Scattering matrices of regular coverings of graphs

Discrete Mathematics 310 (2010) 782–791

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Scattering matrices of regular coverings of graphs Hirobumi Mizuno a , Iwao Sato b,∗ a

Iond University, Tokyo, Japan

b

Oyama National College of Technology, Oyama, Tochigi 323–0806, Japan

article

abstract

info

Article history: Received 24 May 2008 Received in revised form 3 September 2009 Accepted 8 September 2009 Available online 24 September 2009

We give a decomposition formula for the determinant on the bond scattering matrix of a regular covering of G. Furthermore, we define an L-function of G, and give a determinant expression of it. As a corollary, we express the determinant on the bond scattering matrix of a regular covering of G by means of its L-functions. © 2009 Elsevier B.V. All rights reserved.

Keywords: Laplacian matrix Scattering matrix Graph covering L-function

1. Introduction Graphs treated here are finite. Let G be a connected graph (possibly with multiple edges and loops) with the set V (G) of vertices and the set E (G) of unoriented edges. We write uv for an edge joining two vertices u and v . For uv ∈ E (G), an arc (u, v) is the oriented edge from u to v . Set D(G) = {(u, v), (v, u) | uv ∈ E (G)}. For e = (u, v) ∈ D(G), set u = o(e) and v = t (e). Furthermore, let e−1 = (v, u) be the inverse arc of e = (u, v). A path P of length n in G is a sequence P : (v0 , e1 , v1 , e2 , v2 , . . . , vn−1 , en , vn ) of n + 1 vertices and n arcs such that v0 ∈ V (G), vi ∈ V (G), ei ∈ D(G) and ei = (vi−1 , vi ) for 1 ≤ i ≤ n. We write P = (e1 , . . . , en ). Set |P | = n, o(P ) = v0 and 1 t (P ) = vn . Also, P is called an (o(P ), t (P ))-path. We say that a path P = (e1 , . . . , en ) has a backtracking if e− i+1 = ei for some i. A (v, w)-path is called a v -cycle (or v -closed path) if v = w . The inverse cycle of a cycle C = (e1 , . . . , en ) is the cycle C −1 = −1 1 (e− n , . . . , e1 ). As standard terminologies of graph theory, a path and a cycle are a diwalk and a closed diwalk, respectively. We introduce an equivalence relation on the set of cycles. Two cycles C1 = (e1 , . . . , em ) and C2 = (f1 , . . . , fm ) are equivalent if there exists k such that fj = ej+k for all j. The inverse cycle of C is in general not equivalent to C . Let [C ] be the equivalence class that contains a cycle C . Let Br be the cycle obtained by going r times around a cycle B. Such a cycle is called a power 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 power 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 π1 (G, v) of G at a vertex v of G. The Ihara zeta function of a graph G is defined to be a function of u ∈ C with |u| sufficiently small, by Z(G, u) = ZG (u) =

Y

(1 − u|C | )−1 ,

[C ]

where [C ] runs over all equivalence classes of prime, reduced cycles of G (see [9]).

∗

Corresponding author. Tel.: +81 285 20 2176; fax: +81 285 20 2880. E-mail address: [email protected] (I. Sato).

0012-365X/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2009.09.009

H. Mizuno, I. Sato / Discrete Mathematics 310 (2010) 782–791

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Zeta functions of graphs were originally defined for regular graphs by Ihara [9]. In [9], he showed that their reciprocals are explicit polynomials. A zeta function of a regular graph G associated with a unitary representation of the fundamental group of G was developed by Sunada [15,16]. Hashimoto [8] 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 and showed that its reciprocal is again a polynomial. Theorem 1 (Bass). If G is a connected graph, then the reciprocal of the zeta function of G is given by Z(G, u)−1 = (1 − u2 )r −1 det(I − uA(G) + u2 (D − I)), where r and A(G) are the Betti number and the adjacency matrix of G, respectively, and D = DG = (dij ) is the diagonal matrix with dii = deg vi where V (G) = {v1 , . . . , vp }. Stark and Terras [14] gave an elementary proof of Theorem 1 and discussed three different zeta functions of any graph. Other proofs of Bass’ Theorem were given by Foata and Zeilberger [5] and Kotani and Sunada [10]. The spectral determinant of the Laplacian on a quantum graph is closely related to the Ihara zeta function of a graph (see [3,4,7,13]). Smilansky [13] considered spectral zeta functions and trace formulas for (discrete) Laplacians on ordinary graphs and expressed the determinant on the bond scattering matrix of a graph G by using the characteristic polynomial of its Laplacian. 1 −1 Let G be a connected graph with V (G) = {v1 , . . . , vp } and D(G) = {e1 , . . . , eq , e− 1 , . . . , eq }. The Laplacian (matrix) L = L(G) of G is defined by L = −A(G) + D. Let λ be a eigenvalue of L, and let φ = (φ1 , . . . , φp ) be the eigenvector corresponding to λ. For each arc b = (vj , vl ), one associates a bond wave function

