A new weighted Ihara zeta function for a graph

Linear Algebra and its Applications 571 (2019) 154–179

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Linear Algebra and its Applications www.elsevier.com/locate/laa

A new weighted Ihara zeta function for a graph Norio Konno a , Hideo Mitsuhashi b , Hideaki Morita c , Iwao Sato d,∗ a

Department of Applied Mathematics, Faculty of Engineering, Yokohama National University, Hodogaya, Yokohama 240-8501, Japan b Department of Applied Informatics, Faculty of Science and Engineering, Hosei University, Koganei, Tokyo, 184-8584, Japan c Division of System Engineering for Mathematics, Muroran Institute of Technology, Muroran, Hokkaido 050-8585, Japan d National Institute of Technology, Oyama College, Oyama, Tochigi 323-0806, Japan

a r t i c l e

i n f o

Article history: Received 13 November 2018 Accepted 24 February 2019 Available online 28 February 2019 Submitted by R. Brualdi MSC: 05C50 15A15 Keywords: Zeta function Non-backtracking random walk Transition probability matrix Spectra

a b s t r a c t We define a new weighted Ihara zeta function of a graph G, and present its determinant expression. We present a decomposition formula for the new weighted Ihara zeta function of a regular covering of G. Furthermore, we introduce a new weighted Ihara L-function of G, and give determinant expressions of it. As a corollary, we present a decomposition formula for the new weighted Ihara zeta function of a regular covering of G by its new weighted Ihara L-functions. As applications, we give new proofs for the results of Kempton on the spectrum of the transition probability matrices for nonbacktracking random walks for regular graphs and semiregular bipartite graphs. © 2019 Elsevier Inc. All rights reserved.

* Corresponding author. E-mail addresses: [email protected] (N. Konno), [email protected] (H. Mitsuhashi), [email protected] (H. Morita), [email protected] (I. Sato). https://doi.org/10.1016/j.laa.2019.02.022 0024-3795/© 2019 Elsevier Inc. All rights reserved.

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1. Introduction Zeta functions of graphs started from the Ihara zeta functions of 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 [19,20]. Hashimoto [7] generalized Ihara’s result on the Ihara zeta function of a regular graph to an irregular graph, and showed that its reciprocal is again a polynomial by a determinant containing the edge matrix. Bass [3] presented another determinant expression for the Ihara zeta function of an irregular graph by using its adjacency matrix. For weighted zeta functions of graphs, Hashimoto [8] defined a weighted zeta function of a graph with weights indicated to its edges. Stark and Terras [17] introduced the edge zeta function of a graph with weights indicated to its arcs, and gave a determinant expression for the edge zeta function. Mizuno and Sato [15] defined the first weighted zeta function of a graph as a specialization of the edge zeta function, and gave its determinant expression. A non-backtracking random walk on a graph is closely related to the Ihara zeta function of a graph (see [2,10]). Angel, Friedman and Hoory [2] studied the non-backtracking spectrum of the universal cover of a graph. The non-backtracking spectrum of a graph is exactly the spectrum of its edge matrix. Fitzner and Hofstad [4] treated convergence of non-backtracking random walks on lattices and tori. Krzakala et al. [12] used the edge matrix to study spectral clustering algorithms. Alon, Benjamini, Lubetzky and Sodin [1] treated the mixing rate of a non-backtracking random walk for regular graphs, and proved that a non-backtracking random walk on a regular graph has a faster mixing rate than a random walk allowing backtracking. Kempton [10] proved a weighted version of Ihara’s formula, and obtained the spectrum of the transition probability matrix for a non-backtracking random walk for regular and semiregular bipartite graphs. Furthermore, Kempton [10] gave another proof of the result of Alon et al. [1] on the mixing rate of a non-backtracking random walk on a regular graph, and generalized this result to a semiregular bipartite graph. In this paper, we introduce a new weighted Ihara zeta function of a graph, and present new proofs for the results of Kempton [10]. In Section 2, we review for the Ihara zeta function and the weighted zeta functions of a graph G and a non-backtracking random walk on a graph. Furthermore, we state a weighted version of Ihara’s formula by Kempton. In Section 3, we define a new weighted Ihara zeta function of a graph, and present its determinant expression. In Section 4, we present a decomposition formula for the new weighted Ihara zeta function of a regular covering of G. In Section 5, we introduce a new weighted Ihara L-function of G, and present a determinant expression of it. As a corollary, we present a decomposition formula for the new weighted Ihara zeta function of a regular covering of G by its new weighted Ihara L-functions. In Section 6, we present the Euler product for the new weighted Ihara zeta function of a graph. In Section 8, we give another proof for the result of Kempton on

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the spectrum of the transition probability matrix for a non-backtracking random walk for a regular graph. In Section 9, we present another proof for the result of Kempton on the spectrum of the transition probability matrix for a non-backtracking random walk for a semiregular bipartite graph. 2. Preliminaries 2.1. Zeta functions of graphs Graphs and digraphs treated here are finite. Let G be a connected graph and DG the symmetric digraph corresponding to G. 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 of e = (u, v). For v ∈ V (G), the degree deg G v = deg v = dv is the number of vertices adjacent to v in G. A path P of length n in G is a sequence P = (e1 , · · · , en ) of n arcs such that ei ∈ D(G), t(ei ) = o(ei+1 )(1 ≤ i ≤ n − 1). If ei = (vi−1 , vi ) for i = 1, · · · , n, then we write P = (v0 , v1 , · · · , vn−1 , vn ). Set | P |= n, o(P ) = o(e1 ) and t(P ) = t(en ). Also, P is called an (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. −1 The inverse cycle of a cycle C = (e1 , · · · , en ) is the cycle C −1 = (e−1 n , · · · , e1 ). We introduce an equivalence relation between cycles. Two cycles C1 = (e1 , · · · , em ) and C2 = (f1 , · · · , fm ) are called equivalent if there exists a positive number k such that fj = ej+k for all j, where the subscripts are considered by modulo m. The inverse cycle of C is in general 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 at a vertex v of G. The Ihara(-Selberg) zeta function of G is defined by Z(G, u) =



(1 − u|C| )−1 ,

[C]

where [C] runs over all equivalence classes of prime, reduced cycles of G. Let G be a connected graph with n vertices and m edges. Then two 2m × 2m matrices B = B(G) = (Be,f )e,f ∈D(G) and J0 = J0 (G) = (Je,f )e,f ∈D(G) are defined as follows:  Be,f =

 1 if t(e) = o(f ), , Je,f = 0 otherwise

The matrix B − J0 is called the edge matrix of G.

1 0

if f = e−1 , otherwise.

