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Weighted zeta functions of graphs
ARTICLE IN PRESS
Journal of Combinatorial Theory, Series B 91 (2004) 169–183
http://www.elsevier.com/locate/jctb
Weighted zeta functions of graphs Hirobumi Mizunoa and Iwao Satob,1 a
Department of Electronics and Computer Science, Meisei University, 2-590, Nagabuti, Ome, Tokyo 198-8655, Japan b Oyama National College of Technology, Oyama, Tochigi 323-0806, Japan Received 15 April 2002
Abstract We give a decomposition formula for the weighted zeta function of a regular covering of a graph. r 2003 Elsevier Inc. All rights reserved. Keywords: Zeta function; Graph covering; L-function
1. Introduction Graphs and digraphs treated here are finite and simple. Let G ¼ ðV ðGÞ; EðGÞÞ be a connected graph with vertex V ðGÞ and edge set EðGÞ; and D the symmetric digraph corresponding to G: Set DðGÞ ¼ fðu; vÞ; ðv; uÞ j uvAEðGÞg: For e ¼ ðu; vÞADðGÞ; let oðeÞ ¼ u and tðeÞ ¼ v: The inverse arc of e is denoted by e%: A path P of length n in G is a sequence P ¼ ðv0 ; v1 ; y; vn1 ; vn Þ of n þ 1 vertices and n arcs such that consecutive vertices share an arc (we do not require that all vertices are distinct). Also, P is called a ðv0 ; vn Þ-path. If ei ¼ ðvi1 ; vi Þ for i ¼ 1; y; n; then we can write P ¼ ðe1 ; y; en Þ: We say that a path has a backtracking if a subsequence of the form y; x; y; x; y appears. A ðv; wÞ-path is called a cycle (or closed path) if v ¼ w: The inverse cycle of a cycle C ¼ ðv; v1 ; y; vn ; vÞ is the cycle C 1 ¼ ðv; vn ; y; v1 ; vÞ: A subpath ðv1 ; y; vm1 ; vm ; vm1 ; y; v1 Þ of a cycle C ¼ ðv1 ; y; v1 Þ is called a tail if degC v1 X3; degC vi ¼ 2ð2pipm 1Þ and degC vm ¼ 1; where degC v is the degree of v in the graph C: Let Br be the cycle obtained by going r times around a cycle B: Such a cycle is called a multiple of B: Note that any backtracking-less, tail-less cycle C is a
1
E-mail address: [email protected] (I. Sato). Supported by Grant-in-Aid for Science Research (C).
0095-8956/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jctb.2003.12.003
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cycle such that both C and C 2 have no backtracking (see [6,12]). A cycle C is reduced if C has no backtracking nor tail. The notion of tail-less cycle is used in the proof of Theorems 3 and 5. We introduce an equivalence relation between cycles. Such two cycles C1 ¼ ðv1 ; y; vm Þ and C2 ¼ ðw1 ; y; wm Þ are called equivalent if wj ¼ vjþk for all j: Let ½C be the equivalence class which contains a cycle C: A cycle C is prime if C is not obtained by going r times around some other cycle B; i.e., CaBr : Note that each equivalence class of prime, reduced cycles of a graph G corresponds to a unique conjugacy class of the fundamental group p1 ðG; vÞ of G at a vertex v of G: The (Ihara) zeta function of a graph G is defined to be a function of uAC with juj sufficiently small, by Y ZðG; uÞ ¼ ZG ðuÞ ¼ ð1 ujCj Þ1 ; ½C
where ½C runs over all equivalence classes of prime, reduced cycles of G; and jCj is the length of C: This definition is given in [13], and is equivalent to the one in [12]. Zeta functions of graphs started from zeta functions of regular graphs by Ihara [8]. In [8], he showed that their reciprocals are 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 [15,16]. Hashimoto [6] treated multivariable zeta functions of bipartite graphs. Bass [1] generalized Ihara’s result on the zeta function of a regular graph to an irregular graph, and showed that its reciprocal is a polynomial: ZðG; uÞ1 ¼ ð1 u2 Þr1 detðI uAðGÞ þ u2 ðD IÞÞ; where r and AðGÞ are the Betti number and the adjacency matrix of G; respectively, and D ¼ ðdij Þ is the diagonal matrix with dii ¼ deg vi ðV ðGÞ ¼ fvi ; y; vn gÞ: Stark and Terras [13] gave an elementary proof of Bass’s theorem, and discussed three different zeta functions of any graph. Furthermore, various proofs of Bass’s theorem were given by Foata and Zeilberger [3] and Kotani and Sunada [9]. Let G be a graph and G a finite group. Then a mapping a : DðGÞ-G is called an ordinary voltage assignment if aðv; uÞ ¼ aðu; vÞ1 for each ðu; vÞADðGÞ: Furthermore, let r be a representation of G: The L-function of G associated to r and a is defined to be the function of uAC with juj sufficiently small, given by Y Zðu; r; aÞ ¼ ZG ðu; r; aÞ ¼ detðIf rðaðCÞÞujCj Þ1 ; ½C
where f ¼ deg r and ½C runs over all equivalence classes of prime, reduced cycles of G (cf., [7,8,15]). Moreover, the net voltage aðCÞ of C ¼ ðe1 ; y; en Þ is defined by aðCÞ ¼ aðe1 Þyaðen Þ (see [5]). Let r ¼ 1 be the identity representation of G: Then, note that the L-function of G associated to 1 and a is the (Ihara) zeta function ZðG; uÞ of a graph G: Mizuno and Sato [11] showed that the zeta function of a regular covering of G is a product of L-functions of G:
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Let G be a connected graph, and let NðvÞ ¼ fwAV ðGÞ j ðv; wÞADðGÞg for any vertex v in G: A graph H is called a covering of G with projection p : H-G if there is a surjection p : V ðHÞ-V ðGÞ such that pjNðv0 Þ : Nðv0 Þ-NðvÞ is a bijection for all vertices vAV ðGÞ and v0 Ap1 ðvÞ: When a finite group P acts on a graph G; the quotient graph G=P is a simple graph whose vertices are the P-orbits on V ðGÞ; with two vertices adjacent in G=P if and only if some two of their representatives are adjacent in G: A covering p : H-G is said to be regular if there is a subgroup B of the automorphism group Aut H of H acting freely on H such that the quotient graph H=B is isomorphic to G: Let G be a graph and G a finite group. Furthermore, let a : DðGÞ-G be an ordinary voltage assignment. The pair ðG; aÞ is called an ordinary voltage graph. The derived graph Ga of the ordinary voltage graph ðG; aÞ is defined as follows: V ðG a Þ ¼ V ðGÞ G
and
ððu; hÞ; ðv; kÞÞADðG a Þ
if and only if ðu; vÞADðGÞ
and k ¼ haðu; vÞ: The natural projection p : Ga -G is defined by pðu; hÞ ¼ u; ðu; hÞAV ðGa Þ: The graph G a is called a derived graph covering of G with voltages in G or a G-covering of G: The natural projection p commutes with the right multiplication action of the aðeÞ; eADðGÞ and the left action of gAG on the fibers: g3ðu; hÞ ¼ ðu; ghÞ; gAG; which is free and transitive. Thus, the G-covering G a is a jGj-fold regular covering of G with covering transformation group G: Furthermore, every regular covering of a graph G is a G-covering of G for some group G (see [4]). Theorem 1 (Mizuno and Sato). Let G be a connected graph, G a finite group and a : DðGÞ-G an ordinary voltage assignment. Suppose that G a is connected. Then we have Y Zðu; r; aÞf ; ZðGa ; uÞ ¼ r
where r runs over all irreducible representations of G; and f ¼ deg r: Stark and Terras [14] introduced three Artin L-function of graphs based on vertex, edge and path. In particular, they obtained a factorization formula for the multiedge zeta function of a regular covering of a graph G as a product of multiedge Artin Lfunctions. Let G be a connected graph with 2m arcs e1 ; y; e2m ; and let W be a 2m 2m matrix with ij entry the variable wij if tðei Þ ¼ oðej Þ; ej ae%i ; and wij ¼ 0 otherwise. For a path C ¼ ðei1 ; y; eil Þ of G; let wðCÞ ¼ wi1 i2 wi2 i3 ?wil1 il : The multiedge zeta function of G is defined by Y ð1 wðCÞÞ1 ; zE ðW; GÞ ¼ ½C
where ½C runs over all equivalence classes of prime, reduced cycles of G:
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Let H be a regular covering of G with covering transformation group G and r a representation of G: Then the multiedge Artin L-function is defined by Y LE ðW; r; H=GÞ ¼ detðI rðsðCÞÞwðCÞÞ1 ; ½C
where sðCÞ is the Frobenius automorphism of C (see [14]). Theorem 2 (Stark and Terras). Y * HÞ ¼ zE ðW; LE ðW; r; H=GÞdeg r ; r
* is the matrix derived from W; and r runs over all inequivalent irreducible where W representations of G: In Section 2, we define a weighted zeta function of a graph G; and give a determinant expression of it. In Section 3, we give a decomposition formula of the weighted zeta function of a regular covering of G: In Section 4, we define a weighted L-function of G; and present a determinant expression for the weighted L-function of G: Furthermore, we show that the weighted zeta function of a regular covering of G is a product of weighted L-functions of G: A general theory of the representation of groups and graph coverings, the reader is referred to [2] and [5], respectively.
