of order 6. This result can also be obtained from Theorem 9 in Section 4 later in a different way. In particular, if G is the diamond graph, i.e., the complete graph K 4 minus an edge, then an easy computation gives that Z G u) 1 = 1 u )1 u)1 + u )1 + u + u )1 u u 3 ).

9 Zeta functions of graph bundles 177 Consider an S 3 -voltage assignment φ which is defined as in Figure 1. u 4 3) u 1) 1 u u 1 Figure 1: An S 3 -voltage assignment φ on the diamond graph and Then the covering G φ is connected. Moreover, A G φ,1) ) = A G φ,13)) ) = A G φ,13)) ) = A G φ,13)) ) = A G φ,3)) ) =, A G φ,1)) ) = Thus one can have Z G φu) 1 = Z G u) 1 1 u ) 1 u u 3 + u 4 ) 1 u + u u 3 + u 4 )1 + u + u 3 + u 4 ) 1 + u + u + u 3 + u 4 ). Note that this formula is given in Example 53) in [13] with an aid of Mathematica. However we compute it here with a permutation voltage assignment., 3. Graph bundles having voltages in an abelian group In this section, we consider the zeta function of a graph bundle G φ F when the images of φ lie in an abelian subgroup Γ of AutF ) and the graph F is regular. In this case, for any γ 1, γ Γ, the permutation matrices P γ 1 ) and P γ ) are commutative and D F = k F I n, where k F denotes the degree of the graph F.

10 178 Rongquan Feng and Jin Ho Kwak It is well-known [] that every permutation matrix P γ) commutes with the adjacency matrix AF ) of F for all γ AutF ). Since the matrices P γ), γ Γ, and AF ) are all diagonalizable and commute with each other, they are simultaneously diagonalizable. That is, there exists an invertible matrix M Γ such that M 1 Γ P γ)m Γ and M 1 Γ AF )M Γ are diagonal matrices for all γ Γ. Let λ γ,1),..., λ γ,n) be the eigenvalues of the permutation matrix P γ) and let λ F,1),..., λ F,n) be the eigenvalues of the adjacency matrix AF ). Then λ γ,1) 0 M 1 Γ P γ)m Γ =... and 0 λ γ,n) Therefore, M 1 Γ AF )M Γ = λ F,1) λ F,n). I m M Γ ) 1 A G φ,γ) ) P γ) + I m AF ) I m M Γ ) γ Γ = A G φ,γ) ) γ Γ = and Thus λ F,1) 0... λ γ,1) λ γ,n) +I m 0 λ F,n) n λ γ,i) A G φ,γ) ) + λ F,i) I m γ Γ I m M Γ ) 1 Q G I n + I m D F )I m M Γ ) = n Q G + k F I m ). deti mn γ Γ A G φ,γ) ) P γ) + I m AF ) )u +Q G I n + I m D F )u )

11 = det Zeta functions of graph bundles 179 n I m λ γ,i) A G φ,γ) ) + λ F,i) I m )u γ Γ +Q G + k F I m )u )), and then Z G φ F u) 1 = 1 u ) s m+ mk F )n n λ γ,i) A G φ,γ) ) + λ F,i) I m )u + Q G + k F I m )u ). deti m γ Γ Since the cartesian product G F of two graphs G and F is the F -bundle over G associated with the trivial voltage assignment φ, i.e., φe) = 1 for all e E G) and AG) = A G), we get the following corollary. Corollary 8. For any connected graph G and a connected k F - regular graph F, the reciprocal of the zeta function of the cartesian product G F is s m+ mk F )n Z G F u) 1 = 1 u ) n det I m AG) + λ F,i) I m )u + Q G + k F I m )u ), where V G) = m, EG) = s and V F ) = n. In particular, if G is regular of degree k G, then the reciprocal of the zeta function of the cartesian product G F is ) kg +k F 1 mn Z G F u) 1 = 1 u ) n m 1 λg,j) + λ F,i) )u + k G + k F 1)u ), j=1 where λ G,j), 1 j m, are the eigenvalues of the graph G. Example. Let G = F = K 3, the complete graph with 3 vertices. Then ZK 3, u) 1 = 1 u 3 ). The eigenvalues of K 3 are, 1 and 1. By Corollary 8, we get Z K3 K 3 u) 1 = 1 u ) 9 1 4u + 3u )1 u + 3u ) u + 3u ) 4. It is clear that ZK 3, u) 1 is not a divisor of ZK 3 K 3, u) 1. This example says that in general, Z G u) 1 or Z F u) 1 is not necessarily to be a divisor of Z G F u) 1.