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Theorem 1 (Hashimoto; Bass). Let G be a connected graph with n vertices and m edges. Then the reciprocal of the Ihara zeta function of G is given by Z(G, u)−1 = det(I2m − u(B − J0 )) = (1 − u2 )m−n det(In − uA(G) + u2 (DG − In )), where DG = (dij ) is the diagonal matrix with dii = deg G vi (V (G) = {v1 , · · · , vn }). The first identity in Theorem 1 was also obtained by Hashimoto [7]. Bass [3] proved the second identity by using a linear algebraic method. Stark and Terras [17] gave an elementary proof of this formula, and discussed three different zeta functions of any graph. Various proofs of Bass’ Theorem were given by Kotani and Sunada [11], and Foata and Zeilberger [5]. 2.2. The weighted zeta functions of a graph Stark and Terras [17] defined the edge zeta function of a graph G with n vertices and m edges. Let G be a connected graph and D(G) = {e1 , . . . , em , em+1 , . . . , e2m }(em+i = e−1 i (1 ≤ i ≤ m)). We introduce 2m variables u1 , . . . , u2m , and set g(C) = ui1 · · · uik for each cycle C = (ei1 , . . . , eik ) of G. Then the edge zeta function ζ G (u) of G is defined by ζ G (u) =



(1 − g(C))−1 ,

[C]

where [C] runs over all equivalence classes of prime, reduced cycles of G. Theorem 2 (Stark and Terras). Let G be a connected graph with m edges. Then ζ G (u)−1 = det(I2m − (B − J0 )U), where U is the diagonal matrix U = diag(u1 , . . . , u2m ), D(G) = {e1 , . . . , em , em+1 , . . . , e2m } (em+j = e−1 j (1 ≤ j ≤ m). Mizuno and Sato [15] defined the first weighted zeta function of a graph G, and give a determinant expression of it. Let G be a connected graph and V (G) = {v1 , · · · , vn }. Then we consider an n × n matrix W(G) = (wij )1≤i,j≤n with ij entry the complex variable wij if (vi , vj ) ∈ D(G), and wij = 0 otherwise. The matrix W(G) is called the weighted matrix of G. Set w(vi , vj ) = wij , vi , vj ∈ V (G) and w(e) = wij , e = (vi , vj ) ∈ D(G). Furthermore, assume that w(e−1 ) = w(e)−1 for each e ∈ D(G). For each path P = (e1 , · · · , er ) of G, the norm w(P ) of P is defined as follows: w(P ) = w(e1 ) · · · w(er ). The first weighted zeta function of G is defined by Z(G, w, u) =

 [C]

(1 − w(C)u|C| )−1 ,

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where [C] runs over all equivalence classes of prime, reduced cycles of G. Note that the first weighted zeta function is the edge zeta function such that ui = w(ei )u, 1 ≤ i ≤ 2m, where D(G) = {e1 , . . . , e2m } and em+j = e−1 (1 ≤ j ≤ m). j Theorem 3 (Mizuno and Sato). Let G be a connected graph, and let W = W(G) be the weighted matrix of G. Then the reciprocal of the weighted zeta function of G is given by Z(G, w, u)−1 = (1 − u2 )m−n det(I − uW(G) + u2 (DG − I)), where n =| V (G) | and m =| E(G) |. 2.3. A non-backtracking random walk on a graph Let G be a connected graph with n vertices and m edges. Then a non-backtracking random walk on G is a sequence (v0 , v1 , . . . , vk ) of vertices vi ∈ V (G), where vi+1 is chosen randomly among the neighbors of vi such that vi+1 = vi−1 for i = 1, . . . , k − 1. A non-backtracking random walk on G is not a Markov chain since, in any given state, we need to remember the previous step in order to take the next step. Let P(k) be the n × n transition probability matrix for a k-step non-backtracking random walk on the vertices. Note that P(1) = P = D−1 A is the transition probability matrix for the simple random walk on G. In general, it is not easy to calculate the P(k) since a non-backtracking random walk is not a Markov chain. Thus, we change the state space from the vertices of G to the arcs of the symmetric digraph DG corresponding to G. A non-backtracking random walk is a sequence (e1 , e2 , . . . , ek ) such that vk = vr and vs = vj if ei = (vj , vk ) and ei+1 = (vr , vs ).  be the 2m × 2m transition probability matrix for this process. That is, Let P   P((u, v), (x, y)) =

1/(dv − 1) if v = x and y = u, 0 otherwise

˜ k is the transition probability matrix for a non-backtracking random walk with Then P k steps on the arcs. Next, we state a weighted version of Ihara’s formula by Kempton [10]. Let G be a connected graph with n vertices and m edges. Furthermore, let w : V (G) −→ R≥0 be a function, and let an n × n matrix W = (wuv )u,v∈V (G) be defined as follows:  w(u) if u = v, wuv = 0 otherwise.  = P(G)   = J(G)  Then, let two 2m × 2m matrices P = (Pe,f )e,f ∈D(G) and J = (Je,f )e,f ∈D(G) are defined as follows:

N. Konno et al. / Linear Algebra and its Applications 571 (2019) 154–179

 Pe,f =

w(t(e))2 0

if t(e) = o(f ) and f = e−1 , otherwise,

 Je,f =

w(t(e))2 0

159

if f = e−1 , otherwise.

 = (auv )u,v∈V (G) and D ˜ = (duv )u,v∈V (G) be defined Furthermore, two n ×n matrices A as follows:  w(u)w(v) if (u, v) ∈ D(G), auv = 0 otherwise,  2 2 if u = v, o(e)=u w(u) w(t(e))) d˜uv = 0 otherwise. Kempton [10] presented the following formula for the determinants of four matrices  J,  A  and D.  P, Theorem 4 (Kempton). Let G be a connected graph with n vertices and m edges, and let  J,  A  and D  are defined w : V (G) −→ R≥0 be a function. Furthermore, four matrices P, as above. Then  2m − uJ)  + u2 J  + u2 D). 2 ) = det(In − uA  det((I2m − uP)(I By this Theorem, Kempton [10] obtained the spectrum of the transition probability matrices for non-backtracking random walks for regular graphs and semiregular bipartite graphs. 3. A weighted Ihara zeta function of a graph Let G be a connected graph with n vertices and m edges. Furthermore, let w : V (G) −→ C be a function, and let an n × n matrix W = (wuv )u,v∈V (G) be defined as follows:  w(u) if u = v, wuv = 0 otherwise. Then we define a new weighted Ihara zeta function of a graph: ˜ −1 . ζ w (G, u) = det(I2m − uP) If w = 1, i.e., w(v) = 1 for each v ∈ V (G), then the new weighted Ihara zeta function is the Ihara zeta function. Let G be a connected graph with n vertices and m edges, let u be a complex variable, and let w : V (G) −→ C be a function. Then two n × n matrices A = A(G) = (Axy )x,y∈V (G) and D = D(G) = (dxy )x,y∈V (G) be defined as follows:

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 Axy =

w(x)w(y)/(1 − u2 w(x)2 w(y)2 ) if (x, y) ∈ D(G), 0 otherwise,

 dxy =

o(e)=x

w(x)2 w(t(e)))2 /(1 − u2 w(x)2 w(t(e))2 )

0

if x = y, otherwise.