φb (x) = ab eiπ x/4 + ab−1 e−iπ x/4 ,

x = ±1, i =

√ −1

under the condition

φb (1) = φj and φb (−1) = φl . We consider the following three conditions: 1. uniqueness: The value φj of the eigenvector at the vertex vj computed in the terms of the bond wave functions, is the same for all the arcs emanating from vj . 2. φ is an eigenvector of L; 3. consistency: The linear relation between the incoming and the outgoing coefficients must be satisfied simultaneously at all vertices. By uniqueness, we have ab1 eiπ/4 + ab−1 e−iπ/4 = ab2 eiπ/4 + ab−1 e−iπ/4 = · · · = abd eiπ /4 + ab−1 e−iπ /4 , 1

j

2

dj

where b1 , b2 , . . . , bdj are arcs emanating from vj , and dj = deg vj . By condition 2, we have

−

dj dj X 1 X (abk e−iπ/4 + ab−1 eiπ/4 ) = (λ − dj ) (abk eiπ /4 + ab−1 e−iπ /4 ).

dj k=1

k

k=1

k

Thus, for each arc b with o(b) = vj , v

X

ab =

σbcj (λ)ac ,

(1)

t (c )=vj

where



vj

σbc (λ) = i δb−1 c −

2

1



dj 1 − i(1 − λ/dj )

and δb−1 c is the Kronecker delta. The bond scattering matrix U(λ) = (Uef )e,f ∈D(G) of G is defined by Uef =

 t (f ) σef 0

if t (f ) = o(e), otherwise.

By consistency, we have U(λ)a = a, where a = t (a1 , a2 , . . . , a2q ). This holds if and only if det(I2q − U(λ)) = 0.

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Theorem 2 (Smilansky). If G is a connected graph with p vertices and q edges, then, for the bond scattering matrix of G, det(I2q − U(λ)) =

2q ip det(λIp + A(G) − D) p

Q

=

(deg vj − i deg vj + λi)

Y

(1 − aC (λ)),

[C ]

j=1

where [C ] runs over all equivalence classes of prime cycles of G, and aC (λ) = σeo1(een1 ) σeon(eenn−) 1 · · · σeo2(ee12 ) ,

C = (e1 , e2 , . . . , en ).

