L-FUNCTIONS AND THE SELBERG TRACE ...

PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 24 – 27, 2002, Wilmington, NC, USA

pp. 638–646

L-FUNCTIONS AND THE SELBERG TRACE FORMULAS FOR SEMIREGULAR BIPARTITE GRAPHS

Hirobumi Mizuno Department of Electronics and Computer Science, Meisei University 2-590, Nagabuti, Ome, Tokyo 198-8655, JAPAN

Iwao Sato Oyama National College of Technology Oyama, Tochigi 323-0806, JAPAN Abstract. We give a decomposition formula for the L-function of a semiregular bipartite graph G. Furthermore, we present the Selberg trace formula for the above L-function of G.

1. Introduction. Graphs and digraphs treated here are finite. Let G be a connected graph and D the symmetric digraph corresponding to G. Set D(G) = {(u, v), (v, u) | uv ∈ E(G)}. We also refer D as a graph G. For e = (u, v) ∈ D(G), set u = o(e) and v = t(e). Furthermore, let e−1 = (v, u) be the inverse of e = (u, v). A path P of length n in D(or G) is a sequence P = (e1 , · · · , en ) of n arcs such that ei ∈ D(G), t(ei ) = o(ei+1 )(1 ≤ i ≤ n − 1). Set | P |= n, o(P ) = o(e1 ) and t(P ) = t(en ). Also, P is called (o(P ), t(P ))-path. We say that a path P = (e1 , · · · , en ) has a backtracking if e−1 i+1 = ei for some i(1 ≤ i ≤ n − 1). A (v, w)-path is called a v-cycle (or v-closed path) if v = w. The inverse cycle of a cycle C = (e, e1 , · · · , en−1 , e) is −1 −1 the cycle C −1 = (e−1 , e−1 ). n−1 , · · · , e1 , e We introduce an equivalence relation between cycles. Such two cycles C1 = (e1 , · · · , em ) and C2 = (f1 , · · · , fm ) are called equivalent if fj = ej+k for all j. The inverse cycle of C is not equivalent to C. Let [C] be the equivalence class which contains a cycle C. Let B r be the cycle obtained by going r times around a cycle B. Such a cycle is called a multiple of B. A cycle C is reduced if both C and C 2 have no backtracking. Furthermore, a cycle C is prime if it is not a multiple of a strictly smaller cycle. Note that each equivalence class of prime, reduced cycles of a graph G corresponds to a unique conjugacy class of the fundamental group π 1 (G, v) of G for a vertex v of G. Let G be a graph and A a finite group. Let D(G) be the arc set of the symmetric digraph corresponding to G. Then a mapping α : D(G) −→ A is called an ordinary voltage assignment if α(v, u) = α(u, v)−1 for each (u, v) ∈ D(G). Moreover, we define the net voltage α(P ) of each path P = (e1 , · · · , el ) of G by α(P ) = α(e1 ) · · · α(el )(see [6]). Furthermore, let ρ be an irreducible representation of A. The L-function of G associated to ρ and α is defined to be the function of 1991 Mathematics Subject Classification. Primary: 05C50, 05C25, 11F72, 11M41; Secondary: 15A15. Key words and phrases. zeta function, the Selberg trace formula, semiregular graph The second author is supported by Grant-in-Aid for Science Research (C).

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THE SELBERG TRACE FORMULAS FOR SEMIREGULAR GRAPHS

u ∈ C with u sufficiently small, given by Y Z(u, ρ, α) = ZG (u, ρ, α) = det(If − ρ(α(C))u|C| )−1 ,

639

(1)

[C]

where f = deg ρ and [C] runs over all equivalence classes of prime, reduced cycles of G(c.f., [7], [8], [9], [11], [16]). Let ρ = 1 be the identity representation of A. Then, note that the L-function of G associated to 1 and α is the (Ihara) zeta function Z(G, u) of a graph G (see [11]): Y ZG (u, 1, α) = Z(G, u) = (1 − u|C| )−1 . [C]