Note that A and D are matrices with indeterminants as entries. A determinant expression for a new weighted Ihara zeta function ζ w (G, u) of G is given as follows. Theorem 5. Let G be a connected graph with n vertices and m edges, and let w : V (G) −→ ± C be a function. Set D(G) = {e± 1 , . . . , em }. Then the reciprocal of the new weighted Ihara zeta function ζ w (G, u) of G is −1

ζ w (G, u)

=

m 

(1 − u2 w(o(ej ))2 w(t(ej ))2 ) det(In − uA + u2 D).

j=1

Proof. The argument is an analogue of the method of Watanabe and Fukumizu [21]. −1 Let V (G) = {v1 , . . . , vn } and D(G) = {e1 , . . . , em , e−1 1 , . . . , em }. Arrange arcs of G −1 −1 as follows: e1 , e1 , . . . , em , em . Furthermore, arrange vertices of G as follows: v1 , . . . , vn .  =  = Now, we define two 2m × n matrices S (Sev )e∈D(G);v∈V (G) and T  (Tev )e∈D(G);v∈V (G) as follows:  Sev :=

 w(v) if t(e) = v, 0 otherwise,

w(v) 0

Tev :=

if o(e) = v, otherwise.

 and T  under the above order. Here we consider two matrices S Furthermore, we define two 2m × n matrices S = (Sev )e∈D(G);v∈V (G) and T = (Tev )e∈D(G);v∈V (G) as follows:  Sev :=

 1 if t(e) = v, 0 otherwise,

Tev :=

1 if o(e) = v, 0 otherwise.

Then we have  =S  tT  − J,  S  = SW, T  = TW. P Thus, we have   = SWW t T − J. P Therefore, it follows that

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ζ w (G, u)−1 = = = =

161

 det(I2m − uP)  tT  − J))  det(I2m − u(S t  − uS  T)  det(I2m + uJ t   −1 ) det(I2m + uJ).  det(I2m − uS T(I2m + uJ)

Let A and B be an m × n and n × m matrix, respectively. Then we have det(Im − AB) = det(In − BA). Thus, we have  −1 S)   2m + uJ)  det(I2m + uJ). ζ w (G, u)−1 = det(In − u t T(I But, we have  = det(I2m + uJ)

m 

(1 − u2 w(o(ej ))2 w(t(ej ))2 ).

j=1

Let xej = xj = 1 − u2 w(o(ej ))2 w(t(ej ))2 (1 ≤ j ≤ n). Then we have ⎡

 −1 (I2m + uJ)

1/x1 ⎢ −uw(o(e1 ))2 /x1 =⎣

−uw(t(e1 ))2 /x1 1/x1

0

⎤ 0 0 ⎥ ⎦. .. .

Thus, for (u, v) ∈ D(G),  −1 S)  2m + uJ)  uv = w(u)w(v)/(1 − u2 w(u)2 w(v)2 ). (t T(I Furthermore, for each v ∈ V (G),  −1 S)  2m + uJ)  vv = −u (t T(I



w(v)2 w(t(e))2 /(1 − u2 w(v)2 w(t(e))2 ).

o(e)=v

Hence, ζ w (G, u)−1 =

m 

(1 − u2 w(o(ej ))2 w(t(ej ))2 ) det(In − uA + u2 D).

j=1

By the proof of Theorem 5, we obtain Theorem 4 by Kempton.

2

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Corollary 1 (Kempton). Let G be a connected graph with n vertices and m edges, and let  J,  A  and D  are defined w : V (G) −→ R≥0 be a function. Furthermore, four matrices P, as above. Then  2m − uJ)  + u2 J  + u2 D). 2 ) = det(In − uA  det((I2m − uP)(I −1 Proof. Let V (G) = {v1 , . . . , vn } and D(G) = {e1 , . . . , em , e−1 1 , . . . , em }. Arrange arcs −1 −1 of G as follows: e1 , e1 , . . . , em , em . Furthermore, arrange vertices of G as follows: v1 , . . . , vn .  = (S˜ev )e∈D(G);v∈V (G) and T  = Now, we define two 2m × n matrices S (T˜ev )e∈D(G);v∈V (G) as Theorem 5. Furthermore, we define two 2m × n matrices S = (Sev )e∈D(G);v∈V (G) and T = (Tev )e∈D(G);v∈V (G) as Theorem 5. Then we have

= = = = =

 2m − uJ)  + u2 J 2 ) det((I2m − uP)(I t 2 2  T   − J))(I  det((I2m − u(S 2m − uJ) + u J )  t T(I  + u2 J  2 − uS  2m − uJ) ˜2) det(I2m − u2 J t  T(I   2m − uJ)) det(I2m − uS t   det(In − u T(I2m − uJ)S) S  + u2 t T J S). ˜ det(In − u t T

But, we have t

 and TS = A

t  

 TJS = D.

Therefore, it follows that  2m − uJ)  + u2 J  + u2 D). 2 ) = det(In − uA  det((I2m − uP)(I

2

If w = 1, i.e., w(v) = 1 for each v ∈ V (G), then Theorem 5 implies the second formula of Theorem 1. Corollary 2. Let G be a connected graph with n vertices and m edges. Then the reciprocal of the Ihara zeta function of G is given by Z(G, u)−1 = (1 − u2 )m−n det(In − uA(G) + u2 (DG − In )). Proof. If w = 1, then we have A= Thus,

1 1 A(G) and D = DG . 2 1−u 1 − u2

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Z(G, u)−1 = ζ 1 (G, u)−1 = (1 − u2 )m det(In −

u 1−u2 A(G)

u2 1−u2 DG ) u2 (DG − In )).