2. Weighted zeta functions of graphs Let G be a connected graph and V ðGÞ ¼ fv1 ; y; vn g: Then we consider an n n matrix W ¼ ðwij Þ1pi; jpn with ij entry the complex variable wij if ðvi ; vj ÞADðGÞ; and wij ¼ 0 otherwise. Furthermore, assume that w1 ji ¼ wij if ðvi ; vj ÞADðGÞ: The matrix W ¼ WðGÞ is called the weighted matrix of G: For each path P ¼ ðvi1 ; y; vir Þ of G; the norm wðPÞ of P is defined as follows: wðPÞ ¼ wi1 i2 wi2 i3 ?wir1 ir : Furthermore, let wðvi ; vj Þ ¼ wij ; vi ; vj AV ðGÞ: The (vertex) weighted zeta function of G is defined by Y ZG ðw; uÞ ¼ ð1 wðCÞujCj Þ1 ; ½C
where ½C runs over all equivalence classes of prime, reduced cycles of G: Theorem 3. Let G be a connected graph, and let W ¼ WðGÞ be the weighted matrix of G: Then the reciprocal of the weighted zeta function of G is given by ZG ðw; uÞ1 ¼ ð1 u2 Þmn detðI uWðGÞ þ u2 QÞ; where n ¼ jV ðGÞj;m ¼ jEðGÞj and Q ¼ D I: Proof. The proof is an analogue of Stark and Terras’s method [13].
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Let G be a connected graph with n vertices and m edges. At first, X log ZG ðw; uÞ ¼ logð1 wðCÞujCj Þ ½C
N X X 1 wðCÞs ujCjs : ¼ s ½C s¼1
Thus, we have N X X d log ZG ðw; uÞ ¼ u1 jCjwðCÞs ujCjs du ½C s¼1
¼ u1
N X X s¼1
¼ u1
X
wðCÞs ujCjs
C
wðCÞujCj ;
C
P
where the last C runs over all reduced cycles C of G: Note that there exist jCj elements in ½C : Let Cs be the set of all reduced cycles in G with length s: Set X Ns ¼ wðCÞ: CACs
Then, X d log ZG ðw; uÞ ¼ u1 Ns u s : du sX1
ð1Þ
Let V ðGÞ ¼ fv1 ; y; vn g: For sX1; the n n matrix As ¼ ððAs Þi; j Þ1pi; jpn is defined as follows: X ðAs Þi; j ¼ wðPÞ; P
where ðAs Þi; j is the ði; jÞ-component of As ; and P runs over all backtracking-less paths of length s from vi to vj in G: Note that A1 ¼ WðGÞ: Furthermore, let A0 ¼ In : Similarly to the proof of Lemma 1 in [13], we have A2 ¼ ðA1 Þ2 ðQ þ In Þ and As ¼ As1 A1 As2 Q
for sX3:
Here, note that wðu; vÞwðv; uÞ ¼ 1: Thus, ! X s As u ðI uA1 þ u2 QÞ ¼ ð1 u2 ÞI: sX0
ð2Þ
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Since ð1 u2 Þ1 ¼ X
I¼
P
Ak u
jX0
! k
u2j ; u
2j
ðI uA1 þ u2 QÞ
jX0
kX0
¼
!
X
X
½s=2
X
sX0
j¼0
!
As2j u
s
ðI uA1 þ u2 QÞ:
ð3Þ
For sX1; let C0s be the set of all backtracking-less cycles of length s with tails in G; and X ts ¼ wðCÞ: CAC0s
Then t1 ¼ t2 ¼ 0: Similarly to the proof of Lemma 2 in [13], we have ts ¼ Tr½ðQ IÞAs2 þ ts2
for sX2:
Since Ns ¼ TrðAs Þ ts ; Ns ¼ Tr½As ðQ IÞ
½ðs1Þ=2
X
As2j ;
sX3
j¼1
and N2 ¼ N1 ¼ Tr A2 ¼ Tr A1 ¼ 0: Next, set N s ¼ As ðQ IÞ
½s=2
X
As2j
j¼1
¼ QAs ðQ IÞ
½s=2
X
As2j :
j¼0
Then (2) and (3) imply that ! X
s Ns u ðI uA1 þ u2 QÞ ¼ ð1 u2 ÞQ ðQ IÞI sX0
¼ I Qu2 : Since N 0 ¼ A0 ¼ In ; ! X
s Ns u ðI uA1 þ u2 QÞ ¼ I u2 Q IðI uA1 þ u2 QÞ sX1
¼ A1 u 2Qu2 : Therefore it follows that X N s us ¼ ðA1 u 2Qu2 ÞðI uA1 þ u2 QÞ1 : sX1
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175
By [13, Lemma 3], we have X d N s us ¼ u logðI uA1 þ u2 QÞ: du sX1 For sX1; we have TrðN s Þ ¼ Ns
0 if s is odd; TrðQ IÞ if s is even:
Thus, Tr
X
! N s us
¼
sX1
X
s
Ns u TrðQ IÞ
sX1
¼
X
X
! u
2j
jX1
Ns us TrðQ IÞ
sX1
u2 : 1 u2
Eq. (1) implies that d d u2 2 u log ZG ðw; uÞ ¼ Tr u logðI uA1 þ u QÞ þ TrðQ IÞ du du 1 u2 d d ¼ Tr u logðI uA1 þ u2 QÞ u logð1 u2 ÞTrðQIÞ=2 : du du Both functions are 0 at u ¼ 0; and so log ZG ðw; uÞ ¼ TrðlogðI uA1 þ u2 QÞÞ þ logð1 u2 ÞTrðQIÞ=2 : Hence the fact that TrðlogðI BÞÞ ¼ log detðI BÞ implies that ZG ðw; uÞ ¼ ð1 u2 ÞðmnÞ detðI uA1 þ u2 QÞ1 ¼ ð1 u2 ÞðmnÞ detðI uWðGÞ þ u2 QÞ1 :
&
3. Weighted zeta functions of regular coverings of graphs Let G be a connected graph, G a finite group and a : DðGÞ-G an ordinary voltage assignment. In the G-covering Ga ; set vg ¼ ðv; gÞ and eg ¼ ðe; gÞ; where vAV ðGÞ; eADðGÞ; gAG: For e ¼ ðu; vÞADðGÞ; the arc eg emanates from ug and terminates at vgaðeÞ : Note that e%g ¼ ðe%ÞgaðeÞ : Let W ¼ WðGÞ be the weighted matrix of G: Then we define the weighted matrix * ˜ g ; vh ÞÞ of Ga derived from W as follows: W ¼ WðGa Þ ¼ ðwðu wðu; vÞ if ðu; vÞADðGÞ and h ¼ gaðu; vÞ; wðu ˜ g ; vh Þ :¼ 0 otherwise:
ARTICLE IN PRESS H. Mizuno, I. Sato / Journal of Combinatorial Theory, Series B 91 (2004) 169–183
176
ðgÞ
For gAG; let the matrix Wg ¼ ðwuv Þ be defined by wðu; vÞ if aðu; vÞ ¼ g and ðu; vÞADðGÞ; :¼ wðgÞ uv 0 otherwise: Let M1 "?"Ms be the block diagonal sum of square matrices M1 ; y; Ms : If M1 ¼ M2 ¼ ? ¼ Ms ¼ M; then we write s3M ¼ M1 "?"Ms : The Kronecker product A#B of matrices A and B is considered as the matrix A having the element aij replaced by the matrix aij B: Theorem 4. Let G be a connected graph with n vertices and l unoriented edges, G a finite group and a : DðGÞ-G an ordinary voltage assignment. Furthermore, let r1 ¼ 1; r2 ; y; rt be all inequivalent irreducible representations of G; and fi the degree of ri for each i; where f1 ¼ 1: Then the reciprocal of the weighted zeta function of the connected G-covering G a of G is ZGa ðw; ˜ uÞ1 ¼ ZG ðw; uÞ1