12 180 Rongquan Feng and Jin Ho Kwak In general, for the complete graph K m, its eigenvalues are m 1 with multiplicity 1 and 1 with multiplicity m 1. Therefore Z Km K n u) 1 = 1 u ) m+n )mn 1 m + n )u + m + n 3)u ) 1 m )u + m + n 3)u ) n 1 1 n )u + m + n 3)u ) m u + m + n 3)u ) m 1)n 1). Example 3. Let C n be the n-cycle. It is known [15] that the eigenvalues of C n are cosπk/n), 0 k n 1. By Corollary 8, we get Z Cm C n u) 1 n 1 = 1 u ) mn m 1 k =0 k 1 =0 1 cosπk 1 /m) + cosπk /n))u + 3u ). 4. Graph bundles having voltages in a dihedral group In this section, we compute the zeta function of a graph bundle having voltages in a dihedral group. Assume that F is k F -regular with n vertices again and let AutF ) contains the dihedral group D n of order n, which is described below, as a subgroup in addition. Let S n denote the symmetric group on V F ). Set V F ) = {1,,..., n} for a notational convenience. Let a = 1 n 1 n) be an n-cycle and let b = { 1 n) n 1) n 1 n+ n+3 n+1 ) ) if n is odd, ) if n is even 1 n) n 1) n be a permutation in S n. Note that the permutations a and b generate the dihedral subgroup D n of S n, where D n = a, b a n = b = 1, bab = a 1 = {1, a,..., a n 1, b, ab,..., a n 1 b}. Let µ = expπi/n) and let x k = 1, µ k, µ k,..., µ n 1)k ) T be a column vector in the complex n-space C n. Then 1, µ,..., µ n 1 are distinct eigenvalues of the permutation matrix P a) and x k is an eigenvector of P a) belonging to the eigenvalue µ k for 0 k n 1. Let P b) be the permutation matrix of b and let { [x0 x 1 P b)x 1 x P b)x x n 1)/ P b)x n 1)/ ] if n is odd, M = [x 0 x 1 P b)x 1 x P b)x x n )/ P b)x n )/ x n/ ] if n is even.

13 Zeta functions of graph bundles 181 From P a)p b)x k =P ab)x k =P ba 1 )x k =P b)p a 1 )x k = µ n k P b)x k, we know that for any 0 k n 1, P b)x k is an eigenvector of P a) associated with the eigenvalue µ n k. Thus the matrix M is invertible because its column vectors are eigenvectors of P a) associated with distinct eigenvalues. Moreover, P a) and AF ) are commutative since a AutF ) from the assumption. Thus P a) and AF ) are simultaneously diagonalizable. Note that the columns of M are eigenvectors of P a) associated with distinct eigenvalues, so each column of M is again an eigenvector of AF ). Since P b)af ) = AF )P b) for 1 k n 1)/ when n is odd and for 1 k n )/ when n is even, x k and P b)x k are eigenvectors of AF ) belonging to the same eigenvalue, say it λ F,k). The all-one vector x 0 is an eigenvector of AF ) belonging to the eigenvalue k F, the degree of the graph F. When n is even, denote by λ F,n/) the eigenvalue of AF ) associated with the eigenvector x n/. From [8], we know that I m M) 1 A G φ,γ) ) P γ) + I m AF ) I m M) γ D n n 1)/ AG) + k F I m ) A t + λ F,t) I m ) if n is odd, t=1 = n )/ AG) + k F I m ) A t + λ F,t) I m ) t=1 B + λ F,n/) I m ) if n is even, where [ n 1 ) A t = k=0 µ tk A G φ,a k )) µ tk A G φ,a k b)) µ n t)k A G φ,a k b)) µ n t)k A G φ,a k )) ] is a m m matrix and n 1 3) B = 1) k A G φ,a k )) + 1) k+1 A G ) φ,a k b)) k=0 is an m m matrix. Again we have I m M) 1 Q G I n + I m D F )I m M) = n Q G + k F I m ).