+

= (1 − u2 )m−n det(In − uA(G) +

2

4. Weighted Ihara zeta functions of regular coverings Let G be a connected graph and Γ a finite group. Then a mapping α : D(G) −→ Γ is called an ordinary voltage assignment if α(v, u) = α(u, v)−1 for each (u, v) ∈ D(G). 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) × Γ 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 [6]). The graph Gα is called a derived graph covering of G with voltages in Γ or a Γ-covering of G. The Γ-covering Gα is a | Γ |-fold regular covering of G. Furthermore, every regular covering of a graph G is a Γ-covering of G for some group Γ (see [6]). 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) ∈ E(G), the arc eg emanates from ug and terminates at vgα(e) . Note that −1 e−1 )gα(e) . g = (e Let G be a connected graph with n vertices and m edges, and let w be a vertex weight of G. Then we define the vertex weight w ˜ : V (Gα ) −→ R+ derived from w as follows: w(u, ˜ g) = w(u) , (u, g) ∈ V (Gα ).  α) = Furthermore, let |Γ| = r. Then we have two 2mr × 2mr matrices P(G  α ) = (Je f )e ,f ∈D(Gα ) are defined as follows: (Peg fh )eg ,fh ∈D(Gα ) and J(G g h g h  P eg f h =

w(t(e))2 0

if t(eg ) = o(fh ) and fh = e−1 g , otherwise,

 Jeg fh =

w(t(e))2 0

if fh = e−1 g , otherwise.

Note that t(eg ) = o(fh ) if and t(e) = o(f ) and h = α(u, v). Also, fh = e−1 g if and only if −1 f = e and h = α(u, v). For g ∈ Γ, let the matrix n × n matrices Ag = (Axy,g )x,y∈V (G) be defined as follows:  Axy,g =

w(x)w(y)/(1 − u2 w(x)2 w(y)2 ) 0

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

 g = (P (g) )e ,f ∈D(Gα ) and Furthermore, for g ∈ Γ, let the matrices 2m × 2m matrices P eh f l h l g = (J (g) )e ,f ∈D(Gα ) be defined as follows: J eh fl

h

l

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(g) P eh f l

=

(g)

Jeh fl

w(t(e))2 if t(eg ) = o(fh ) and α(e) = g, 0 otherwise,  w(t(e))2 if fh = e−1 g and α(e) = g, = 0 otherwise.

If M1 , M2 , · · · , Ms are square matrices, then let M1 ⊕ · · · ⊕ Ms be the block diagonal sum of M1 , · · · , Ms and if M1 = M2 = · · · = Ms = M, then we write 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. A determinant expression for a new weighted zeta function ζ w˜ (Gα , u) of Gα is given as follows. Theorem 6. Let G be a connected graph with n vertices and m edges, Γ a finite group, α : D(G) −→ Γ an ordinary voltage assignment, and w : V (G) −→ C a function. Set ± |Γ| = r and D(G) = {e± 1 , . . . , em }. 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 the reciprocal of a new weighted Ihara zeta function ζ w˜ (Gα , u) of Gα is k 

 fi ζ w˜ (Gα , u)−1 = ζ w (G, u)−1 · i=2 det(I2mfi − u g∈Γ ρi (g) P g)   k m = ζ w (G, u)−1 i=2 { j=1 (1 − u2 w(o(ej ))2 w(t(ej ))2 )fj 

× det(Infj − u g∈Γ ρi (g) Ag + u2 Ifj D(G))}fj . Proof. By Theorem 5, we have  α )) ζ w˜ (Gα , u)−1 = det(I2mr − uP(G =

m 

(1 − u2 w(o(ej ))2 w(t(ej ))2 )r det(Inr − uA(Gα ) + u2 D(Gα )).

j=1

At first, let D(G) = {e1 , · · · , e2m } such that em+j = e−1 j (1 ≤ j ≤ m). Furthermore, let Γ = {1 = g1 , g2 , · · · , gr }. Arrange arcs of Gα in r blocks: (e1 , 1), · · · , (e2m , 1); (e1 , g2 ), · · · , (e2m , g2 ); · · · ; (e1 , gr ), · · · , (e2m , gr ). We consider two matrices  α ) and J(G  α ) under this order. P(G (g) For g ∈ Γ, the matrix Pg = (phl )h,l∈Γ is defined as follows:  (g) phl

(g)

=

1 0

if l = hg, otherwise.

Suppose that phl = 1, i.e., l = hg. Then (t(e), hα(e)) = t(e, h) = o(f, l) = (o(f ), l) if and only if t(e) = o(f ) and l = hα(e), i.e., α(e) = h−1 l = h−1 hg = g. Furthermore, fl = e−1 h if and only if f = e−1 or l = hα(e) = hg. Thus, if t(eh ) = o(fl ) and fl = e−1 h , then we have

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165

 α ))e ,f = w(t(eh ))2 = w(t(e))2 . (P(G h l Therefore, we have  α) = P(G



Pg



 g. P

g∈Γ

Let ρ be the right regular representation of Γ. 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) f or each g ∈ Γ, where d is the number of irreducible representations of Γ(see [16]). Putting F = (P−1



˜ α )(P I2m )P(G



I2m ),

we have F=



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



 g. P

g∈Γ

  ˜ g and 1 + f 2 + · · · + f 2 = r. Therefore, it follows that Note that P(G) = g∈Γ P 2 k  α )) ζ w˜ (Gα , u)−1 = det(I2mr − uP(G  k det(I2mf − u  ρi (g) P  g )fi = det(I2m − uP) i g i=2 

 k  g )fi . = ζ w˜ (G, u)−1 i=2 det(I2mfi − u g ρi (g) P Next, set V (G) = {v1 , · · · , vn }. Arrange vertices of Gα in r blocks: (v1 , 1), · · · , (vn , 1); (v1 , g2 ), · · · , (vn , g2 ); · · · ; (v1 , gr ), · · · , (vn , gr ). We consider two matrices A(Gα ) and D(Gα ) under this order. (g) Suppose that pij = 1, i.e., gj = gi g. Then ((u, gi ), (v, gj )) ∈ D(Gα ) if and only if (u, v) ∈ D(G) and gi α(u, v) = gj , i.e., α(u, v) = gi−1 gj = g. Thus we have A(Gα ) =



Pg



Ag .

g∈Γ

Putting E = (P−1 we have



In )A(Gα )(P



In ),

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N. Konno et al. / Linear Algebra and its Applications 571 (2019) 154–179

E=



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



Ag .

g∈Γ

Furthermore, we have D(Gα ) = Ir Note that A(G) =

= = × × = ×

 g∈Γ



D(G).