In this paper, we consider the bond scattering matrix of a regular covering of a graph and express it as a product of L-functions over a representation of the covering transformation group of the covering. In Section 2, we give a decomposition formula for the determinant on the bond scattering matrix of a regular covering of G. In Section 3, we define an L-function of G and give a determinant expression for it. As a corollary, we express the determinant on the bond scattering matrix of a regular covering of G by means of its L-functions. A general theory of the representation of groups and graph coverings, the reader is referred to [12,6], respectively. 2. Scattering matrix of a regular covering of a graph Let G be a connected graph, and let N (v) = {w ∈ V (G) | (v, w) ∈ D(G)} denote the neighbourhood of a vertex v in G. A graph H is 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 G, the quotient graph G/Π is a graph whose vertices are the Π -orbits on V (G), with two vertices adjacent in G/Π if and only if some two of their representatives are adjacent in G. A covering π : H −→ G is 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 Γ a finite group. Then a mapping α : D(G) −→ Γ is an ordinary voltage assignment if α(v, u) = α(u, v)−1 for each (u, v) ∈ D(G). The pair (G, α) is an ordinary voltage graph. The derived graph Gα of the ordinary voltage graph (G, α) is defined as follows: V (Gα ) = V (G) × Γ and ((u, h), (v, k)) ∈ D(Gα ) if and only if (u, v) ∈ D(G) and k = hα(u, v). The natural projection π : Gα −→ G is defined by π (u, h) = u. The graph Gα is a derived graph covering of G with voltages in Γ or a Γ -covering of G. The natural projection π commutes with the right multiplication action of the α(e), e ∈ D(G) and the left action of Γ on the fibers: g (u, h) = (u, gh), g ∈ Γ , which is free and transitive. Thus, the Γ covering Gα is a |Γ |-fold regular covering of G with covering transformation group Γ . Furthermore, every regular covering of a graph G is a Γ -covering of G for some group Γ (see [6]). Let G be a connected graph, Γ be a finite group and α : D(G) −→ Γ be an ordinary voltage assignment. In the Γ -covering Gα , set vg = (v, g ) and eg = (e, g ), where v ∈ V (G), e ∈ D(G), g ∈ Γ . For e = (u, v) ∈ D(G), the arc eg emanates from ug 1 and terminates at vg α(e) . Note that e− = (e−1 )g α(e) . g We consider the bond wave function of the regular covering Gα of G. Let V (G) = {v1 , . . . , vp }, D(G) = 1 −1 α α α {e1 , . . . , eq , e− 1 , . . . , eq } and Γ = {g1 = 1, g2 , . . . , gm }. Let λ be a eigenvalue of L(G ) = −A(G ) + DG , and let

φ˜ = (φv1 ,g1 , . . . , φv1 ,gm , . . . , φvp ,g1 , . . . , φvp ,gm ) be the eigenvector corresponding to λ, where φvi ,gj corresponds to the vertex (vi , gj ) (1 ≤ i ≤ p; 1 ≤ j ≤ m) of Gα . Furthermore let bg = (vg , zg α(b) ) be any arc of Gα , where b = (v, z ) ∈ D(G), g ∈ Γ . Then the bond wave function of Gα is φbg (x) = abg eiπ x/4 + ab−g 1 e−iπ x/4 ,

x = ±1, i =

√ −1

under the condition

φbg (1) = φv,g and φbg (−1) = φz ,g α(b) . By (1), we have abg =

v

X

σbggch (λ)ach

t (ch )=vg

for each arc bg with o(bg ) = vg , where

 v σbggch (λ) = i δb−g 1 ch −

2

1

deg vg 1 − i(1 − λ/ deg vg )



.

˜ (λ) = (U (eg , fh ))eg ,f ∈D(Gα ) of Gα is given by Thus, the bond scattering matrix U h U ( eg , fh ) =

i(δe−1 f − xo(eg ) )



g

0

h

if t (fh ) = o(eg ), otherwise,

H. Mizuno, I. Sato / Discrete Mathematics 310 (2010) 782–791

785

where xvg =

1

2

for vg ∈ V (Gα ).

deg vg 1 − i(1 − λ/ deg vg )

Since δe−1 f = δe−1 f and xo(eg ) = xo(e) , we have g

o(eg ) eg fh

σ

h

= σefo(e) ,

i.e., U (eg , fh ) = Uef .

Therefore, it follows that



U (eg , fh ) =

i(δe−1 f − xo(e) ) 0

if t (fh ) = o(eg ), otherwise.

(2)

(g )

For g ∈ Γ , let the matrix Ag = (auv ) be defined by a(ugv) =

if α(u, v) = g and (u, v) ∈ D(G), otherwise.



1 0

Furthermore, let Ug = (U (g ) (e, f )) be given by U

(g )

i(δe−1 f − xo(e) ) 0

if t (f ) = o(e) and α(f ) = g , otherwise.