Ihara [11] defined zeta functions of graphs, and showed that the reciprocals of zeta functions of regular graphs are explicit polynomials. Hashimoto [7] treated multivariable zeta functions of bipartite graphs. Bass [2] generalized Ihara’s result on the zeta function of a regular graph to an irregular graph G. Different proof of Bass’ Theorem were given by Stark and Terras [14], Kotani and Sunada [12] and Foata and Zeilberger [4]. Theorem 1. [Ihara] Let G be a connected (q + 1)-regular graph with n vertices. Set Spec(G) = {λ1 , · · · , λn }. Then the reciprocal of the zeta function of G is n Y Z(G, u)−1 = (1−u2 )²−n det(In −uA(G)+qu2 In ) = (1−u2 )(1−q)n/2 (1−λj u+qu2 ) j=1

where ² =| E(G) |. Theorem 2. [Hashimoto] Let G = (V1 , V2 ) be a connected (q1 +1, q2 +1)-semiregular bipartite graph with ν vertices and ² edges, | V1 |= n and | V2 |= m(n ≤ m). Then n Y Z(G, u)−1 = (1 − u2 )²−ν (1 + q2 u2 )m−n (1 − (λ2j − q1 − q2 )u2 + q1 q2 u4 ) j=1

= (1 − u2 )²−ν (1 + q2 u2 )m−n det(In − (A[1] − (q2 − 1)In )u2 + q1 q2 u4 In ) = (1 − u2 )²−ν (1 + q1 u2 )n−m det(Im − (A[2] − (q1 − 1)Im )u2 + q1 q2 u4 Im ), where Spec(G) = {±λ1 , · · · , ±λn , 0, · · · , 0} and A[i] = A(G[i] )(i = 1, 2). Theorem 3. [Bass] Let G be a connected with n vertices and m edges. Then the reciprocal of the zeta function of G is Z(G, u)−1 = (1 − u2 )m−n det(In − uA(G) + u2 (D − In )). We state the background about L-functions of graphs. Let G be a connected graph, and let N (v) = {w ∈ V (G) | vw ∈ E(G)} for any vetex v in G. A graph H is called a covering of G with projection π : H −→ G if there is a surjection π : V (H) −→ V (G) such that π|N (v0 ) : N (v 0 ) −→ N (v) is a bijection for all vertices v ∈ V (G) and v 0 ∈ π −1 (v). When a finite group Π acts on a graph (digraph) G, the quotient graph (digraph) G/Π is a simple graph (digraph) whose vertices are the Π-orbits on V (G), with two vertices being adjacent in G/Π if and only if some two of their representatives are adjacent in G. A covering π : H −→ G is said to be

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a regular covering of G if there is a subgroup B of the automorphism group AutH of H such that the quotient graph H/B is isomorphic to G. Let G be a graph, A a finite group and α : D(G) −→ A an ordinary voltage assignment. The pair (G, α) is called an ordinary voltage graph. The derived graph Gα of the ordinary voltage graph (G, α) is defined as follows: V (Gα ) = V (G) × A and ((u, h), (v, k)) ∈ D(Gα ) if and only if (u, v) ∈ D(G) and k = hα(u, v), where V (G) is the vertex set of G(see [5]). Also, Gα is called an A-covering of G. The natural projection π : Gα −→ G is defined by π(v, h) = v for all (v, h) ∈ V (G) × A. Then the covering π : Gα −→ G is a regular covering of G. Hashimoto [8] showed that the zeta function of a regular covering of G is a product of L-functions of G(c.f., [13], [15]). Theorem 4. [Hashimoto] Z(Gα , u) =