Ag and 1 + f22 + · · · + fk2 = q. Therefore, it follows that

ζ w˜ (Gα , u)−1 m 2 2 2 r α 2 α j=1 (1 − u w(o(ej )) w(t(ej )) ) det(Inr − uA(G ) + u D(G )) m 2 2 2 2 j=1 (1 − u w(o(ej )) w(t(ej )) ) det(In − uA + u D) k m 2 2 2 fj i=2 { j=1 (1 − u w(o(ej )) w(t(ej )) ) 

det(Infi − u g∈Γ ρi (g) Ag + u2 Ifi D(G))}fi   k m ζ w (G, u)−1 i=2 { j=1 (1 − u2 w(o(ej ))2 w(t(ej ))2 )fi 

det(Infi − u g∈Γ ρi (g) Ag + u2 Ifi D(G))}fi . 2

In the case of w = 1, we obtain a decomposition formula for the Ihara zeta function of a regular covering of a graph by Stark and Terras [18], and independently, Mizuno and Sato [14]. Corollary 3 (Stark and Terras; Mizuno and Sato). Let G be a connected graph with n vertices and m edges, Γ a finite group, α : D(G) −→ Γ an ordinary voltage assignment. 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 the reciprocal of the Ihara zeta function Z(Gα , u) of Gα is Z(Gα , u)−1 = Z(G, u)−1 ·

k 



{(1 − u2 )(l−n)fi det(Ifi n − u(

i=2

ρi (g)



Ag ) + u2 Ifi



g∈A (g)

where the matrix n × n matrices Ag = (Axy )x,y∈V (G) be defined as follows:  (g) Axy

=

1 0

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

for g ∈ Γ, Proof. Similar to the proof of Corollary 2.

2

Q)}fi ,

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5. L-functions of graphs Let G be a connected graph with m edges, Γ a finite group, α : D(G) −→ Γ an ordinary voltage assignment, and w a vertex weight of G. For each path P = (e1 , · · · , er ) of G, set α(P ) = α(e1 ) · · · α(er ). This is called the net voltage of P . Furthermore, let ρ be a representation of Γ and d its degree. The new weighted Ihara L-function of G associated with ρ and α is defined by ζ w (G, u, ρ, α) = det(I2md − u



ρ(g)



 g )−1 . P

g∈Γ

A determinant expression for the new weighted Ihara L-function of G associated with ρ and α is given as follows: Let 1 ≤ i, j ≤ n. Then the (i, j)-block Fij of a dn × dn matrix F is the submatrix of F consisting of d(i − 1) + 1, . . . , di rows and d(j − 1) + 1, . . . , dj columns. Theorem 7. Let G be a connected graph with n vertices and m edges, Γ a finite group, α : D(G) −→ Γ an ordinary voltage assignment, and w a vertex weight of G. Furthermore, let ρ be a representation of Γ and d its degree. Then the reciprocal of the new weighted Ihara L-function of G is given by ζ w (G, u, ρ, α)−1 m 

= j=1 (1 − u2 w(o(ej ))2 w(t(ej ))2 )d det(Ind − u g∈Γ ρ(g) Ag + u2 Id D(G)). Proof. Let V (G) = {v1 , · · · , vn }, and let D(G) = {e1 , . . . , em , em+1 , . . . , e2m } such that em+j = e−1 j (1 ≤ j ≤ m). Arrange arcs of G as follows: e1 , . . . , em , em+1 , . . . , e2m . Furthermore, let ρ = P



g P



ρ = ρ(g) and J

g∈Γ

Then we have  2  ρ )e,f = w(t(e)) ρ(α(e)) (P 0d



g J



ρ(g).

g∈Γ

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

ρ )e,f = (J

w(t(e))2 ρ(α(e)) 0d

if f = e−1 , otherwise.

 ρ = (Sρ,ev )e∈D(G);v∈V (G) and T ρ = Now, we define two 2md × nd matrices S (Tρ,ev )e∈D(G);v∈V (G) as follows:  Sρ,ev :=

 w(v)ρ(e) 0

if t(e) = v, otherwise,

Tρ,ev :=

w(v)Id 0

 and T  under the above order. Here we consider two matrices S

if o(e) = v, otherwise.

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Furthermore, we define two 2md × nd matrices Sρ = (Sρ,ev )e∈D(G);v∈V (G) and Tρ = (Tρ,ev )e∈D(G);v∈V (G) as follows:  Sρ,ev :=

 ρ(e) if t(e) = v, otherwise, 0d

Tρ,ev :=

Id 0d

if o(e) = v, otherwise.

Then we have ρ = S ρ tT ρ − J ρ , S  ρ = Sρ Wd , T ˜ ρ = Tρ Wd , P where Wd = W

Id . Thus, we have  ρ = Sρ Wd Wd t Tρ − J ρ . P

Therefore, it follows that ζ w (G, u, ρ, α)−1 = = = = =

 ρ) det(I2md − uP ρ − J ρ tT ρ )) det(I2md − u(S t   det(I2md + uJρ − uSρ Tρ )  ρ (I2md + uJ ρ tT ρ )−1 ) det(I2md + uJ ρ ) det(I2md − uS  ρ (I2md + uJ ρ )−1 S  ρ ) det(I2md + uJ ρ ). det(Ind − u t T

But, we have ρ ) = det(I2md + uJ

m 

(1 − u2 w(o(ej ))2 w(t(ej ))2 )d .

j=1

Let xej = xj = 1 − u2 w(o(ej ))2 w(t(ej ))2 (1 ≤ j ≤ n). Then we have ⎡

ρ )−1 (I2md + uJ

1/x1 Id ⎢ −uw(o(e1 ))2 /x1 ρ(e−1 ) =⎣

−uw(t(e1 ))2 /x1 ρ(e) 1/x1 Id

0d

⎤ 0d 0d ⎥ ⎦. .. .

 ρ (I2md + uJ ρ )−1 S  ρ is Thus, for (x, y) ∈ D(G), the (x, y)-block of t T  ρ (I2md + uJ ρ )−1 S  ρ )xy = w(x)w(y)/(1 − u2 w(x)2 w(y)2 )ρ(α(x, y)). (t T  ρ (I2md + uJ  ρ is ρ )−1 S Furthermore, for each v ∈ V (G), the (v, v)-block of t T

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 −1 S)  2m + uJ)  vv = −u (t T(I



169

w(v)2 w(t(e))2 /(1 − u2 w(v)2 w(t(e))2 )Id .

o(e)=v

Hence, ζ w (G, u, ρ, α)−1 m 

= j=1 (1 − u2 w(o(ej ))2 w(t(ej ))2 )d det(Ind − u g∈Γ Ag ρ(g) + u2 D(G)) Id ) 

m

= j=1 (1 − u2 w(o(ej ))2 w(t(ej ))2 )d det(Ind − u g∈Γ ρ(g) Ag + u2 Id D(G)).