(e, f ) =

Let M1 ⊕ · · · ⊕ Ms be the block diagonal sum of square matrices M1 , . . . , Ms . If M1 = M2 = · · · = Ms = M, then we write N s ◦ M = 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 3. Let G be a connected graph with p vertices and q unoriented edges, Γ be a finite group and α : D(G) −→ Γ be an ordinary voltage assignment. Set |Γ | = m. Furthermore, let ρ1 = 1, ρ2 , . . . , ρk be the irreducible representations of Γ , and fi be the degree of ρi for each i, where f1 = 1. If the Γ -covering Gα of G is connected, then, for the bond scattering matrix of Gα ,

˜ (λ)) = det(I2q − U(λ)) det(I2qm − U

k Y

det I2qfi −

t X

i=2

=

ρ i ( h)

Uh

h

k 2qm ipm det(λIp + A(G) − D) Y p Q

!f i O

(dj − idj + λi)m

!f i det λIpfi +

X

ρi (h)

O

Ah − Ifi

O

D

.

h∈Γ

i=2

j =1

Proof. Let |Γ | = m. By Theorem 2, for the scattering matrix of Gα , we have

˜ (λ)) = det(I2qm − U

2qm ipm det(λIpm + A(Gα ) − DGα ) p Q

.

(dj − idj + λi)m

j =1

= {e1 , . . . , eq , eq+1 , . . . , e2q } and Γ = {1 = g1 , g2 , . . . , gm }. Arrange arcs of Gα in m blocks: (e1 , 1), . . . , (e2q , 1); (e1 , g2 ), . . . , (e2q , g2 ); . . . ; (e1 , gm ), . . . , (e2q , gm ). We consider the matrix U˜ (λ) under this order. For (h) h ∈ Γ , the matrix Ph = (pij ) is defined as follows:  1 if gi h = gj , (h) pij = 0 otherwise. Let D(G)

(h)

= 1, i.e., gj = gi h. Then U (egi , fgj ) 6= 0 if and only if t (f , gj ) = o(e, gi ). Furthermore, t (f , gj ) = o(e, gi ) if and only if (o(e), gi ) = o(e, gi ) = t (f , gj ) = (t (f ), gj α(f )). Thus, t (f ) = o(e) and α(f ) = gj−1 gi = gj−1 gj h−1 = h−1 . Now, by (2), Suppose that pij

˜ (λ) = U

X h∈Γ

Ph

O

Uh−1 =

X g ∈Γ

Pg −1

O

Ug =

X

t

Pg

O

Ug .

g ∈Γ

Here, note that Pg −1 = t Pg for each g ∈ Γ . Let ρ be the right regular representation of Γ . Furthermore, let ρ1 = 1, ρ2 , . . . , ρk be all inequivalent irreducible representations of Γ , and fi the degree of ρi for each i, where f1 = 1. Then we have ρ(g ) = Pg for g ∈ Γ . Furthermore, there exists a nonsingular matrix P such that P−1 ρ(g )P = (1)⊕ f2 ◦ρ2 (g )⊕· · ·⊕ fk ◦ρk (g ) for each g ∈ Γ (see [12]). Thus, we have t

Pt Pg t P−1 = (1) ⊕ f2 ◦ t ρ2 (g ) ⊕ · · · ⊕ fk ◦ t ρk (g ).

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H. Mizuno, I. Sato / Discrete Mathematics 310 (2010) 782–791

Putting F = (t P F=

X

N

˜ (λ)(t P I2q )U

−1 N

I2q ), we have

{(1) ⊕ f2 ◦ t ρ2 (g ) ⊕ · · · ⊕ fk ◦ t ρk (g )}

O

Ug .

g ∈Γ

Note that U(λ) =

Ug and 1 + f22 + · · · + fk2 = m. Therefore it follows that

P

g ∈Γ

˜ (λ)) = det(I2q − U(λ)) det(I2qm − U

k Y

det I2qfi −

t X

!f i ρi (g )

O

.