Y

ZG (u, ρ, α)deg ρ ,

ρ

where ρ runs over all inequivalent irreducible representations of A. Let G be a connected graph with n vertices v1 , · · · , vn . Let D = (dij ) beNthe diagonal matrix with dii = deg G vi , and Q = D−I. The Kronecker product A B of matrices A and B is considered as the matrix A having the Nelement aij replaced by the matrix aij B. For a positive integer m, set Qm = Im Q, where Im is the identity matrix of order m. Mizuno and Sato [13] showed that the reciprocal of any L-function of G is an explicit polynomial. Theorem 5. [Mizuno and Sato] Let G be a connected graph with n vertices and l edges, A a finite group and α : D(G) −→ A an ordinary voltage assignment. Furthermore, let ρ be an irreducible representation of A, and f the degree of ρ. For (g) g ∈ A, the matrix Ag = (auv ) is defined as follows: ½ | {e | e = (u, v), α(e) = g} | if there exists an arc (u, v) ∈ D(G), a(g) := uv 0 otherwise. Then the reciprocal of the L-function of G associated to ρ and α is X O ZG (u, ρ, α)−1 = (1 − u2 )(l−n)f det(If n − u ρ(g) Ag + u2 Qf ). g∈A

Originally, Theorem 1 is proved for simple graphs, but holds for multigraphs.

2. The Selberg trace formulas for graphs. The Selberg trace formula for a connected graph G is closely related to the zeta function of G. Ahumada [1] gave the Selberg trace formula for a regular graph. For a semiregular bipartite graph G, Hashizume [10] presented the Selberg trace formula. Let G be a connected graph with n vertices v1 , · · · , vn , and n ∈ N. Then we denote the number of prime, reduced cycles with length n of G by π(n). The adjacency matrix A = A(G) = (aij ) is the square matrix such that aij =| {e ∈ D(G) | e = (vi , vj )} | if vi and vj are adjacent, and aij = 0 otherwise. Let Spec(G) be the set of all eigenvalues of A(G). Now, let h(n) be an even complex function on Z which is support finite(or ”rapidly decreased”). For this h(n), we define its Fourier transform

THE SELBERG TRACE FORMULAS FOR SEMIREGULAR GRAPHS

ˆ h(v) =

X

641

h(n)v n .

n∈Z

ˆ ˆ Note that h(v) = h(v ). The degree deg G v = deg v of a vertex v in G is defined by deg G v =| {w | vw ∈ E(G)} |. A graph H is called k-regular if deg H v = k for each vertex v ∈ V (H). −1

Theorem 6. [Ahumada] Let G be a connected (q + 1)-regular graph with n vertices, P and let Nm = k|m kπ(k) for each m ∈ N. Set Spec(G) = {λ1 , · · · , λn }. Furthermore, for each λi , let zi be a root of q 1/2 (t + t−1 ) = λi . Then the following trace formula holds: n ∞ X X ˆ h(zi ) = n(Sh)(0) + q −m/2 Nm h(m), i=1

where (Sh)(0) = h(0) − (q − 1)

m=1

P j≥1

q

−j

h(2j).

Venkov and Nikitin [18] showed that Theorem 6 is obtained from Theorem 1 by doing the logarithmic differentiation and a suitable complex integration. A graph G is called bipartite, denoted by G = (V1 , V2 ) if there exists a partition V (G) = V1 ∪ V2 of V (G) such that uv ∈ E(G) if and only if u ∈ V1 and v ∈ V2 . A bipartite graph G = (V1 , V2 ) is called (q1 +1, q2 +1)-semiregular if deg G v = qi +1 for each v ∈ Vi (i = 1, 2). For a (q1 + 1, q2 + 1)-semiregular bipartite graph G = (V1 , V2 ), let G[i] be the graph with vertex set Vi and edge set {P : reduced path | | P |= 2; o(P ), t(P ) ∈ Vi } for i = 1, 2. Theorem 7. [Hashizume] Let GP = (V1 , V2 ) be a connected (q1 +1, q2 +1)-semiregular [1] 0 bipartite graph, and let Nm = k|m kπ(2k) for each m ∈ N. Set Spec(G ) = {λ1 , · · · , λn }. Furthermore, for each λi , let zi be a root of (q1 q2 )1/2 (t + t−1 ) = λi − q2 + 1. Then the following trace formula holds: n ∞ X X 0 ˆ i ) = n(Sh)(0) + h(z (q1 q2 )−m/2 Nm h(m), i=1

where (Sh)(0) = h(0) − 1/(q2 + 1)

P

m=1

−j/2 (q1 q2 j≥1 (q1 q2 )

− 1 + (−q2 )j (q1 − q2 ))h(j).