2

By Theorems 6 and 7, we obtain a decomposition formula for the new weighted Ihara zeta function of a regular covering of a graph by the product of its new weighted Ihara L-functions. Corollary 4. Let G be a connected graph, Γ a finite group and α : D(G) −→ Γ an ordinary voltage assignment. Furthermore, let w be a vertex weight of G. Suppose that the Γ-covering Gα of G is connected. Then the new weighted Ihara zeta function of Gα is ζ w˜ (Gα , u) =



ζ w (G, u, ρ, α)deg ρ ,

ρ

where ρ runs over all inequivalent irreducible representations of Γ. In the case of w = 1, we obtain a determinant expression for the Ihara L-function of a graph by Stark and Terras [18], and independently, Mizuno and Sato [14]. Corollary 5 (Stark and Terras; Mizuno and Sato). Let G be a connected graph with n vertices and m edges, Γ a finite group, α : D(G) −→ Γ an ordinary voltage assignment. Furthermore, let ρ be a representation of Γ and d its degree. Then the reciprocal of the Ihara L-function of G is given by

ZG (u, ρ, α)−1 = (1 − u2 )(m−n)d det(Idn − u(

ρ(g)



Ag ) + u2 Id

 (D − In )).

g∈Γ

In the case of w = 1, we obtain a decomposition formula for the Ihara zeta function of a regular covering of a graph by Stark and Terras [18], and independently, Mizuno and Sato [14]. Corollary 6 (Stark and Terras; Mizuno and Sato). Let G be a connected graph, Γ a finite group and α : D(G) −→ Γ an ordinary voltage assignment. Suppose that the Γ-covering Gα of G is connected. Then we have

170

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Z(Gα , u) =



ZG (u, ρ, α)deg ρ ,

ρ

where ρ runs over all inequivalent irreducible representations of Γ. 6. The Euler product for the new weighted Ihara zeta function We present the Euler product for the new weighted Ihara zeta function of a graph. Foata and Zeilberger [5] gave a new proof of Bass’s Theorem by using the algebra of Lyndon words. Let X be a finite nonempty set, < a total order in X, and X ∗ the monoid generated by X. Then the total order < on X derive the lexicographic order < on X ∗ . A Lyndon word in X is defined to a nonempty word in X ∗ which is prime, i.e., not the power lr of any other word l for any r ≥ 2, and which is also minimal in the class of its cyclic rearrangements under < (see [13]). Let L denote the set of all Lyndon words in X. Let F be a square matrix whose entries b(x, x )(x, x ∈ X) form a set of commuting variables. If w = x1 x2 · · · xm is a word in X ∗ , define β(w) = b(x1 , x2 )b(x2 , x3 ) · · · b(xm−1 , xm )b(xm , x1 ). Furthermore, let β(L) =



(1 − β(l)).

l∈L

The following theorem played a central role in [5]. Theorem 8 (Foata and Zeilbereger). β(L) = det(I − F). Let G be a connected graph and w a vertex weight of G. Then, let w(e, f ) be the  (e, f )-array of the matrix P:  w(e, f ) =

w(t(e))2 0

if t(e) = o(f ) and f = e−1 , otherwise.

Note that w(e1 e2 ) · · · w(er , e1 ) = 0 for C = ((e1 , . . . , er ) is not a reduced cycle in G. Theorem 9. Let G be a connected graph, and let w a vertex weight of G. Then the reciprocal of the new weighted Ihara zeta function of G is given by ζ w (G, u) =

 (1 − wC u|C| )−1 , [C]

where [C] runs over all equivalence classes of prime, reduced cycles of G, and

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171

wC = w(t(e1 )2 w(t(e2 )2 · · · w(t(en ))2 , C = (e1 , e2 , . . . , en ) Proof. Let V (G) = {v1 , · · · , vn } and v1 < v2 < · · · < vn a total order of V (G). We consider the free monid V (G)∗ generated by V (G), and the lexicographic order on V (G)∗ derived from <. If C is a prime, reduced cycle, then there exists a unique cycle in [C] which is a Lyndon word in V (G). For z ∈ V (G)∗ , let  β(z) =

wz u|z| 0

if z is a prime, reduced cycle, otherwise.

Then we have β(L) =



(1 − β(l)) =

l∈L



(1 − wC u|C| ),

[C]

where [C] runs over all equivalence classes of prime, reduced cycles of G. Furthermore, we define variables b(x, x )(x, x ∈ V (G)) as follows:  

b(x, x ) =

w(x, x ) 0

if (x, x ) ∈ D(G), otherwise.

Theorem 6 implies that   (1 − wC u|C| ) = det(I − uF) = det(I − uP).

2

[C]

7. Example Let G = K3 be the complete graph with vertices 1,2,3, and let w be a vertex weight of G as follows: w(1) = a, w(2) = b, w(3) = c. Then we have ⎡ ⎢ A(K3 ) = ⎢ ⎣ ⎡ ⎢ D(K3 ) = ⎢ ⎣

2 2

a b 1−u2 a2 b2

+ 0 0

0

ab 1−u2 a2 b2

ab 1−u2 a2 b2 ac 1−u2 a2 c2

0

ac 1−u2 a2 c2 bc 1−u2 b2 c2

bc 1−u2 b2 c2

0

2 2

a c 1−u2 a2 c2

⎤ ⎥ ⎥, ⎦

0 a2 b2 1−u2 a2 b2

+ 0

⎤

0 b2 c2 1−u2 b2 c2

0 a2 c2 1−u2 a2 c2

+

b2 c2 1−u2 b2 c2

⎥ ⎥. ⎦

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By Theorem 5, we have ζ w (G, u)−1 = (1 − u2 a2 b2 )(1 − u2 a2 c2 )(1 − u2 b2 c2 ) det(I3 − uA(K3 ) + u2 D(K3 )) = (1 − u2 a2 b2 )(1 − u2 a2 c2 )(1 − u2 b2 c2 ) ⎡ 2 a b a c 2

× det ⎣ ⎡ ⎢ = det ⎢ ⎣

2

1 + u ( 1−u2 a2 b2 +

2

2

1−u2 a2 c2

−u 1−uab 2 a2 b2

)

2

−u 1−uab 2 a2 b2

2

a b 1 + u2 ( 1−u 2 a2 b2 +

−u 1−uac 2 a2 c 2 4 2 2

b2 c2 1−u2 b2 c2

−uab

−uac

−uab

1−u4 a2 b4 c2 1−u2 a2 c2

−ubc

−uac

−ubc

1−u4 a2 b2 c4 1−u2 a2 b2

1+u

⎤

⎦

−u 1−ubc 2 b2 c2

)

−u 1−ubc 2 b2 c2

1−u4 a b c 1−u2 b2 c2

⎤

−u 1−uac 2 a2 c 2 2

a2 c 2 ( 1−u 2 a2 c 2

+

b2 c2 1−u2 b2 c2

)

⎥ ⎥ = (1 − u2 a2 b2 c3 )2 .@ ⎦

Next, let Γ = Z3 = {1, τ, τ 2 }(τ 3 = 1) be the cyclic group of order 3 and α : D(K3 ) −→ Z3 the ordinary voltage assignment such that α(1, 2) = τ, α(2, 3) = 1 and α(3, 1) = 1. Then the Z3 -covering (K3 )α is isomorphic to the cycle graph C9 of length 9, and the weighted matrix A((K3 )α ) derived from A(G) is given as follows: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ α A((K3 ) ) = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0 0