Ug

g

i=2

Next, let V (G) = {v1 , . . . , vp }. Arrange vertices of Gα in m blocks: (v1 , 1), . . . , (vp , 1); (v1 , g2 ), . . . , (vp , g2 ); . . . ; (v1 , gm ), . . . , (vp , gm ). We consider the adjacency matrix A(Gα ) under this order. (h) Suppose that pij = 1, i.e., gj = gi h. Then ((u, gi ), (v, gj )) ∈ D(Gα ) if and only if (u, v) ∈ D(G) and gj = gi α(u, v). If gj = gi α(u, v), then α(u, v) = gi−1 gj = gi−1 gi h = h. Thus we have X O A(Gα ) = Ph Ah . h∈Γ

Putting E = (P−1

Ip )A(Gα )(P

Ip ), we have O X {(1) ⊕ f2 ◦ ρ2 (h) ⊕ · · · ⊕ fk ◦ ρk (h)} Ah . E=

N

N

h∈Γ

Note that A(G) =

P

h∈Γ

Ah . Therefore it follows that

α

det(λIpm + A(G ) − DGα ) = det(λIp + A(G) − D)

k Y

!fi X

det λIpfi +

ρi (h)

O

Ah − Ifi

O

Ah − Ifi

O

D

.

h∈Γ

i=2

Hence, it follows that

˜ (λ)) = det(I2q − U(λ)) det(I2qm − U

k Y

!f i det I2qfi −

X

i=2

=

Uh

h

k 2qm ipm det(λIp + A(G) − D) Y p Q

ρi (h)

O

(dj − idj + λi)m

!f i det λIpfi +

X

ρi (h)

O

D

. 

h∈Γ

i=2

j =1

3. L-functions of graphs Let G be a connected graph with p vertices and q unoriented edges, Γ be a finite group and α : D(G) −→ Γ be an ordinary voltage assignment. Furthermore, let ρ be a unitary representation of Γ and d its degree. The L-function of G associated with ρ and α is defined by

! −1 ZS (G, λ, ρ, α) = det I2qd −

X

t

ρ(h)

O

Uh

.

h∈Γ

If ρ = 1 is the identity representation of Γ , then the L-function of G is a determinant on the bond scattering matrix of G. A determinant expression for the L-function of G associated with ρ and α is given as follows. For 1 ≤ i, j ≤ n, the (i, j)-block Fi,j of a dn × dn matrix F is the submatrix of K consisting of d(i − 1) + 1, . . . , di rows and d(j − 1) + 1, . . . , dj columns. Theorem 4. Let G be a connected graph with p vertices and q unoriented edges, Γ be a finite group and α : D(G) −→ Γ be an ordinary voltage assignment. If ρ is a representation of Γ and d is the degree of ρ , then the reciprocal of the L-function of G associated with ρ and α is ZS (G, λ, ρ, α)

−1

=

!

2qd ipd p Q

(dj − idj + λi)

det λIp + d

X

ρ(g )

O

Ag − Id

O

D .

g ∈Γ

j =1

Proof. The argument is an analogue of Bass’ method [2]. P Nt 1 Let D(G) = {e1 , . . . , eq , eq+1 , . . . , e2q } such that eq+i = e− ρ(g ))ef i (1 ≤ i ≤ q). Note that the (e, f )-block ( g ∈Γ Ug P Nt of g ∈Γ Ug ρ(g ) is given by

H. Mizuno, I. Sato / Discrete Mathematics 310 (2010) 782–791

! X

O

Ug

t

t

ρ(g )

=

g ∈Γ

ρ(α(f ))σefo(e)

if t (f ) = o(e), otherwise.

0d

ef

787

(g )

(g )

For g ∈ Γ , two 2q × 2q matrices Bg = (Bef )e,f ∈D(G) and Jg = (Jef )e,f ∈D(G) are defined as follows: if t (e) = o(f ) and α(e) = g , , otherwise



xt (e) 0

(g )

Bef =

(g )

Jef =



1 0

if f = e−1 and α(e) = g , otherwise.

Now for g ∈ Γ .

it Ug = Bg − Jg

Let K = (Kij )1≤i≤2q;1≤j≤p be the 2qd × pd matrix defined as follows:

 Kij :=

ρ(α(ei ))

if t (ei ) = vj , otherwise.

0d

Furthermore, we define two 2qd × pd matrices L = (Lij )1≤i≤2q;1≤j≤p and H = (Hij )1≤i≤2q;1≤j≤p as follows:

 Lij :=

if o(ei ) = vj , , otherwise.

xvj Id 0d

 Hij :=

Id 0d

if o(ei ) = vj , otherwise.