t Y i¼2
(
2 ðlnÞfi
ð1 u Þ
det Infi u
X
! ) fi 2
ri ðgÞ#Wg þ u ð fi 3QÞ
:
gAG
Proof. The argument is an analogue of Mizuno and Sato’s method [10]. Set V ðGÞ ¼ fv1 ; y; vn g and G ¼ f1 ¼ g1 ; g2 ; y; gm g: Arrange vertices of G a in m blocks: ðv1 ; 1Þ; y; ðvn ; 1Þ; ðv1 ; g2 Þ; y; ðvn ; g2 Þ; y; ðv1 ; gm Þ; y; ðvn ; gm Þ: We consider the weighted matrix WðG a Þ under this order. For gAG; the matrix ðgÞ Pg ¼ ðpij Þ is defined as follows: 1 if gi g ¼ gj ; ðgÞ pij ¼ 0 otherwise: ðgÞ
Suppose that pij ¼ 1; i.e., gj ¼ gi g: Then ððu; gi Þ; ðv; gj ÞÞADðG a Þ if and only if ðu; vÞADðGÞ and gi aðu; vÞ ¼ gj ; i.e., aðu; vÞ ¼ g1 i gj ¼ g: Thus we have X Pg #Wg : WðG a Þ ¼ gAG
Let r be the right regular representation of G: Then we have rðgÞ ¼ Pg for gAG: Furthermore, there exists a nonsingular matrix P such that P1 rðgÞP ¼ ð1Þ"f2 3r2 ðgÞ"?"ft 3rt ðgÞ for each gAG: Putting B ¼ ðP1 #In ÞWðGa ÞðP#In Þ;
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177
we have B¼
X
fð1Þ"f2 3r2 ðgÞ"?"ft 3rt ðgÞg#Wg :
gAG
Note that WðGÞ ¼ that
P
gAG
Wg and 1 þ f22 þ ? þ ft2 ¼ m: By Theorem 3, it follows
ZGa ðw; ˜ uÞ1 ¼ ð1 u2 ÞðlnÞm detðInm WðG a Þu þ u2 ðm3QÞÞ ¼ ð1 u2 Þln detðIn WðGÞu þ Qu2 Þ ( ! ) fi t X Y ðlnÞf i ð1 u2 Þ det Ifi n u ri ðgÞ#Wg þ u2 ð fi 3QÞ g
i¼2
¼ ZG ðw; uÞ1
t Y i¼2
(
2 ðlnÞfi
ð1 u Þ
detðIfi n u
X
)fi 2
ri ðgÞ#Wg þ u ð fi 3QÞÞ
:
&
gAG
Under the hypothesis of Theorem 4, ZG ðw; uÞ1 divides ZGa ðw; ˜ uÞ1 :
Let G be a connected graph, G a finite abelian group and G the character group of G: Corollary 1. Let G be a connected graph with n vertices and l edges, G a finite abelian group and a : DðGÞ-G an ordinary voltage assignment. Then the reciprocal of the weighted zeta function of the connected regular covering G a of G is ( ) Y X 1 1 2 ln 2 ˜ uÞ ¼ ZG ðw; uÞ ð1 u Þ detðIn u wðgÞWg þ u QÞ : ZGa ðw; wa1AG
g
Proof. Each irreducible representation of G is a linear representation, and these constitute the character group G : By Theorem 4, the result follows. &
4. Weighted L-functions of graphs Let G be a connected graph, G a finite group and a : DðGÞ-G an ordinary voltage assignment. Furthermore, let r be a representation of G and d its degree. For each path P ¼ ðv1 ; y; vr Þ of G; let aðPÞ ¼ aðv1 ; v2 Þaðv2 ; v3 Þ?aðvr1 ; vr Þ:
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Let W ¼ WðGÞ be the weighted matrix of G: Then the weighted L-function of G associated to r and a is defined by Y Zðu; r; a; wÞ ¼ ZG ðu; r; a; wÞ ¼ detðId rðaðCÞÞwðCÞujCj Þ1 ; ½C
where ½C runs over all equivalence classes of prime, reduced cycles of G: Theorem 5. Let G be a connected graph with n vertices and l unoriented edges, G a finite group and a : DðGÞ-G an ordinary voltage assignment. Furthermore, let r be a representation of G; and d the degree of r: Then the reciprocal of the weighted Lfunction of G associated to r and a is ! ! X 1 2 ðlnÞd 2 ZG ðu; r; a; wÞ ¼ ð1 u Þ det Idn u rðgÞ#Wg þ u ðd3QÞ : gAG
Proof. The argument is an analogue of Stark and Terras’s method [13]. At first, since TrðlogðI BÞÞ ¼ log detðI BÞ; X logZðu; r; a; wÞ ¼ log detðId rðaðCÞÞwðCÞujCj Þ ½C
¼
N X X 1 trðrðaðCÞÞm ÞwðCÞm ujCjm : m ½C m¼1
Thus, Z0 ðu; r; a; wÞ=Zðu; r; a; wÞ ¼ u1
N X X ½C
¼ u1
N X X m¼1
¼ u1
jCj tr ðrðaðCÞÞm ÞwðCÞm ujCjm
½C
N X X m¼1
jCj trðrðaðCÞÞm ÞwðCÞm ujCjm
m¼1
trðrðaðCÞÞm ÞwðCÞm ujCjm ;
C
where C runs over all prime, reduced cycles of G: Note that there exist jCj elements in ½C ; and trðBAB1 Þ ¼ trðAÞ: Let Cm be the set of all reduced cycles in G with length m: Set X Nm ¼ trðrðaðCÞÞÞwðCÞ: CACm
Then, Z0 ðu; r; a; wÞ=Zðu; r; a; wÞ ¼ u1
X
N m um :
ð4Þ
mX1
Let 1pi; jpn: Then, let the ði; jÞ-block Bi; j of a dn dn matrix B be the submatrix of B consisting of rows dði 1Þ þ 1; y; di and columns dðj 1Þ þ 1; y; dj: Set
ARTICLE IN PRESS H. Mizuno, I. Sato / Journal of Combinatorial Theory, Series B 91 (2004) 169–183
179
V ðGÞ ¼ fv1 ; y; vn g: For mX1; the dn dn matrix Am ¼ ððAm Þi; j Þ1pi; jpn is defined as follows: X ðAm Þi; j ¼ rðaðPÞÞwðPÞ; P
where ðAm Þi; j is the ði; jÞ-block of Am ; and P runs over all backtracking-less paths of P length m from vi to vj in G: Note that A1 ¼ g Wg #rðgÞ: Furthermore, let A0 ¼ Idn and Q0d ¼ Q#Id : Similarly to the proof of Lemma 1 in [13], we have A2 ¼ ðA1 Þ2 ðQ0d þ Idn Þ and Am ¼ Am1 A1 Am2 Q0d
for mX3:
Here, note that rðaðvk ; vj ÞÞwðvk ; vj Þrðaðvj ; vk ÞÞwðvj ; vk Þ ¼ Id : Thus, ! X Am um ðIdn uA1 þ u2 Q0d Þ ¼ ð1 u2 ÞIdn :
ð5Þ
mX0
Since ð1 u2 Þ1 ¼ Idn ¼
X
P
jX0
u2j ;
!
Ak u
k
jX0
kX0
¼
X
X
½m=2
X
mX0
j¼0
Am2j u
! u
2j
ðIdn uA1 þ u2 Q0d Þ
!