0 0

ac 1−u2 a2 c2

bc 1−u2 b2 c2

0

0 0 0

ab 1−u2 a2 b2

0 0 0 0 0 0 0

ac 1−u2 a2 c2 bc 1−u2 b2 c2

0 0 0 0

ac 1−u2 a2 c2 bc 1−u2 b2 c2

0 0 0 0 0 0 0

ab 1−u2 a2 b2

0 0 0

0 0 0

ab 1−u2 a2 b2

0 0 0 0 0 0 ac 1−u2 a2 c2

ab 1−u2 a2 b2

0 0 0 0 bc 1−u2 b2 c2

0 0 0 0 0

ab 1−u2 a2 b2

ac 1−u2 a2 c2

bc 1−u2 b2 c2

0

0 0 0

ab 1−u2 a2 b2

0 0 0 0 0 0 ac 1−u2 a2 c2 bc 1−u2 b2 c2

0 ⎤

0 0 0 0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0

The characters of Z3 are given as follows: χi (τ j ) = (ρi )j , 0 ≤ i, j ≤ 2, ρ = exp Then we have

√ √ −1 + −3 2π −1 = . 3 2

N. Konno et al. / Linear Algebra and its Applications 571 (2019) 154–179

⎡ ⎢ A1 = ⎢ ⎣ ⎡

0 ⎢ Aτ = ⎣ 0 0

0

0

0

0

ac 1−u2 a2 c2 ab 1−u2 a2 c2

⎤

bc 1−u2 b2 c2

ac 1−u2 a2 c2 bc 1−u2 b2 c2

⎡

0 ⎢ ⎥ 0 ⎦ , Aτ 2 = ⎣ 0

0 0

0 0 ab 1−u2 a2 c2

0

173

⎤ ⎥ ⎥, ⎦ ⎤ 0 0 ⎥ 0 0⎦. 0 0

By Theorem 7, we have ζ w (G, u, χ1 , α)−1 = (1 − u2 a2 b2 )(1 − u2 a2 c2 )(1 − u2 b2 c2 ) 2 × det(I3 − u j=0 χ1 (τ j )Aτ j + u2 D(K3 )) ⎤ ⎡ 1−u4 a4 b2 c2 −uabρ −uac 1−u2 b2 c2 ⎥ ⎢ 1−u4 a2 b4 c2 = det ⎣ −uabρ2 −ubc ⎦ = 1 + u3 a2 b2 c2 + u6 a4 b4 c4 .@ 1−u2 a2 c2 4 2 2 4 1−u a b c −uac −ubc 1−u2 a2 b2 Similarly, we have ζ w (G, u, χ1 , α)−1 = (1 − u2 a2 b2 )(1 − u2 a2 c2 )(1 − u2 b2 c2 ) 2 × det(I3 − u j=0 χ2 (τ j )Aτ j + u2 D(K3 )) ⎤ ⎡ 1−u4 a4 b2 c2 −uabρ2 −uac 1−u2 b2 c2 ⎥ ⎢ 1−u4 a2 b4 c2 = det ⎣ −uabρ −ubc ⎦ = 1 + u3 a2 b2 c2 + u6 a4 b4 c4 .@ 1−u2 a2 c2 4 2 2 4 1−u a b c −uac −ubc 1−u2 a2 b2 By Corollary 4, we have ζw˜ (Gα , u)−1 = ζ w (G, u)−1 ζ w (G, u, χ1 , α)−1 ζ w (G, u, χ2 , α)−1 = (1 − u3 a2 b2 c2 )2 (1 + u3 a2 b2 c2 + u6 a4 b4 c4 )2 = (1 − u9 a6 b6 c6 )2 . 8. An application to the non-backtracking random walk on a regular graph We present spectra for the transition matrix of the non-backtracking random walk on a regular graph. Let G be a connected d-regular graph with n vertices and m edges, where d ≥ 3. Furthermore, let w : V (G) −→ C be given by w(u) = √

1 f or u ∈ V (G), d−1

and let P = (Pef )e,f ∈D(G) be defined as follows:  Pe,f =

1 d−1

0

if t(e) = o(f ) and f = e−1 , otherwise.

N. Konno et al. / Linear Algebra and its Applications 571 (2019) 154–179

174

Then the matrix P is the transition matrix of the non-backtracking random walk on G. Furthermore, we have  (A)xy = (D)xx =

o(e)=x

1 d−1 /(1

1 − u2 (d−1) 2)

0

if (x, y) ∈ D(G), otherwise,

d 1 1 1 /(1 − u2 )= /(1 − u2 ). 2 2 2 (d − 1) (d − 1) (d − 1) (d − 1)2

By Theorem 5, we have  det(I2m − uP) 1 1 d 1 1 m 2 2 = (1 − u2 (d−1) det(In − u d−1 /(1 − u2 (d−1) 2) 2 )A(G) + u (d−1)2 /(1 − u (d−1)2 )In ) 1 m−n = (1 − u2 (d−1) det(In − 2)

u d−1 A(G)

1 + u2 d−1 In ).

Now, let λ = 1/u. Then we have  = (λ2 − det(λI2m − P) = (λ2 −

1 m−n (d−1)2 ) 1 m−n (d−1)2 )

det(λ2 In − 

λ d−1 A(G)

λA ∈Spec(A(G)) (λ

2

−

+

1 d−1 In )

λA d−1 λ

+

1 d−1 ).

Thus, solving λ2 −

1 λA λ+ = 0, d−1 d−1

we obtain λ=

λA ±

 λA 2 − 4(d − 1) . 2(d − 1)

Theorem 10 (Kempton). Let G be a connected d(≥ 3)-regular graph with n vertices and  has 2n eigenvalues of the form m edges. The transition matrix P λ=

λA ±

 λA 2 − 4(d − 1) , 2(d − 1)

where λA is an eigenvalue of the matrix A(G). The remaining 2(m − n) eigenvalues of ˜ are ±1/(d − 1) with equal multiplicities. P 9. An application to the non-backtracking random walk on a semiregular bipartite graph We present spectra for the transition matrix of the non-backtracking random walk on a semiregular bipartite graph. Hashimoto [7] presented a determinant expression for the

N. Konno et al. / Linear Algebra and its Applications 571 (2019) 154–179

175

Ihara zeta function of a semiregular bipartite graph. We use an analogue of the method in the proof of Hashimoto’s result. A bipartite graph G = (V1 , V2 ) is called (q1 , q2 )-semiregular if deg G v = qi 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. Then G[1] is (q1 + 1)q2 -regular, and G[2] is (q2 + 1)q1 -regular. Let G = (V1 , V2 ) be a connected (c, d)-semiregular bipartite graph with ν vertices and  edges. Set | V1 |= m and | V2 |= n(m ≤ n). Furthermore, let w : V (G) −→ C be given by w(x) = √