Now

X

Lt K =

Bh

O

t

Ag

O

ρ(g ),

t

ρ(h) = t B

(3)

h∈Γ

and t

X

HK =

(4)

g ∈Γ

where B=

X

Bg

O

ρ(g ).

g ∈Γ

Furthermore, t

HH = D

O

Id ,

(5)

t

where F is the transpose of F. If



xv1



0

..

X=



 ,

.

0

xvp

then

 O 

L=H X where Xd = X t

Id

N

= HXd ,

Id . By (4) and (5), we have

KL = t KHXd = t (t HK)Xd =

X

t

Ah X

O

t

ρ(h)

(6)

h∈Γ

and t

HL = t HHXd = DX

Next, let

 J=

0 N



M . 0

and T = B − J,

O

Id .

(7)

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H. Mizuno, I. Sato / Discrete Mathematics 310 (2010) 782–791

where M = xo(e−1 ) ρ(α(e1 )) ⊕ · · · ⊕ xo(e−1 ) ρ(α(eq )) q 1 and 1 −1 N = xo(e1 ) ρ(α(e− 1 )) ⊕ · · · ⊕ xo(eq ) ρ(α(eq )).

Now Lt H = t Tt Jρ + (xo(e1 ) Id ⊕ · · · ⊕ xo(e−1 ) Id ),

(8)

q

where Jρ = ρ(g ). g ∈ Γ Jg We introduce two (2q + p)d × (2q + p)d matrices as follows:

P

 P=

N

(1 − u2 )Ipd

−t K + u t H

0

I2qd





Ipd Q= uL

,

t

K − u tH . (1 − u2 )I2qd



By (6) and (7), we have PQ =

 (1 − u2 )Ipd − u t KL + u2 t HL

0 (1 − u2 )I2qd

uL



 X O O t t (1 − u2 )Ipd − u Ag X ρ(g ) + u2 DX Id = g ∈Γ



0

(1 − u2 )I2qd

uL

.

Furthermore, we have

 QP =

(1 − u2 )Ipd u(1 − u2 )L



0 . −uLt K + u2 Lt H + (1 − u2 )I2qd

Note that xo(e1 ) Id ⊕ · · · ⊕ xo(e−1 ) Id = t Jt Jρ q and (t Jρ )2 = I2qd . By (3) and (8), we have

−uLt K + u2 Lt H + (1 − u2 )I2qd = I2qd − u(t T + t J) + u2 (t Tt Jρ + t Jt Jρ − t Jρ t Jρ ) = (I2qd − u(t T + t J − t Jρ ))(I2qd − u t Jρ ). Thus,

 QP =

(1 − u2 )Ipd u(1 − u2 )L



0

(I2qd − u(t T + t J − t Jρ ))(I2qd − u t Jρ )

.

Since det(PQ) = det(QP), we have

(1 − u )

2 2qd

det Ipd − u

X

t

Ag X

O

t

 O  ρ(g ) + (DX − Ip ) I d u2

!

g ∈Γ

= (1 − u2 )pd det(I2qd − u(t T + t J − t Jρ )) det(I2qd − u t Jρ ). But, det(I2qd − u Jρ ) = det t



Iqd 0

uS Iqd



 det

 (1 − u2 )Iqd = det ∗

0 Iqd

Iqd −uS−1



 −uS Iqd

= (1 − u2 )qd ,

1 t −1 where S = t ρ(α(e− 1 )) ⊕ · · · ⊕ ρ(α(eq )). Therefore it follows that

! (1 − u )

2 2qd

det Ipd − u

X

t

Ag X

O

t

ρ(g ) + ((DX − Ip )

g ∈Γ

= (1 − u2 )(q+p)d det(I2qd − u(t T + t J − t Jρ )).