m
ðIdn uA1 þ u2 Q0d Þ:
ð6Þ
For mX1; let C0m be the set of all backtracking-less cycles of length m with tails in G; and X tm ¼ trðrðaðCÞÞÞwðCÞ: CAC0m
Then t1 ¼ t2 ¼ 0: Similarly to the proof of Lemma 2 in [13], we have tm ¼ Tr½ðQ0d Idn ÞAm2 þ tm2 Since Nm ¼ TrðAm Þ tm ; " Nm ¼ Tr Am
ðQ0d
Idn Þ
½ðm1Þ=2
X j¼1
and N2 ¼ N1 ¼ Tr A2 ¼ Tr A1 ¼ 0:
for mX2:
# Am2j ;
mX3
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180
Next, set N m ¼ Am ðQ0d Idn Þ
½m=2
X
Am2j
j¼1
¼ Q0d Am ðQ0d Idn Þ
½m=2
X
Am2j :
j¼0
Then (5) and (6) imply that ! X
m Nm u ðIdn uA1 þ u2 Q0d Þ ¼ ð1 u2 ÞQ0d ðQ0d Idn ÞIdn mX0
¼ Idn Q0d u2 : Since N 0 ¼ A0 ¼ Idn ; ! X N m um ðIdn uA1 þ u2 Q0d Þ ¼ Idn u2 Q0d Idn ðIdn uA1 þ u2 Q0d Þ mX1
¼ A1 u 2Q0d u2 : Therefore it follows that X N m um ¼ ðA1 u 2Q0d u2 ÞðIdn uA1 þ u2 Q0d Þ1 : mX1
By Stark and Terras, [13, Lemma 3], we have ! X d
m 2 0 Nm u ¼ Tr u logðIdn uA1 þ u Qd Þ : Tr du mX1 For mX1; we have TrðN m Þ
¼ Nm
if m is odd;
0
TrðQ0d Idn Þ if m is even:
Thus, Tr
X
! N m um
¼
mX1
X
m
Nm u
TrðQ0d
Idn Þ
mX1
¼
X mX1
X m:even; mX2
Nm um TrðQ0d Idn Þ
u2 : 1 u2
! u
m
ARTICLE IN PRESS H. Mizuno, I. Sato / Journal of Combinatorial Theory, Series B 91 (2004) 169–183
181
(4) implies that u
d logZðu; r; a; wÞ du d u2 ¼ Tr u logðIdn uA1 þ u2 Q0d Þ þ TrðQ0d Idn Þ du 1 u2 0 d d ¼ Tr u logðIdn uA1 þ u2 Q0d Þ u logð1 u2 ÞTrðQd Idn Þ=2 : du du
Both functions are 0 at u ¼ 0; and so 0
log Zðu; r; a; wÞ ¼ TrðlogðIdn uA1 þ u2 Q0d ÞÞ þ logð1 u2 ÞTrðQd Idn Þ=2 : Hence Zðu; r; a; wÞ1 ¼ ð1 u2 ÞðlnÞd detðIdn uA1 þ u2 Q0d Þ 2 ðlnÞd
¼ ð1 u Þ
det Idn u
X
rðgÞ#Wg
!
! þ u2 ðd3QÞ :
&
gAG
By Theorems 4 and 5, the following result holds. Corollary 2. Let G be a connected graph, G a finite group and a : DðGÞ-G an ordinary voltage assignment. Suppose that G a is connected. Then we have ZGa ðw; ˜ uÞ ¼
Y
ZG ðu; r; a; wÞd ;
r
where r runs over all inequivalent irreducible representations of G; and d ¼ deg r: If wðu; vÞ ¼ 1 for each ðu; vÞADðGÞ; then Corollary 2 implies Theorem 1. By Theorem 5 and Corollary 1, the following result holds. Corollary 3. Let G be a connected graph, G a finite abelian group and a : DðGÞ-G an ordinary voltage assignment. Suppose that G a is connected. Then we have ZGa ðw; ˜ uÞ ¼
Y wAG
ZG ðu; w; a; wÞ:
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5. Example We give an example. Let G ¼ K3 be the complete graph with vertices 1,2,3, and let the following matrix W be the weighted matrix of K3 : 2 3 0 w12 w13 6 7 0 w23 5: W ¼ 4 w1 12 w1 13
w1 23
0
Then all equivalence classes of prime, reduced cycles of K3 are ½C ; ½C 1 ; where C ¼ ð1; 2; 3; 1Þ: Thus, ZK3 ðw; uÞ1 ¼ ð1 W ðCÞu3 Þð1 W ðC 1 Þu3 Þ 3 1 1 3 ¼ ð1 w12 w23 w1 13 u Þð1 w12 w23 w13 u Þ:
By Theorem 3, we have
2
1 þ u2
6 ZK3 ðw; uÞ1 ¼ detðI3 uW þ u2 I3 Þ ¼ det4 uw1 12 uw1 13
uw12 1 þ u2 uw1 23
uw13
3
7 uw23 5 1 þ u2
2 3
3 1 1 3 ¼ ð1 þ u Þ 3u2 ð1 þ u2 Þ w12 w23 w1 13 u w12 w23 w13 u 3 1 1 3 ¼ ð1 w12 w23 w1 13 u Þð1 w12 w23 w13 u Þ:
Next, let G ¼ Z3 ¼ f1; t; t2 gðt3 ¼ 1Þ the cyclic group of order 3 and a : DðK3 Þ-Z3 the ordinary voltage assignment such that að1; 2Þ ¼ t; að2; 3Þ ¼ 1 and að3; 1Þ ¼ 1: Then the Z3 -covering ðK3 Þa is isomorphic to the cycle graph C9 of length 9, and the a * weighted matrix WððK 3 Þ Þ is given as follows: 3 2 0 w12 0 0 0 0 0 0 w13 6 0 0 w23 0 0 0 w1 0 0 7 7 6 12 7 6 1 6 w13 w1 0 0 0 0 0 0 0 7 23 7 6 6 0 0 0 0 0 w13 0 w12 0 7 7 6 7 6 1 a 7: 6 * w 0 0 0 0 w 0 0 0 WððK Þ Þ ¼ 23 3 7 6 12 7 6 1 1 0 0 w13 w23 0 0 0 0 7 6 0 7 6 6 0 w12 0 0 0 0 0 0 w13 7 7 6 7 6 0 0 w1 0 0 0 0 w23 5 4 0 12 0 0 0 0 0 0 w1 w1 0 13 23 By the definition of the weighted zeta function, we have 9 3 3 3 9 ZK3a ðw; ˜ uÞ1 ¼ ð1 w312 w323 w3 13 u Þð1 w12 w23 w13 u Þ:
The characters of Z3 are given as follows: j
i j
wi ðt Þ ¼ ðz Þ ;
0pi;
jp2;
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2p 1 1 þ 3 ¼ : z ¼ exp 3 2
ARTICLE IN PRESS H. Mizuno, I. Sato / Journal of Combinatorial Theory, Series B 91 (2004) 169–183
183
By Corollary 1, we have ZK3a ðw; ˜ uÞ1 ¼ ZK3 ðw; uÞ1
2 Y i¼1
detðI3 u 2
1 þ u2
w1 13 u
1 þ u2
6 det4 w1 12 zu w1 13 u
wi ðt j ÞWt j þ u2 I3 Þ
j¼0
6 2 ¼ ZK3 ðw; uÞ1 det4 w1 12 z u 2
2 X
w12 z2 u 1 þ u2 w1 23 u
w12 zu
w13 u
3
7 1 þ u2 w23 u 5 w1 1 þ u2 23 u 3 w13 u 7 w23 u 5 1 þ u2
¼ ð1 wðCÞu3 Þð1 wðCÞzu3 Þð1 wðCÞz2 u3 Þð1 wðC 1 Þu3 Þ ð1 wðC 1 Þzu3 Þð1 wðC 1 Þz2 u3 Þ ¼ ð1 wðCÞ3 u9 Þð1 wðC 1 Þ3 u9 Þ: Acknowledgments We thank the referee for many valuable comments and suggestions.
References [1] H. Bass, The Ihara–Selberg zeta function of a tree lattice, Internat. J. Math. 3 (1992) 717–797. [2] M. Burrow, Representation Theory of Finite Group, Academic Press, New York, 1965. [3] D. Foata, 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. [4] J.L. Gross, T.W. Tucker, Generating all graph coverings by permutation voltage assignments, Discrete Math. 18 (1977) 273–283. [5] J.L. Gross, T.W. Tucker, Topological Graph Theory, Wiley-Interscience, New York, 1987. [6] K. Hashimoto, Zeta functions of finite graphs and representations of p-adic groups, Adv. Stud. Pure Math. 15 (1989) 211–280. [7] K. Hashimoto, Artin-type L-functions and the density theorem for prime cycles on finite graphs, Internat. J. Math. 3 (1992) 809–826. [8] 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. [9] M. Kotani, T. Sunada, Zeta functions of finite graphs, J. Math. Sci. Univ. Tokyo 7 (2000) 7–25. [10] H. Mizuno, I. Sato, Characteristic polynomials of some graph coverings, Discrete Math. 142 (1995) 295–298. [11] H. Mizuno, I. Sato, Zeta functions of graph coverings, J. Combin. Theory Ser. B 80 (2000) 247–257. [12] S. Northshield, A note on the zeta function of a graph, J. Combin. Theory Ser. B 74 (1998) 408–410. [13] H.M. Stark, A.A. Terras, Zeta functions of finite graphs and coverings, Adv. Math. 121 (1996) 124–165. [14] H.M. Stark, A.A. Terras, Zeta functions of finite graphs and coverings, Part II, Adv. Math. 154 (2000) 132–195. [15] T. Sunada, L-functions in geometry and some applications, in: Lecture Notes in Mathematics, Vol. 1201, Springer, New York, 1986, pp. 266–284. [16] T. Sunada, Fundamental Groups and Laplacians, Kinokuniya, Tokyo, 1988 (in Japanese).