In this paper, we give a decomposition formula for the L-function of a semiregular bipartite graph G. Futhermore, we present the Selberg trace formula for the above L-function of G. 3. L-functions of semiregular bipartite graphs. We present a decomposition formula for the L-function of a semiregular bipartite graph G. Let G = (V1 , V2 ) be a connected (q1 + 1, q2 + 1)-semiregular bipartite graph, A a finite group and α : D(G) −→ A an ordinary voltage assignment. Furthermore, let ρ be a representation of A and d = deg ρ. Set | V1 |= n and | V2 |= m(n ≤ m). P N [1] [1] Let Aρ = g∈A Ag ρ(g). Then the nd×nd matrix Aρ = ((Aρ )uv )u,v∈V (G[1] ) is defined as follows: X (A[1] {ρ(α(P )) | P : reduced (u, v) − path with length 2}. ρ )uv = [1]

If there does not exist such P , then let (Aρ )uv = 0d . Similarly, the md×md matrix [2] [2] Aρ = ((Aρ )uv )u,v∈V (G[2] ) is defined. Let Spec(B) be the set of all eigenvalues of a square matrix B.

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Theorem 8. Let G = (V1 , V2 ) be a connected (q1 + 1, q2 + 1)-semiregular bipartite graph with ν vertices and ² edges, A a finite group and α : D(G) −→ A an ordinary voltage assignment. Furthermore, let ρ be a unitary representation of A and d = deg ρ. Set | V1 |= n and | V2 |= m(n ≤ m). Then ZG (u, ρ, α)−1 = (1 − u2 )(²−ν)d (1 + q2 u2 )(m−n)d

nd Y

(1 − (λ2j − q1 − q2 )u2 + q1 q2 u4 )

j=1 2 (²−ν)d

= (1 − u )

2 (m−n)d

(1 + q2 u )

2 4 det(Ind − (A[1] ρ − (q2 − 1)Ind )u + q1 q2 u Ind )

2 4 = (1 − u2 )(²−ν)d (1 + q1 u2 )(n−m)d det(Imd − (A[2] ρ − (q1 − 1)Imd )u + q1 q2 u Imd ), where Spec(Aρ ) = {±λ1 , · · · , ±λnd , 0, · · · , 0}.

Proof. . The argument is similar to the proof of Theorem 2(see [7]). By Theorem 1, we have

X

ZG (u, ρ, α)−1 = (1 − u2 )(²−ν)d det(Iνd − u

ρ(g)

g∈A 2 (²−ν)d

= (1 − u )

X

det(Iνd − u

Ag

O

O

Ag + u2 Qd )

ρ(g) + u2 Q

O

Id ).

g∈A

P N Let Aρ = g∈A Ag ρ(g). For the brevity, let G be simple (for a multi-graph, the similar result holds). When A(G) = (ai,j ), we have Aρ = (ai,j ρ(α(i, j))). Since ρ is unitary, we can let ¸ · 0 Bρ . Aρ = t ¯ Bρ 0 For (u, v) ∈ D(G), we have ρ(α(v, u)) = ρ(α(u, v))−1 = t ρ(α(u, v)). Thus, Aρ is Hermitian. There exists a unitary matrix W ∈ U (md) such that   µ1 0 0 ··· 0 £ ¤  .. ..  . .. Bρ W = Cρ 0 =  . . .  ? µnd 0 · · · 0 Now, let

· P=

Ind 0

0 W

¸ .