O

Id ) u

2

H. Mizuno, I. Sato / Discrete Mathematics 310 (2010) 782–791

789

Hence

! 2 (q−p)d

det(I2qd − u( B − Jρ )) = (1 − u ) t

t

det Ipd − u

X

t

Ag X

O

t

ρ(g ) + ((DX − Ip )

O

Id )u

2

,

g ∈Γ

i.e.,

! ! X O X O O t t t 2 (q−p)d 2 det I2qd − u ( Bh − Jh ) ρ(h) = (1 − u ) det Ipd − u ρ(g ) XAg + (Id (DX − Ip ))u . g ∈Γ

h

Letting u = −i,

! det I2qd −

X

t

ρ(g )

O

Ug

! =2

(q−p)d

det 2Ipd + i

g ∈Γ

X

ρ(g )

O

XAg − Id

O

DX

g ∈Γ

=2

(q−p)d pd

i

 O 

det Id

X det 2i

−1

 O  −1 Id

X

! +

X

ρ(g )

O

Ag + iId

O

D .

g ∈Γ

Since

 O 

det Id

X =

!d

p Y

2

j =1

dj − idj + λi

and

 O −1

2i−1 Id

X

O

= Id

(−i(1 − i)D + λIp ),

it follows that

! ZS (G, λ, ρ, α)

−1

= det I2qd −

X

t

ρ(g )

O

Ug

g ∈Γ

=

!

2qd ipd p Q

det λIpd +

(dj − idj + λi)

d

X

ρ(g )

O

Ag − Id

O

D . 

g ∈Γ

j =1

By Theorems 3 and 4, the following result holds. Corollary 1. If G is a connected graph with q edges, Γ is a finite group and α : D(G) −→ Γ is an ordinary voltage assignment, then we have

˜ (λ)) = det(I2qm − U

Y

ZS (G, λ, ρ, α)− deg ρ ,

ρ

where ρ runs over all inequivalent irreducible representations of Γ and m = |Γ |. 4. The Euler product for the L-function ZS (G , λ, ρ, α) of a graph We present the Euler product for the L-function of a graph introduced in Section 3. Foata and Zeilberger [5] gave a new proof of Bass’ Theorem by using the algebra of Lyndon words. Let X be a finite nonempty set, < a total order in X , and X ∗ the free monoid generated by X . Then the total order < on X derives the lexicographic order <∗ on X ∗ . A Lyndon word in X is defined to a nonempty word in X ∗ that is prime (not the power lr of any other word l for any r ≥ 2) and that is also minimal in the class of its cyclic rearrangements under <∗ (see [11]). Let L denote the set of all Lyndon words in X . Foata and Zeilberger [5] gave a short proof of Amitsur’s identity [1]. Theorem 5 (Amitsur). For square matrices A1 , . . . , Ak , det(I − (A1 + · · · + Ak )) =

Y

det(I − Al ),

l∈L

where the product runs over all Lyndon words in {1, . . . , k}, and Al = Ai1 · · · Air for l = i1 · · · ir . Theorem 6. Let G be a connected graph with p vertices and q unoriented edges, Γ be a finite group and α : D(G) −→ Γ be an ordinary voltage assignment. For each path P = (e1 , . . . , en ) of G, set α(P ) = α(e1 ) · · · α(en ). If ρ is a representation of Γ and d is the degree of ρ , then

790

H. Mizuno, I. Sato / Discrete Mathematics 310 (2010) 782–791

ZS (G, λ, ρ, α) =

Y

det(Id − t ρ(α(C ))aC (λ))−1 ,

[C ]

where [C ] runs over all equivalence classes of prime cycles of G, and aC (λ) = σeo1(een1 ) σeon(eenn−) 1 · · · σeo2(ee12 ) ,

C = (e1 , e2 , . . . , en ).

Proof. At first, let D(G) = {e1 , . . . , eq , eq+1 , . . . , e2q } and consider the lexicographic order on D(G) × D(G) derived from a total order of D(G): e1 < e2 < · · · < e2q . If (ei , ej ) is the mth pair under the above order, then we define the 2qd × 2qd matrix Tm = ((Tm )r ,s )1≤r ,s≤2q as follows:

(Tm )r ,s =

t

ρ(α(ej ))σeoi(eej i )

0

if r = ei , s = ej and o(ei ) = t (ej ), otherwise,

where

σefo(e) = i(δe−1 f − xo(e) ). If F = T1 + · · · + Tk and k = 4q2 , then F=

X

Uh

O

t

ρ(h).