14 18 Rongquan Feng and Jin Ho Kwak We denote the m m matrix Q G + k F I m ) I by L G. Thus when n is odd, deti mn γ D n A G φ,γ) ) P γ) + I m AF ))u +Q G I n + I m D F )u ) = deti m AG) + k F I m )u + Q G + k F I m )u ) n 1)/ t=1 and when n is even, deti mn deti m A t + λ F,t) I m )u + L G u ), γ D n A G φ,γ) ) P γ) + I m AF ))u +Q G I n + I m D F )u ) = deti m AG) + k F I m )u + Q G + k F I m )u ) deti m B + λ F,n/) I m )u + Q G + k F I m )u ) n )/ t=1 deti m A t + λ F,t) I m )u + L G u ). The following theorem comes from Theorem 3. Theorem 9. Let G be a connected graph with m vertices and s edges, and let F be a connected k F -regular graph with n vertices such that AutF ) contains the dihedral group D n. Then for any D n -voltage assignment φ on G, the reciprocal of the zeta function of the graph bundle G φ F is ) Z G φ F u) 1 = 1 u s m+ ) mk n 1)/ F n fg,f u) g G,F,t u) when n is odd, and ) Z G φ F u) 1 = 1 u s m+ ) mk n )/ F n fg,f u)h G,F u) g G,F,t u) t=1 t=1

15 Zeta functions of graph bundles 183 when n is even, where the polynomials f G,F u), g G,F,t u) and h G,F u) are f G,F u) = deti m AG) + k F I m )u + Q G + k F I m )u ), g G,F,t u) = deti m A t + λ F,t) I m )u + L G u ), h G,F u) = deti m B + λ F,n/) I m )u + Q G + k F I m )u ). 5. Applications For a graph G with m vertices, let ω : E G) C be a function on the set of directed edges of G such that ωe 1 ) = ωe), the complex conjugate of ωe) for each e E G). Such a function ω is called a symmetric weight function on the graph G. Define an m m matrix AG ω ) = a ij ) as { ωui u j ) if u i u j E G), a ij = 0 otherwise. Note that AG ω ) is a Hermitian matrix and AG ω ) = AG) when ωe) = 1 for all e E G). For any D n -voltage assignment φ on G, define a new Z -voltage assignment ψ φ on G by { 1 if φe) = a k for some 0 k n 1, ψ φ e) = 1 if φe) = a k b for some 0 k n 1, for e = u i u j E G). The derived double covering of G by ψ φ is denoted by G ψ φ. For any t, 1 t n 1)/, define a function ω t φ) : E G ψ φ) C as follows: for any e = u i, g)u j, ψ φ u i u j )g) E G ψ φ), { µ tk if g = 1 and φu i u j ) = a k or a k b), ω t φ)e) = µ n t)k if g = 1 and φu i u j ) = a k or a k b), where µ = expπi/n). Then ω t φ) is a symmetric weight function on the graph G ψ φ. Define another function ω 1 φ) : E G) C on the graph G by { 1) k if φu i u j ) = a k, ω 1 φ)u i u j ) = 1) k+1 if φu i u j ) = a k b for u i u j E G). Then ω 1 φ) is a symmetric weight function on G. The following lemma was obtained by Kwak and Kwon.

16 184 Rongquan Feng and Jin Ho Kwak Lemma 10. [8]) Let A t and B be the matrices in Eq ) and 3), respectively. Then 1) for any t = 1,,..., n 1)/, AG ψ φ ω tφ) ) = A t as m m matrices under vertex order u 1, 1), u, 1),..., u m, 1), u 1, 1), u, 1),..., u m, 1), and ) if n is even, AG ω 1 φ)) = B as m m matrices. Let G = C m be the m-cycle with consecutive vertices u 1, u,..., u m. The digraph C m is an edge-disjoint union of two directed cycles C m + = u 1, u,..., u m, u 1 ) and Cm = u 1, u m,..., u, u 1 ). Now, let F = C n be another cycle with vertices v 1, v,..., v n so that AutF ) = AutC n ) = D n. Let φ C 1 C m, D n ) be a D n -voltage assignment on C m. Define the net voltage on C m + by φc m) + = φu 1 u ) φu m 1 u m )φu m u 1 ). The graph bundle C m φ C n is called a discrete torus if φc m) + = a k for some 0 k n 1 and a discrete Klein bottle if φc m) + = a k b for some 0 k n 1. As an application of our formula, we compute the zeta functions of a discrete torus and of a discrete Klein bottle in this section. Let H 1 x) = x and H x) = x 1 be polynomials in x and let H j x) be a sequence of polynomials satisfying the recurrence relation Set H j+ x) = xh j+1 x) H j x). 4) P j x) = H j x) H j x). Then a straightforward calculation gives the following lemma. Lemma 11. Let ω be any symmetric weight function on C m. Then det I m AC m ) ω ) + λi m )u + δi m )u ) ) 1 = u m P m u λ + δu ωc m) + + ωc m)) ) +, where ωc + m) = ωu 1 u ) ωu m 1 u m )ωu m u 1 ). Now, we are ready to compute the zeta functions of a discrete torus and of a discrete Klein bottle. To do this, one needs to compute three polynomials f Cm,Cn u), h Cm,Cn u) and g Cm,Cn,tu) defined in Theorem 9. For G = C m and F = C n, we have k F = and Q G + k F I m = 3I m. From Lemma 11, it is easy to show that ) ) 1 f Cm,Cn u) = u m P m u + 3u.