Journal of Combinatorial Theory, Series B 91 (2004) 169–183
http://www.elsevier.com/locate/jctb
Weighted zeta functions of graphs Hirobumi Mizunoa and Iwao Satob,1 a
Department of Electronics and Computer Science, Meisei University, 2-590, Nagabuti, Ome, Tokyo 198-8655, Japan b Oyama National College of Technology, Oyama, Tochigi 323-0806, Japan Received 15 April 2002
Abstract We give a decomposition formula for the weighted zeta function of a regular covering of a graph. r 2003 Elsevier Inc. All rights reserved. Keywords: Zeta function; Graph covering; L-function
1. Introduction Graphs and digraphs treated here are finite and simple. Let G ¼ ðV ðGÞ; EðGÞÞ be a connected graph with vertex V ðGÞ and edge set EðGÞ; and D the symmetric digraph corresponding to G: Set DðGÞ ¼ fðu; vÞ; ðv; uÞ j uvAEðGÞg: For e ¼ ðu; vÞADðGÞ; let oðeÞ ¼ u and tðeÞ ¼ v: The inverse arc of e is denoted by e%: A path P of length n in G is a sequence P ¼ ðv0 ; v1 ; y; vn1 ; vn Þ of n þ 1 vertices and n arcs such that consecutive vertices share an arc (we do not require that all vertices are distinct). Also, P is called a ðv0 ; vn Þ-path. If ei ¼ ðvi1 ; vi Þ for i ¼ 1; y; n; then we can write P ¼ ðe1 ; y; en Þ: We say that a path has a backtracking if a subsequence of the form y; x; y; x; y appears. A ðv; wÞ-path is called a cycle (or closed path) if v ¼ w: The inverse cycle of a cycle C ¼ ðv; v1 ; y; vn ; vÞ is the cycle C 1 ¼ ðv; vn ; y; v1 ; vÞ: A subpath ðv1 ; y; vm1 ; vm ; vm1 ; y; v1 Þ of a cycle C ¼ ðv1 ; y; v1 Þ is called a tail if degC v1 X3; degC vi ¼ 2ð2pipm 1Þ and degC vm ¼ 1; where degC v is the degree of v in the graph C: Let Br be the cycle obtained by going r times around a cycle B: Such a cycle is called a multiple of B: Note that any backtracking-less, tail-less cycle C is a
1
E-mail address: [email protected] (I. Sato). Supported by Grant-in-Aid for Science Research (C).
0095-8956/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jctb.2003.12.003
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cycle such that both C and C 2 have no backtracking (see [6,12]). A cycle C is reduced if C has no backtracking nor tail. The notion of tail-less cycle is used in the proof of Theorems 3 and 5. We introduce an equivalence relation between cycles. Such two cycles C1 ¼ ðv1 ; y; vm Þ and C2 ¼ ðw1 ; y; wm Þ are called equivalent if wj ¼ vjþk for all j: Let ½C be the equivalence class which contains a cycle C: A cycle C is prime if C is not obtained by going r times around some other cycle B; i.e., CaBr : Note that each equivalence class of prime, reduced cycles of a graph G corresponds to a unique conjugacy class of the fundamental group p1 ðG; vÞ of G at a vertex v of G: The (Ihara) zeta function of a graph G is defined to be a function of uAC with juj sufficiently small, by Y ZðG; uÞ ¼ ZG ðuÞ ¼ ð1 ujCj Þ1 ; ½C
where ½C runs over all equivalence classes of prime, reduced cycles of G; and jCj is the length of C: This definition is given in [13], and is equivalent to the one in [12]. Zeta functions of graphs started from zeta functions of regular graphs by Ihara [8]. In [8], he showed that their reciprocals are 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 [15,16]. Hashimoto [6] treated multivariable zeta functions of bipartite graphs. Bass [1] generalized Ihara’s result on the zeta function of a regular graph to an irregular graph, and showed that its reciprocal is a polynomial: ZðG; uÞ1 ¼ ð1 u2 Þr1 detðI uAðGÞ þ u2 ðD IÞÞ; where r and AðGÞ are the Betti number and the adjacency matrix of G; respectively, and D ¼ ðdij Þ is the diagonal matrix with dii ¼ deg vi ðV ðGÞ ¼ fvi ; y; vn gÞ: Stark and Terras [13] gave an elementary proof of Bass’s theorem, and discussed three different zeta functions of any graph. Furthermore, various proofs of Bass’s theorem were given by Foata and Zeilberger [3] and Kotani and Sunada [9]. Let G be a graph and G a finite group. Then a mapping a : DðGÞ-G is called an ordinary voltage assignment if aðv; uÞ ¼ aðu; vÞ1 for each ðu; vÞADðGÞ: Furthermore, let r be a representation of G: The L-function of G associated to r and a is defined to be the function of uAC with juj sufficiently small, given by Y Zðu; r; aÞ ¼ ZG ðu; r; aÞ ¼ detðIf rðaðCÞÞujCj Þ1 ; ½C
where f ¼ deg r and ½C runs over all equivalence classes of prime, reduced cycles of G (cf., [7,8,15]). Moreover, the net voltage aðCÞ of C ¼ ðe1 ; y; en Þ is defined by aðCÞ ¼ aðe1 Þyaðen Þ (see [5]). Let r ¼ 1 be the identity representation of G: Then, note that the L-function of G associated to 1 and a is the (Ihara) zeta function ZðG; uÞ of a graph G: Mizuno and Sato [11] showed that the zeta function of a regular covering of G is a product of L-functions of G:
ARTICLE IN PRESS H. Mizuno, I. Sato / Journal of Combinatorial Theory, Series B 91 (2004) 169–183
171
Let G be a connected graph, and let NðvÞ ¼ fwAV ðGÞ j ðv; wÞADðGÞg for any vertex v in G: A graph H is called a covering of G with projection p : H-G if there is a surjection p : V ðHÞ-V ðGÞ such that pjNðv0 Þ : Nðv0 Þ-NðvÞ is a bijection for all vertices vAV ðGÞ and v0 Ap1 ðvÞ: When a finite group P acts on a graph G; the quotient graph G=P is a simple graph whose vertices are the P-orbits on V ðGÞ; with two vertices adjacent in G=P if and only if some two of their representatives are adjacent in G: A covering p : H-G is said to be regular if there is a subgroup B of the automorphism group Aut H of H acting freely on H such that the quotient graph H=B is isomorphic to G: Let G be a graph and G a finite group. Furthermore, let a : DðGÞ-G be an ordinary voltage assignment. The pair ðG; aÞ is called an ordinary voltage graph. The derived graph Ga of the ordinary voltage graph ðG; aÞ is defined as follows: V ðG a Þ ¼ V ðGÞ G
and
ððu; hÞ; ðv; kÞÞADðG a Þ
if and only if ðu; vÞADðGÞ
and k ¼ haðu; vÞ: The natural projection p : Ga -G is defined by pðu; hÞ ¼ u; ðu; hÞAV ðGa Þ: The graph G a is called a derived graph covering of G with voltages in G or a G-covering of G: The natural projection p commutes with the right multiplication action of the aðeÞ; eADðGÞ and the left action of gAG on the fibers: g3ðu; hÞ ¼ ðu; ghÞ; gAG; which is free and transitive. Thus, the G-covering G a is a jGj-fold regular covering of G with covering transformation group G: Furthermore, every regular covering of a graph G is a G-covering of G for some group G (see [4]). Theorem 1 (Mizuno and Sato). Let G be a connected graph, G a finite group and a : DðGÞ-G an ordinary voltage assignment. Suppose that G a is connected. Then we have Y Zðu; r; aÞf ; ZðGa ; uÞ ¼ r
where r runs over all irreducible representations of G; and f ¼ deg r: Stark and Terras [14] introduced three Artin L-function of graphs based on vertex, edge and path. In particular, they obtained a factorization formula for the multiedge zeta function of a regular covering of a graph G as a product of multiedge Artin Lfunctions. Let G be a connected graph with 2m arcs e1 ; y; e2m ; and let W be a 2m 2m matrix with ij entry the variable wij if tðei Þ ¼ oðej Þ; ej ae%i ; and wij ¼ 0 otherwise. For a path C ¼ ðei1 ; y; eil Þ of G; let wðCÞ ¼ wi1 i2 wi2 i3 ?wil1 il : The multiedge zeta function of G is defined by Y ð1 wðCÞÞ1 ; zE ðW; GÞ ¼ ½C
where ½C runs over all equivalence classes of prime, reduced cycles of G:
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Let H be a regular covering of G with covering transformation group G and r a representation of G: Then the multiedge Artin L-function is defined by Y LE ðW; r; H=GÞ ¼ detðI rðsðCÞÞwðCÞÞ1 ; ½C
where sðCÞ is the Frobenius automorphism of C (see [14]). Theorem 2 (Stark and Terras). Y * HÞ ¼ zE ðW; LE ðW; r; H=GÞdeg r ; r
* is the matrix derived from W; and r runs over all inequivalent irreducible where W representations of G: In Section 2, we define a weighted zeta function of a graph G; and give a determinant expression of it. In Section 3, we give a decomposition formula of the weighted zeta function of a regular covering of G: In Section 4, we define a weighted L-function of G; and present a determinant expression for the weighted L-function of G: Furthermore, we show that the weighted zeta function of a regular covering of G is a product of weighted L-functions of G: A general theory of the representation of groups and graph coverings, the reader is referred to [2] and [5], respectively.