h

Let L be the set of all Lyndon words in D(G) × D(G). We can also consider L as the set of all Lyndon words in {1, . . . , k}: (ei1 , ej1 ) · · · (eis , ejs ) corresponds to m1 m2 · · · ms , where (eir , ejr )(1 ≤ r ≤ s) is the mr th pair. Theorem 5 implies that Y det(I2qd − Tt ), det(I2qd − F) = t ∈L

where Tt = Ti1 · · · Tir for t = i1 · · · ir . Note that det(I2qd − Tt ) is the alternating sum of the diagonal minors of Tt . Thus, we have det(I − t ρ(α(C ))aC (λ)) 1



det(I − Tt ) =

if t is a prime cycle C , otherwise,

where aC (λ) = σeo1(een1 ) σeon(eenn−) 1 · · · σeo2(ee12 ) ,

C = (e1 , e2 , . . . , en ).

Therefore, it follows that

! ZS (G, λ, ρ, α)

−1

= det I2qd −

X

t

ρ(h)

O

Uh

h∈Γ

! = det I2qd −

X

Uh

O

t

ρ(h) =

h∈Γ

Y

(Id − t ρ(α(C ))aC (λ)),

[C ]

where [C ] runs over all equivalence classes of prime cycles of G.



5. Example 1 −1 −1 We give an example. Let G = K3 be the complete graph with three vertices v1 , v2 , v3 and six arcs e1 , e2 , e3 , e− 1 , e2 , e3 , where e1 = (v1 , v2 ), e2 = (v2 , v3 ), e3 = (v3 , v1 ). Then we have

xv1 = xv2 = xv3 =

2 2 − 2i + λi

.

1 −1 −1 Set a = 2−2i2+λi . Considering U(λ) under the order e1 , e2 , e3 , e− 1 , e2 , e3 , we have

0 − ia   0  U(λ) =  i(1 − a) 0 0



0 0 −ia 0 i(1 − a) 0

−ia 0 0 0 0

i(1 − a)

i(1 − a) 0 0 0 0 −ia

0

i(1 − a) 0 −ia 0 0

0 0



i(1 − a) .  0  −ia 0



H. Mizuno, I. Sato / Discrete Mathematics 310 (2010) 782–791

791

But, 0 1 1

" A(K3 ) =

1 0 1

1 1 , 0

#

D = I3 .

By Theorem 2, we have det(I6 − U(λ)) =

26 i3

(2 − 2i + λi)3

det(λI3 + A(K3 ) − D) =

2 6 i3

(2 − 2i + λi)3

λ(λ − 3)2 .

Next. let Γ = Z3 = {1, τ , τ 2 }(τ 3 = 1) be the cyclic group of order 3, and let α : D(K3 ) −→ Z3 be the ordinary voltage 1 −1 −1 2 assignment such that α(e1 ) = τ , α(e− 1 ) = τ and α(e2 ) = α(e2 ) = α(e3 ) = α(e3 ) = 1. The characters of Z3 are given √

as follows: χi (τ j ) = (ξ i )j , 0 ≤ i, j ≤ 2, where ξ = −1+2 −3 . But, we have

" A1 =

0 0 1

0 0 1

1 1 , 0

#

0 0 0

1 0 0

" Aτ =

0 0 , 0

#

" Aτ 2 =

0 1 0

0 0 0

0 0 . 0

#

Now, by Theorem 4,

ζS (K3 , λ, χ1 , α)

−1

2 6 i3

=

det λI3 +

(2 − 2i + λi)3  λ−2 ξ =  ξ2 λ−2 1

1

2 X

! χ1 (τ )Aτ i − D i

i =0

1 1

λ−2

 =

26 i3

(2 − 2i + λi)3

(λ3 − 6λ2 + 9λ − 3).

Similarly, we have

ζS (K3 , λ, χ2 , α)−1 =

26 i3

(2 − 2i + λi)3

(λ3 − 6λ2 + 9λ − 3).

By Corollary 1, it follows that

˜ (λ)) = det(I6 − U(λ))ζS (K3 , λ, χ1 , α)−1 ζS (K3 , λ, χ2 , α)−1 det(I18 − U =

218 i

(2 − 2i + λi)9

λ(λ − 3)2 (λ3 − 6λ2 + 9λ − 3)2 .

Acknowledgments We would like to thank the referee for many valuable comments and many helpful suggestions. The second author was supported by Grant-in-Aid for Science Research (C). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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