Then we have 

 0 Cρ 0 t¯ ¯ρ 0 0 . PAρ P =  t C 0 0 0 N N t¯ Furthermore, we have P(Q Id )P = Q Id . Thus, · (1 + q1 u2 )Ind −1 2 (²−ν)d 2 (m−n)d ZG (u, ρ, α) = (1−u ) (1+q2 u ) det ¯ρ −ut C = (1 − u2 )(²−ν)d (1 + q2 u2 )(m−n)d

−uCρ (1 + q2 u2 )Ind

¸

THE SELBERG TRACE FORMULAS FOR SEMIREGULAR GRAPHS

· det

(1 + q1 u2 )Ind ¯ρ −ut C

0 ¯ ρ Cρ (1 + q2 u2 )Ind − (1 + q1 u2 )−1 u2t C

643

¸

¯ ρ Cρ ). = (1 − u2 )(²−ν)d (1 + q2 u2 )(m−n)d det((1 + q1 u2 )(1 + q2 u2 )Ind − u2t C ¯ ρ Cρ is Hermitian and semipositive definite, i.e., the eigenSince Aρ is Hermitian, t C ¯ ρ Cρ are of form: λ2 , · · · , λ2 (λ1 , · · · , λnd ≥ 0). Therefore the first values of t C 1 nd identity follows. The proof of the second and the third equations is omitted. In the case that ρ = 1(the trivial representation of A), we obtain Theorem 2, i.e., Hashimoto’s Theorem [7] for the zeta function of a semiregular bipartite graph. 4. The Selberg trace formulas for semiregular bipartite graphs. Let G = (V1 , V2 ) be a connected (q1 + 1, q2 + 1)-semiregular bipartite graph, A a finite group and α : D(G) −→ A an ordinary voltage assignment. Furthermore, let ρ be a representation of A and f = deg ρ. Set | V1 |= n and | V2 |= m(n ≤ m). Let √ Z(v, ρ, α) = ZG ( v, ρ, α). (2) By Theorem 8, we have Z(v, ρ, α)−1 = (1 − v)(²−ν)d (1 + q2 v)(m−n)d

nd Y

(1 − (λ2j − q1 − q2 )v + q1 q2 v 2 ). (3)

j=1

Now, for m ≥ 1,let Nm =

X

tr(ρ(α(C))),

C

where C runs over all reduced cycles with length m. Theorem 9. Let G = (V1 , V2 ) be a connected (q1 + 1, q2 + 1)-semiregular bipartite graph with ν vertices and ² edges, A a finite group and α : D(G) −→ A an ordinary voltage assignment. Furthermore, let ρ be a unitary representation of A and d = deg ρ. Set | V1 |= n and | V2 |= m(n ≤ m). Furthermore, let h(n) be an even [1] complex function on Z. For µj ∈ Spec(Aρ )(1 ≤ j ≤ nd), let zj be a root of 1/2 −1 (q1 q2 ) (t + t ) = µj − q2 + 1. Then the following trace formula holds: nd X

∞ X N2m ˆ j ) = nd(Sh)(0) + 1 h(z h(m), 2 m=1 (q1 q2 )m/2 j=1

where (Sh)(0) = h(0) − 1/(q2 + 1)

P

l≥1 (q1 q2 )

−l/2

(q1 q2 − 1 + (−q2 )l (q1 − q2 ))h(l).

Proof. . The argument is similar to the the method of Venkov and Nikitin [18]. Let [1] Spec(Aρ ) = {µ1 , · · · , µnd }. By the latter half of the proof of Theorem 8, we have 2 µj = λj − q1 − 1(1 ≤ j ≤ nd), where Spec(Aρ ) = {±λ1 , · · · , ±λnd , 0, · · · , 0}. Thus, λ2j − (q1 + q2 ) = µj − q2 + 1. By (3), we have v , ρ, α)−1 Z( √ q1 q2

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= (1 − √

nd Y v v v )(²−ν)d (1 + q2 √ )(m−n)d (1 − (µj − q2 + 1) √ + v2 ) q1 q2 q1 q2 q q 1 2 j=1

√ √ √ Then we have 1−(µj −q2 +1)v/ q1 q2 +v 2 = v/ q1 q2 ( q1 q2 (v+v −1 )−(µj −q2 +1)). Doing the logarithmic differentiation to the above identity, we obtain nd X d (m − n)d q2 (² − ν)d √ − log( q1 q2 (v + v −1 ) − (µj − q2 + 1)) = − √ +√ dv q q − v q1 q2 + q2 v 1 2 j=1