2. Weighted zeta functions of graphs Let G be a connected graph and V ðGÞ ¼ fv1 ; y; vn g: Then we consider an n n matrix W ¼ ðwij Þ1pi; jpn with ij entry the complex variable wij if ðvi ; vj ÞADðGÞ; and wij ¼ 0 otherwise. Furthermore, assume that w1 ji ¼ wij if ðvi ; vj ÞADðGÞ: The matrix W ¼ WðGÞ is called the weighted matrix of G: For each path P ¼ ðvi1 ; y; vir Þ of G; the norm wðPÞ of P is defined as follows: wðPÞ ¼ wi1 i2 wi2 i3 ?wir1 ir : Furthermore, let wðvi ; vj Þ ¼ wij ; vi ; vj AV ðGÞ: The (vertex) weighted zeta function of G is defined by Y ZG ðw; uÞ ¼ ð1 wðCÞujCj Þ1 ; ½C
where ½C runs over all equivalence classes of prime, reduced cycles of G: Theorem 3. Let G be a connected graph, and let W ¼ WðGÞ be the weighted matrix of G: Then the reciprocal of the weighted zeta function of G is given by ZG ðw; uÞ1 ¼ ð1 u2 Þmn detðI uWðGÞ þ u2 QÞ; where n ¼ jV ðGÞj;m ¼ jEðGÞj and Q ¼ D I: Proof. The proof is an analogue of Stark and Terras’s method [13].
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173
Let G be a connected graph with n vertices and m edges. At first, X log ZG ðw; uÞ ¼ logð1 wðCÞujCj Þ ½C
N X X 1 wðCÞs ujCjs : ¼ s ½C s¼1
Thus, we have N X X d log ZG ðw; uÞ ¼ u1 jCjwðCÞs ujCjs du ½C s¼1
¼ u1
N X X s¼1
¼ u1
X
wðCÞs ujCjs
C
wðCÞujCj ;
C
P
where the last C runs over all reduced cycles C of G: Note that there exist jCj elements in ½C : Let Cs be the set of all reduced cycles in G with length s: Set X Ns ¼ wðCÞ: CACs
Then, X d log ZG ðw; uÞ ¼ u1 Ns u s : du sX1
ð1Þ
Let V ðGÞ ¼ fv1 ; y; vn g: For sX1; the n n matrix As ¼ ððAs Þi; j Þ1pi; jpn is defined as follows: X ðAs Þi; j ¼ wðPÞ; P
where ðAs Þi; j is the ði; jÞ-component of As ; and P runs over all backtracking-less paths of length s from vi to vj in G: Note that A1 ¼ WðGÞ: Furthermore, let A0 ¼ In : Similarly to the proof of Lemma 1 in [13], we have A2 ¼ ðA1 Þ2 ðQ þ In Þ and As ¼ As1 A1 As2 Q
for sX3:
Here, note that wðu; vÞwðv; uÞ ¼ 1: Thus, ! X s As u ðI uA1 þ u2 QÞ ¼ ð1 u2 ÞI: sX0
ð2Þ
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Since ð1 u2 Þ1 ¼ X
I¼
P
Ak u
jX0
! k
u2j ; u
2j
ðI uA1 þ u2 QÞ
jX0
kX0
¼
!
X
X
½s=2
X
sX0
j¼0
!
As2j u
s
ðI uA1 þ u2 QÞ:
ð3Þ
For sX1; let C0s be the set of all backtracking-less cycles of length s with tails in G; and X ts ¼ wðCÞ: CAC0s
Then t1 ¼ t2 ¼ 0: Similarly to the proof of Lemma 2 in [13], we have ts ¼ Tr½ðQ IÞAs2 þ ts2
for sX2:
Since Ns ¼ TrðAs Þ ts ; Ns ¼ Tr½As ðQ IÞ
½ðs1Þ=2
X
As2j ;
sX3
j¼1
and N2 ¼ N1 ¼ Tr A2 ¼ Tr A1 ¼ 0: Next, set N s ¼ As ðQ IÞ
½s=2
X
As2j
j¼1
¼ QAs ðQ IÞ
½s=2
X
As2j :
j¼0
Then (2) and (3) imply that ! X
s Ns u ðI uA1 þ u2 QÞ ¼ ð1 u2 ÞQ ðQ IÞI sX0
¼ I Qu2 : Since N 0 ¼ A0 ¼ In ; ! X
s Ns u ðI uA1 þ u2 QÞ ¼ I u2 Q IðI uA1 þ u2 QÞ sX1
¼ A1 u 2Qu2 : Therefore it follows that X N s us ¼ ðA1 u 2Qu2 ÞðI uA1 þ u2 QÞ1 : sX1
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175
By [13, Lemma 3], we have X d N s us ¼ u logðI uA1 þ u2 QÞ: du sX1 For sX1; we have TrðN s Þ ¼ Ns
0 if s is odd; TrðQ IÞ if s is even:
Thus, Tr
X
! N s us
¼
sX1
X
s
Ns u TrðQ IÞ
sX1
¼
X
X
! u
2j
jX1
Ns us TrðQ IÞ
sX1
u2 : 1 u2
Eq. (1) implies that d d u2 2 u log ZG ðw; uÞ ¼ Tr u logðI uA1 þ u QÞ þ TrðQ IÞ du du 1 u2 d d ¼ Tr u logðI uA1 þ u2 QÞ u logð1 u2 ÞTrðQIÞ=2 : du du Both functions are 0 at u ¼ 0; and so log ZG ðw; uÞ ¼ TrðlogðI uA1 þ u2 QÞÞ þ logð1 u2 ÞTrðQIÞ=2 : Hence the fact that TrðlogðI BÞÞ ¼ log detðI BÞ implies that ZG ðw; uÞ ¼ ð1 u2 ÞðmnÞ detðI uA1 þ u2 QÞ1 ¼ ð1 u2 ÞðmnÞ detðI uWðGÞ þ u2 QÞ1 :
&
3. Weighted zeta functions of regular coverings of graphs Let G be a connected graph, G a finite group and a : DðGÞ-G an ordinary voltage assignment. In the G-covering Ga ; set vg ¼ ðv; gÞ and eg ¼ ðe; gÞ; where vAV ðGÞ; eADðGÞ; gAG: For e ¼ ðu; vÞADðGÞ; the arc eg emanates from ug and terminates at vgaðeÞ : Note that e%g ¼ ðe%ÞgaðeÞ : Let W ¼ WðGÞ be the weighted matrix of G: Then we define the weighted matrix * ˜ g ; vh ÞÞ of Ga derived from W as follows: W ¼ WðGa Þ ¼ ðwðu wðu; vÞ if ðu; vÞADðGÞ and h ¼ gaðu; vÞ; wðu ˜ g ; vh Þ :¼ 0 otherwise:
ARTICLE IN PRESS H. Mizuno, I. Sato / Journal of Combinatorial Theory, Series B 91 (2004) 169–183
176
ðgÞ
For gAG; let the matrix Wg ¼ ðwuv Þ be defined by wðu; vÞ if aðu; vÞ ¼ g and ðu; vÞADðGÞ; :¼ wðgÞ uv 0 otherwise: Let M1 "?"Ms be the block diagonal sum of square matrices M1 ; y; Ms : If M1 ¼ M2 ¼ ? ¼ Ms ¼ M; then we write s3M ¼ M1 "?"Ms : The Kronecker product A#B of matrices A and B is considered as the matrix A having the element aij replaced by the matrix aij B: Theorem 4. Let G be a connected graph with n vertices and l unoriented edges, G a finite group and a : DðGÞ-G an ordinary voltage assignment. Furthermore, let r1 ¼ 1; r2 ; y; rt be all inequivalent irreducible representations of G; and fi the degree of ri for each i; where f1 ¼ 1: Then the reciprocal of the weighted zeta function of the connected G-covering G a of G is ZGa ðw; ˜ uÞ1 ¼ ZG ðw; uÞ1