17 Zeta functions of graph bundles 185 For any even n, since λ Cn,n/ =, we have by Lemmas 10 and 11 ) ) 1 h Cm,C n u) = u m P m u + + 3u ω 1 φ)c m) +, where ω 1 φ)c + m) = { 1) k if φc + m) = a k, 1) k+1 if φc + m) = a k b. For the polynomial g Cm,C n,tu), first note that the eigenvalues of C n are λ Cn,t) = cos πt n 1 n for 1 t. When φc+ m) = a k for some 0 k n 1, C ψ φ m is a disjoint union of two copies of C m but when φc m) + = a k b for some 0 k n 1, C ψ φ m is the cycle C m. By a method similar as in [8] with Lemmas 10 and 11, one can get = g Cm,Cn,tu) { u m P m 1 u u m P m 1 u cos πt n ) πtk + 3u) cos if φc m) + = a k, cos πt n + 3u) ) if φc + m) = a k b. Summarizing our discussions, we have the following theorems. Theorem 1. Let a discrete torus C m φ C n have the net voltage φc + m) = a k for some 0 k n 1. Then the reciprocal of its zeta function is Z Cm φ C n u) 1 1 u ) mn u mn P m 1 u + 3u) ) n 1)/ P m 1 πt cos u n t=1 = 1 u ) mn u mn P m 1 u + 3u) ) P m 1 u + + 3u) 1)k) n )/ P m 1 πt cos u n t=1 where P m x) is defined in Eq 4). n + 3u) cos πtk n + 3u) cos πtk n ) if n is odd, ) if n is even, Theorem 13. Let a discrete Klein bottle C m φ C n have the net voltage φc + m) = a k b for some 0 k n 1. Then the reciprocal of its

18 186 Rongquan Feng and Jin Ho Kwak zeta function is Z Cm φ C n u) 1 1 u ) mn u mn P m 1 u + 3u) ) n 1)/ P m 1 ) πt cos u n + 3u) t=1 = 1 u ) mn u mn P m 1 u + 3u) ) P m 1 u + + 3u) 1)k+1) n )/ t=1 where P m x) is defined in Eq 4). P m 1 πt cos u n + 3u) ) if n is odd, if n is even, References [1] H. Bass, The Ihara-Selberg zeta function of a tree lattice, Internat. J. Math ), no. 6, [] N. Biggs, Algebraic Graph Theory, nd ed., Cambridge University Press, Cambridge, [3] J. L. Gross and T. W. Tucker, Generating all graph coverings by permutation voltage assignments, Discrete Math ), no. 3, [4] K. Hashimoto, Zeta functions of finite graphs and representations of p-adic groups, Adv. Stud. Pure Math ), [5] Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields, J. Math. Soc. Japan ), [6] J. H. Kwak and J. Lee, Isomorphism classes of graph bundles, Canad. J. Math ), no. 4, [7], Characteristic polynomials of some graph bundles II, Linear and Multilinear Algebra 3 199), no. 1, [8] J. H. Kwak and Y. S. Kwon, Characteristic polynomials of graph bundles having voltages in a dihedral group, Linear Algebra Appl ), [9] H. Mizuno and I. Sato, Zeta functions of graph coverings, J. Combin. Theory Ser. B ), no., [10], L-functions for images of graph coverings by some operations, Discrete Math ), no. 1-, [11] S. Northshield, A note on the zeta function of a graph, J. Combin. Theory Ser. B ), no., [1] J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 4, Springer-Verlag, New York, [13] H. M. Stark and A. A. Terras, Zeta functions of finite graphs and coverings, Adv. Math ), no. 1, [14] T. Sunada, L-functions in geometry and some applications, Lecture Notes in Mathematics, Vol. 101, 66 84, Springer-Verlag, New York, 1986.

19 Zeta functions of graph bundles 187 [15] A. Terras, Fourier Analysis on Finite Groups and Applications, London Mathematical Society Student Texts, 43, Cambridge University Press, Cambridge, Rongquan Feng LMAM, School of Mathematical Sciences Peking University Beijing , P. R. China Jin Ho Kwak Department of Mathematics Pohang University of Science and Technology Pohang , Korea