+

d nd √ + log Z(v/ q1 q2 , ρ, α). v dv

Since G is bipartite, we have n(q1 + 1) = m(q2 + 1), i.e., mq2 − nq1 = n − m. Thus, (² − ν)d (m − n)d q2 nd ndq1 q2 (1 − v 2 ) +√ + = √ . −√ √ q1 q2 ) − v q1 q2 + q2 v v ( q1 q2 − v)( q1 q2 + q2 v)v Next, since Tr(log(I − B)) = log det(I − B), (1) and (2) imply that log Z(v, ρ, α) = log ZG (v 1/2 , ρ, α) X =− log det(Id − ρ(α(C))v |C|/2 ) [C]

=

∞ XX 1 tr(ρ(α(C))m )u|C|m/2 . m m=1 [C]

Thus, ∞ XX 1 d log Z(v, ρ, α) = v −1 | C | tr(ρ(α(C))m )v |C|m/2 dv 2 m=1 [C]

=

∞ 1 −1 X X v | C | tr(ρ(α(C))m )v |C|m/2 2 m=1 [C]

=

∞ X

X 1 −1 v tr(ρ(α(C))m )v |C|m/2 , 2 m=1 C

where C runs over all prime, reduced cycles of G. Note that there exist | C | elements in [C], and tr(BAB−1 ) = tr(A). Since G has no cycles with lenght odd, we have N2m+1 = 0. That is, X 1 d Z(v, ρ, α) = v −1 N2m v m . dv 2 m≥1

Thus,

X 1 Nm d √ Z(v/ q1 q2 , ρ, α) = v −1 vm . m/2 dv 2 (q q ) 1 2 m≥1

Therefore, −

nd X ndq1 q2 (1 − v 2 ) d √ log( q1 q2 (v + v −1 ) − (µj − q2 + 1)) = √ √ dv ( q1 q2 − v)( q1 q2 + q2 v)v j=1

(4)

THE SELBERG TRACE FORMULAS FOR SEMIREGULAR GRAPHS

+

1 X Nm v m−1 . m/2 2 (q q ) 1 2 m≥1

645

(5)

[1]

Now, for µj ∈ Spec(Aρ )(1 ≤ j ≤ nd), let zj be a root of (q1 q2 )1/2 (t + t−1 ) = µj − q2 + 1. Moreover, let h(n) be an even complex function on Z. Futhermore, let C be a simple closed curve traced which p in the positive direction(counterclockwise) √ −1 , 0. We contains z1 , · · · , znd , q1 q2 , − q1 /q2 , and does not contain z1−1 , · · · , znd consider the following three contour integrals: I 1 d √ ˆ Q(h, j) = − h(v) log( q1 q2 (v + v −1 ) − (µj − q2 + 1))dv, 2πi −C dv I I 1 q1 q2 (1 − v 2 ) 1 m−1 ˆ ˆ I(h) = h(v) dv, H(h, m) = h(v)v dv. √ √ 2πi −C ( q1 q2 − v)( q1 q2 + q2 v)v 2πi −C Then, by (5), we have nd X

Q(h, j) = ndI(h) +

j=1

1 X Nm H(h, m). m/2 2 (q q 1 2) m≥1

By the property of the residue, we have

Q(h, j) =

1 2πi

I

d √ ˆ ˆ j ). h(v) log q1 q2 /v(v − zj )(v − 1/zj )dv = h(z dv C

Moreover, we have H(h, m) =

1 2πi

I

X

h(n)v n+m−1 dv = h(−m) = h(m).