t Y i¼2
(
2 ðlnÞfi
ð1 u Þ
det Infi u
X
! ) fi 2
ri ðgÞ#Wg þ u ð fi 3QÞ
:
gAG
Proof. The argument is an analogue of Mizuno and Sato’s method [10]. Set V ðGÞ ¼ fv1 ; y; vn g and G ¼ f1 ¼ g1 ; g2 ; y; gm g: Arrange vertices of G a in m blocks: ðv1 ; 1Þ; y; ðvn ; 1Þ; ðv1 ; g2 Þ; y; ðvn ; g2 Þ; y; ðv1 ; gm Þ; y; ðvn ; gm Þ: We consider the weighted matrix WðG a Þ under this order. For gAG; the matrix ðgÞ Pg ¼ ðpij Þ is defined as follows: 1 if gi g ¼ gj ; ðgÞ pij ¼ 0 otherwise: ðgÞ
Suppose that pij ¼ 1; i.e., gj ¼ gi g: Then ððu; gi Þ; ðv; gj ÞÞADðG a Þ if and only if ðu; vÞADðGÞ and gi aðu; vÞ ¼ gj ; i.e., aðu; vÞ ¼ g1 i gj ¼ g: Thus we have X Pg #Wg : WðG a Þ ¼ gAG
Let r be the right regular representation of G: Then we have rðgÞ ¼ Pg for gAG: Furthermore, there exists a nonsingular matrix P such that P1 rðgÞP ¼ ð1Þ"f2 3r2 ðgÞ"?"ft 3rt ðgÞ for each gAG: Putting B ¼ ðP1 #In ÞWðGa ÞðP#In Þ;
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177
we have B¼
X
fð1Þ"f2 3r2 ðgÞ"?"ft 3rt ðgÞg#Wg :
gAG
Note that WðGÞ ¼ that
P
gAG
Wg and 1 þ f22 þ ? þ ft2 ¼ m: By Theorem 3, it follows
ZGa ðw; ˜ uÞ1 ¼ ð1 u2 ÞðlnÞm detðInm WðG a Þu þ u2 ðm3QÞÞ ¼ ð1 u2 Þln detðIn WðGÞu þ Qu2 Þ ( ! ) fi t X Y ðlnÞf i ð1 u2 Þ det Ifi n u ri ðgÞ#Wg þ u2 ð fi 3QÞ g
i¼2
¼ ZG ðw; uÞ1
t Y i¼2
(
2 ðlnÞfi
ð1 u Þ
detðIfi n u
X
)fi 2
ri ðgÞ#Wg þ u ð fi 3QÞÞ
:
&
gAG
Under the hypothesis of Theorem 4, ZG ðw; uÞ1 divides ZGa ðw; ˜ uÞ1 :
Let G be a connected graph, G a finite abelian group and G the character group of G: Corollary 1. Let G be a connected graph with n vertices and l edges, G a finite abelian group and a : DðGÞ-G an ordinary voltage assignment. Then the reciprocal of the weighted zeta function of the connected regular covering G a of G is ( ) Y X 1 1 2 ln 2 ˜ uÞ ¼ ZG ðw; uÞ ð1 u Þ detðIn u wðgÞWg þ u QÞ : ZGa ðw; wa1AG
g
Proof. Each irreducible representation of G is a linear representation, and these constitute the character group G : By Theorem 4, the result follows. &
4. Weighted L-functions of graphs Let G be a connected graph, G a finite group and a : DðGÞ-G an ordinary voltage assignment. Furthermore, let r be a representation of G and d its degree. For each path P ¼ ðv1 ; y; vr Þ of G; let aðPÞ ¼ aðv1 ; v2 Þaðv2 ; v3 Þ?aðvr1 ; vr Þ:
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Let W ¼ WðGÞ be the weighted matrix of G: Then the weighted L-function of G associated to r and a is defined by Y Zðu; r; a; wÞ ¼ ZG ðu; r; a; wÞ ¼ detðId rðaðCÞÞwðCÞujCj Þ1 ; ½C
where ½C runs over all equivalence classes of prime, reduced cycles of G: Theorem 5. Let G be a connected graph with n vertices and l unoriented edges, G a finite group and a : DðGÞ-G an ordinary voltage assignment. Furthermore, let r be a representation of G; and d the degree of r: Then the reciprocal of the weighted Lfunction of G associated to r and a is ! ! X 1 2 ðlnÞd 2 ZG ðu; r; a; wÞ ¼ ð1 u Þ det Idn u rðgÞ#Wg þ u ðd3QÞ : gAG
Proof. The argument is an analogue of Stark and Terras’s method [13]. At first, since TrðlogðI BÞÞ ¼ log detðI BÞ; X logZðu; r; a; wÞ ¼ log detðId rðaðCÞÞwðCÞujCj Þ ½C
¼
N X X 1 trðrðaðCÞÞm ÞwðCÞm ujCjm : m ½C m¼1
Thus, Z0 ðu; r; a; wÞ=Zðu; r; a; wÞ ¼ u1
N X X ½C
¼ u1
N X X m¼1
¼ u1
jCj tr ðrðaðCÞÞm ÞwðCÞm ujCjm
½C
N X X m¼1
jCj trðrðaðCÞÞm ÞwðCÞm ujCjm
m¼1
trðrðaðCÞÞm ÞwðCÞm ujCjm ;
C
where C runs over all prime, reduced cycles of G: Note that there exist jCj elements in ½C ; and trðBAB1 Þ ¼ trðAÞ: Let Cm be the set of all reduced cycles in G with length m: Set X Nm ¼ trðrðaðCÞÞÞwðCÞ: CACm
Then, Z0 ðu; r; a; wÞ=Zðu; r; a; wÞ ¼ u1
X
N m um :
ð4Þ
mX1
Let 1pi; jpn: Then, let the ði; jÞ-block Bi; j of a dn dn matrix B be the submatrix of B consisting of rows dði 1Þ þ 1; y; di and columns dðj 1Þ þ 1; y; dj: Set
ARTICLE IN PRESS H. Mizuno, I. Sato / Journal of Combinatorial Theory, Series B 91 (2004) 169–183
179
V ðGÞ ¼ fv1 ; y; vn g: For mX1; the dn dn matrix Am ¼ ððAm Þi; j Þ1pi; jpn is defined as follows: X ðAm Þi; j ¼ rðaðPÞÞwðPÞ; P
where ðAm Þi; j is the ði; jÞ-block of Am ; and P runs over all backtracking-less paths of P length m from vi to vj in G: Note that A1 ¼ g Wg #rðgÞ: Furthermore, let A0 ¼ Idn and Q0d ¼ Q#Id : Similarly to the proof of Lemma 1 in [13], we have A2 ¼ ðA1 Þ2 ðQ0d þ Idn Þ and Am ¼ Am1 A1 Am2 Q0d
for mX3:
Here, note that rðaðvk ; vj ÞÞwðvk ; vj Þrðaðvj ; vk ÞÞwðvj ; vk Þ ¼ Id : Thus, ! X Am um ðIdn uA1 þ u2 Q0d Þ ¼ ð1 u2 ÞIdn :
ð5Þ
mX0
Since ð1 u2 Þ1 ¼ Idn ¼
X
P
jX0
u2j ;
!
Ak u
k
jX0
kX0
¼
X
X
½m=2
X
mX0
j¼0
Am2j u
! u
2j
ðIdn uA1 þ u2 Q0d Þ
!
m
ðIdn uA1 þ u2 Q0d Þ:
ð6Þ
For mX1; let C0m be the set of all backtracking-less cycles of length m with tails in G; and X tm ¼ trðrðaðCÞÞÞwðCÞ: CAC0m
Then t1 ¼ t2 ¼ 0: Similarly to the proof of Lemma 2 in [13], we have tm ¼ Tr½ðQ0d Idn ÞAm2 þ tm2 Since Nm ¼ TrðAm Þ tm ; " Nm ¼ Tr Am
ðQ0d
Idn Þ
½ðm1Þ=2
X j¼1
and N2 ¼ N1 ¼ Tr A2 ¼ Tr A1 ¼ 0:
for mX2:
# Am2j ;
mX3
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180
Next, set N m ¼ Am ðQ0d Idn Þ
½m=2
X
Am2j
j¼1
¼ Q0d Am ðQ0d Idn Þ
½m=2
X
Am2j :
j¼0
Then (5) and (6) imply that ! X
m Nm u ðIdn uA1 þ u2 Q0d Þ ¼ ð1 u2 ÞQ0d ðQ0d Idn ÞIdn mX0
¼ Idn Q0d u2 : Since N 0 ¼ A0 ¼ Idn ; ! X N m um ðIdn uA1 þ u2 Q0d Þ ¼ Idn u2 Q0d Idn ðIdn uA1 þ u2 Q0d Þ mX1
¼ A1 u 2Q0d u2 : Therefore it follows that X N m um ¼ ðA1 u 2Q0d u2 ÞðIdn uA1 þ u2 Q0d Þ1 : mX1
By Stark and Terras, [13, Lemma 3], we have ! X d
m 2 0 Nm u ¼ Tr u logðIdn uA1 þ u Qd Þ : Tr du mX1 For mX1; we have TrðN m Þ
¼ Nm
if m is odd;
0
TrðQ0d Idn Þ if m is even:
Thus, Tr
X
! N m um
¼
mX1
X
m
Nm u
TrðQ0d
Idn Þ
mX1
¼
X mX1
X m:even; mX2
Nm um TrðQ0d Idn Þ
u2 : 1 u2
! u
m
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181
(4) implies that u
d logZðu; r; a; wÞ du d u2 ¼ Tr u logðIdn uA1 þ u2 Q0d Þ þ TrðQ0d Idn Þ du 1 u2 0 d d ¼ Tr u logðIdn uA1 þ u2 Q0d Þ u logð1 u2 ÞTrðQd Idn Þ=2 : du du
Both functions are 0 at u ¼ 0; and so 0
log Zðu; r; a; wÞ ¼ TrðlogðIdn uA1 þ u2 Q0d ÞÞ þ logð1 u2 ÞTrðQd Idn Þ=2 : Hence Zðu; r; a; wÞ1 ¼ ð1 u2 ÞðlnÞd detðIdn uA1 þ u2 Q0d Þ 2 ðlnÞd
¼ ð1 u Þ
det Idn u
X
rðgÞ#Wg
!
! þ u2 ðd3QÞ :
&
gAG
By Theorems 4 and 5, the following result holds. Corollary 2. Let G be a connected graph, G a finite group and a : DðGÞ-G an ordinary voltage assignment. Suppose that G a is connected. Then we have ZGa ðw; ˜ uÞ ¼
Y
ZG ðu; r; a; wÞd ;
r
where r runs over all inequivalent irreducible representations of G; and d ¼ deg r: If wðu; vÞ ¼ 1 for each ðu; vÞADðGÞ; then Corollary 2 implies Theorem 1. By Theorem 5 and Corollary 1, the following result holds. Corollary 3. Let G be a connected graph, G a finite abelian group and a : DðGÞ-G an ordinary voltage assignment. Suppose that G a is connected. Then we have ZGa ðw; ˜ uÞ ¼
Y wAG
ZG ðu; w; a; wÞ:
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5. Example We give an example. Let G ¼ K3 be the complete graph with vertices 1,2,3, and let the following matrix W be the weighted matrix of K3 : 2 3 0 w12 w13 6 7 0 w23 5: W ¼ 4 w1 12 w1 13
w1 23
0
Then all equivalence classes of prime, reduced cycles of K3 are ½C ; ½C 1 ; where C ¼ ð1; 2; 3; 1Þ: Thus, ZK3 ðw; uÞ1 ¼ ð1 W ðCÞu3 Þð1 W ðC 1 Þu3 Þ 3 1 1 3 ¼ ð1 w12 w23 w1 13 u Þð1 w12 w23 w13 u Þ:
By Theorem 3, we have
2
1 þ u2
6 ZK3 ðw; uÞ1 ¼ detðI3 uW þ u2 I3 Þ ¼ det4 uw1 12 uw1 13
uw12 1 þ u2 uw1 23
uw13
3
7 uw23 5 1 þ u2
2 3
3 1 1 3 ¼ ð1 þ u Þ 3u2 ð1 þ u2 Þ w12 w23 w1 13 u w12 w23 w13 u 3 1 1 3 ¼ ð1 w12 w23 w1 13 u Þð1 w12 w23 w13 u Þ:
Next, let G ¼ Z3 ¼ f1; t; t2 gðt3 ¼ 1Þ the cyclic group of order 3 and a : DðK3 Þ-Z3 the ordinary voltage assignment such that að1; 2Þ ¼ t; að2; 3Þ ¼ 1 and að3; 1Þ ¼ 1: Then the Z3 -covering ðK3 Þa is isomorphic to the cycle graph C9 of length 9, and the a * weighted matrix WððK 3 Þ Þ is given as follows: 3 2 0 w12 0 0 0 0 0 0 w13 6 0 0 w23 0 0 0 w1 0 0 7 7 6 12 7 6 1 6 w13 w1 0 0 0 0 0 0 0 7 23 7 6 6 0 0 0 0 0 w13 0 w12 0 7 7 6 7 6 1 a 7: 6 * w 0 0 0 0 w 0 0 0 WððK Þ Þ ¼ 23 3 7 6 12 7 6 1 1 0 0 w13 w23 0 0 0 0 7 6 0 7 6 6 0 w12 0 0 0 0 0 0 w13 7 7 6 7 6 0 0 w1 0 0 0 0 w23 5 4 0 12 0 0 0 0 0 0 w1 w1 0 13 23 By the definition of the weighted zeta function, we have 9 3 3 3 9 ZK3a ðw; ˜ uÞ1 ¼ ð1 w312 w323 w3 13 u Þð1 w12 w23 w13 u Þ:
The characters of Z3 are given as follows: j
i j
wi ðt Þ ¼ ðz Þ ;
0pi;
jp2;
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2p 1 1 þ 3 ¼ : z ¼ exp 3 2
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183
By Corollary 1, we have ZK3a ðw; ˜ uÞ1 ¼ ZK3 ðw; uÞ1
2 Y i¼1
detðI3 u 2
1 þ u2
w1 13 u
1 þ u2
6 det4 w1 12 zu w1 13 u
wi ðt j ÞWt j þ u2 I3 Þ
j¼0
6 2 ¼ ZK3 ðw; uÞ1 det4 w1 12 z u 2
2 X
w12 z2 u 1 þ u2 w1 23 u
w12 zu
w13 u
3
7 1 þ u2 w23 u 5 w1 1 þ u2 23 u 3 w13 u 7 w23 u 5 1 þ u2
¼ ð1 wðCÞu3 Þð1 wðCÞzu3 Þð1 wðCÞz2 u3 Þð1 wðC 1 Þu3 Þ ð1 wðC 1 Þzu3 Þð1 wðC 1 Þz2 u3 Þ ¼ ð1 wðCÞ3 u9 Þð1 wðC 1 Þ3 u9 Þ: Acknowledgments We thank the referee for many valuable comments and suggestions.
References [1] H. Bass, The Ihara–Selberg zeta function of a tree lattice, Internat. J. Math. 3 (1992) 717–797. [2] M. Burrow, Representation Theory of Finite Group, Academic Press, New York, 1965. [3] D. Foata, 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. [4] J.L. Gross, T.W. Tucker, Generating all graph coverings by permutation voltage assignments, Discrete Math. 18 (1977) 273–283. [5] J.L. Gross, T.W. Tucker, Topological Graph Theory, Wiley-Interscience, New York, 1987. [6] K. Hashimoto, Zeta functions of finite graphs and representations of p-adic groups, Adv. Stud. Pure Math. 15 (1989) 211–280. [7] K. Hashimoto, Artin-type L-functions and the density theorem for prime cycles on finite graphs, Internat. J. Math. 3 (1992) 809–826. [8] 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. [9] M. Kotani, T. Sunada, Zeta functions of finite graphs, J. Math. Sci. Univ. Tokyo 7 (2000) 7–25. [10] H. Mizuno, I. Sato, Characteristic polynomials of some graph coverings, Discrete Math. 142 (1995) 295–298. [11] H. Mizuno, I. Sato, Zeta functions of graph coverings, J. Combin. Theory Ser. B 80 (2000) 247–257. [12] S. Northshield, A note on the zeta function of a graph, J. Combin. Theory Ser. B 74 (1998) 408–410. [13] H.M. Stark, A.A. Terras, Zeta functions of finite graphs and coverings, Adv. Math. 121 (1996) 124–165. [14] H.M. Stark, A.A. Terras, Zeta functions of finite graphs and coverings, Part II, Adv. Math. 154 (2000) 132–195. [15] T. Sunada, L-functions in geometry and some applications, in: Lecture Notes in Mathematics, Vol. 1201, Springer, New York, 1986, pp. 266–284. [16] T. Sunada, Fundamental Groups and Laplacians, Kinokuniya, Tokyo, 1988 (in Japanese).