−C n∈Z

Let C1 be the circle of radius 1 traced in the positive direction. Then we have I 1 (1 − v 2 ) ˆ p I(h) = dv. h(v) √ 2πi C1 (1 − v/ q1 q2 )(1 + v/ q1 /q2 )v p √ Since | v/ q1 q2 |< 1 and | v/ q1 /q2 |< 1 for v ∈ C1 , we have X X p √ 1 − v/ q1 q2 = (q1 q2 )−j/2 v j and 1 − v/ q1 /q2 = (−1)j (q1 /q2 )−j/2 v j . j≤0

Thus,

=

X

j≤0

ˆ h(v)(1 − v2 ) p √ (1 − v/ q1 q2 )(1 + v/ q1 /q2 ) h(n)v n + 1/(q2 + 1)

n∈Z

=

X

XX

(q1 q2 )−l/2 {1 − q1 q2 + (−q2 )l (q2 − q1 )}h(n)v l+n

n∈Z l≥1

{h(n) + 1/(q2 + 1)

n∈Z

X

(q1 q2 )−l/2 (1 − q1 q2 + (−q2 )l (q2 − q1 ))h(n − l)}v n .

l≥1

By the property of the residue and the fact that h(−n) = h(n), we have X I(h) = h(0) + 1/(q2 + 1) (q1 q2 )−l/2 (1 − q1 q2 + (−q2 )l (q2 − q1 ))h(l). l≥1

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H. MIZUNO AND I. SATO

Therefore, the result follows. In the case that ρ = 1(the trivial representation of A), Theorem 7, i.e., Hashizume’s Theorem is obtained as a corollary. REFERENCES [1] G. Ahumada, Fonctions periodiques et formule des traces de Selberg sur les arbres, C. R. Acad. Sci. Parris Ser. I 305, no. 16 (1987), 709–712. [2] H. Bass, The Ihara-Selberg zeta function of a tree lattice, Internat. J. Math. 3 (1992), 717–797. [3] M. Burrow, ”Representation Theory of Finite Groups,” Academic Press, New York, 1965. [4] D. Foata and D. Zeilberger, A combinatorial proof of Bass’s evaluations of the Ihara-Selberg zeta function for graphs, Trans. Amer. Math. Soc. 351 (1999), 2257–2274. [5] J. L. Gross and T. W. Tucker, Generating all graph coverings by permutation voltage assignments, Discrete Math. 18 (1977), 273–283. [6] J. L. Gross and T. W. Tucker, ”Topological Graph Theory,” Wiley-Interscience, New York, 1987. [7] K. Hashimoto, ”Zeta Functions of Finite Graphs and Representations of p-Adic Groups,” Adv. Stud. Pure Math. Vol. 15, pp. 211–280, Academic Press, New York, 1989. [8] K. Hashimoto, On zeta and L-functions of finite graphs, Internat. J. Math. 1 (1990), 381-396. [9] K. Hashimoto, Artin-type L-functions and the density theorem for prime cycles on finite graphs, Internat. J. Math. 3 (1992), 809–826. [10] M. Hashizume, Selberg trace formula for semiregular graphs, preprint. [11] 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. [12] M. Kotani and T. Sunada, Zeta functions of finite graphs, J. Math. Sci. U. Tokyo 7 (2000), 7–25. [13] H. Mizuno and I. Sato, Zeta functions of graph coverings, J. Combin. Theory Ser. B 80 (2000), 247–257. [14] H. M. Stark and A. A. Terras, Zeta functions of finite graphs and coverings, Adv. Math. 121 (1996), 124–165. [15] H. M. Stark and A. A. Terras, Zeta functions of finite graphs and coverings, part II, Adv. Math. 154 (2000), 132–195. [16] T. Sunada, ”L-Functions in Geometry and Some Applications,” in Lecture Notes in Math., Vol. 1201, pp. 266–284, Springer-Verlag, New York, 1986. [17] A. Terras, ”Fourier Analysis on Finite Groups and Applications,” Cambridge Univ. Press, Cambridge 1999. [18] A. B. Venkov and A. M. Nikitin, The Selberg trace formula, Ramanujan graphs, and some problems of Mathematical Physics, St. Petersburg Math. J. 5 no. 3 (1994), 419–484.

Received July 2002; in revised April 2003. E-mail address: [email protected]