1

for any x ∈ V (G).

deg G x − 1

 is given as follows: Then the matrix P  1 −1  ef = deg t(e)−1 if t(e) = o(f ) and f = e , (P) 0 otherwise. Furthermore, we have  (A)xy =

0

(D)xx =  =

−

1  /(1 (c−1)(d−1)

o(e)=x

u2 (c−1)(d−1) )

if (x, y) ∈ D(G),

otherwise, 1 1 /(1 − u2 ) (c − 1)(d − 1) (c − 1)(d − 1)

c/((c − 1)(d − 1)(1 − d/((c − 1)(d − 1)(1 −

u2 (c−1)(d−1) )) u2 (c−1)(d−1) ))

if x ∈ V1 , if x ∈ V2 .

Next, let V1 = {v1 , · · · , vm } and V2 = {w1 , · · · , wn }. Arrange vertices of G as follows: v1 , · · · , vm ; w1 , · · · , wn . We consider the matrix A = A(G) under this order. Then, let 

 E . 0

0 t E

A=

Since A is symmetric, there exists an orthogonal matrix F ∈ O(n) such that  EF = R



⎡

⎢ 0 =⎣

0

μ1 ..

. μm

Now, let  H=

Im 0

 0 . F

⎤ 0 ··· 0 .. .. ⎥ . . .⎦ 0 ··· 0

176

N. Konno et al. / Linear Algebra and its Applications 571 (2019) 154–179

Then we have ⎡

0 ⎢t t HAH = ⎣ R 0

⎤ 0 ⎥ 0⎦. 0

R 0 0

Furthermore, let α = 1/((c − 1)(d − 1)(1 −

u2 )). (c − 1)(d − 1)

Then we have  1 u2 )A(G) = α (c − 1)(d − 1)A. A=  /(1 − (c − 1)(d − 1) (c − 1)(d − 1) and  1 cIm ))D = αD = α D = 1/((c − 1)(d − 1)(1 − u (c − 1)(d − 1) 0 2

 0 dIn

.

Thus, we have ⎡

t

HAH = α



0 ⎢ (c − 1)(d − 1) ⎣ t R 0

R 0 0

⎤ 0 ⎥ 0⎦ 0

and t

HDH = αD = D.

By Theorem 5, ˜ = (1 − det(I2 − uP)

u2  (c−1)(d−1) )

det(Iν − uA + u2 D)

2

u = (1 − (c−1)(d−1) ) ⎛⎡ 2 Im + αcu Im t ⎜⎢ × det ⎝⎣ −α (c − 1)(d − 1)u R 0 2

−α



(c − 1)(d − 1)uR Im + αdu2 Im 0

⎤⎞ 0 ⎥⎟ 0 ⎦⎠ 2 In−m + αdu In−m

u = (1 − (c−1)(d−1) ) (1 + du2 α)n−m    2 )I −α (c − 1)(d − 1)uR (1 + αcu m  × det (1 + αdu2 )Im −α (c − 1)(d − 1)u t R

N. Konno et al. / Linear Algebra and its Applications 571 (2019) 154–179

 Im 0

· det = (1 −

= (1 −



(c−1)(d−1)u R 1+αcu2



Im

u2 −(n−m) (1 (c−1)(d−1) )

 × det

α

−

u2 (c−1)(d−1)

(1 + αcu2 )Im  −α (c − 1)(d − 1)u t R

u2 −(n−m) (1 (c−1)(d−1) )

u2 −(n−m) (1 (c−1)(d−1) )

+

du2 n−m (c−1)(d−1) )



0 (1 + αdu2 )Im −

α2 (c−1)(d−1)u2 t RR 1+αcu2

d−1 + u2 (c−1)(d−1) )n−m

× (1 + αcu2 )m det((1 + αdu2 )Im − = (1 −

177

+

α2 (c−1)(d−1)u2 t RR) 1+αcu2

u2 n−m c−1 )

× det((1 + αcu2 )(1 + αdu2 )Im − α2 (c − 1)(d − 1)u2 t RR). Since A is symmetric, t RR is symmetric and positive semi-definite, i.e., the eigenvalues of t RR are of form: λ21 , · · · , λ2m (λ1 , · · · , λm ≥ 0). Therefore it follows that  det(I2 − uP) 2

2

2

2

u u = (1 − (c−1)(d−1) )−(n−m) (1 + c−1 )n−m m × j=1 ((1 + αcu2 )(1 + αdu2 ) − λ2j α2 (c − 1)(d − 1)u2 ) u u = (1 − (c−1)(d−1) )−(n+m) (1 + c−1 )n−m m 2 1 u × j=1 ((1 − (c−1)(d−1) + cu2 (c−1)(d−1) )(1 −

u2 (c−1)(d−1)

1 + du2 (c−1)(d−1) )

2

u − λ2j (c−1)(d−1) ) 2

2

u u = (1 − (c−1)(d−1) )−ν (1 + c−1 )n−m m λ2j u2 u2 u2 × j=1 ((1 + (d−1) )(1 + (c−1) ) − (c−1)(d−1) ).

Now, let u = 1/λ. Then we have 1 1 n−m )−ν (λ2 + ) (c − 1)(d − 1) c−1

 = (λ2 − det(λI2 − P) ×

m 

((λ2 +

j=1

Solving

λ2j λ2 1 1 )(λ2 + )− ). d−1 c−1 (c − 1)(d − 1)

N. Konno et al. / Linear Algebra and its Applications 571 (2019) 154–179

178

(λ2 +

λ2j λ2 1 1 )(λ2 + )− = 0, d−1 c−1 (c − 1)(d − 1)

i.e., λ4 + (

λ2j 1 1 1 + − )λ2 + = 0, c − 1 d − 1 (c − 1)(d − 1) (c − 1)(d − 1)

we obtain    λ2j 1 1 λ = ± 1 (( − − )± 2 (c − 1)(d − 1) c−1 d−1

(λ2j − (c − 1) − (d − 1))2 4 − ), (c − 1)2 (d − 1)2 (c − 1)(d − 1)

i.e.,  !   λ2 − (c − 1) − (d − 1) ± (λ2 − (c − 1) − (d − 1))2 − 4(c − 1)(d − 1) j j  λ=± . 2(c − 1)(d − 1) Theorem 11 (Kempton). Let G = (V1 , V2 ) be a connected (c, d)-semiregular bipartite graph with ν vertices and  edges, |V1 | = m, |V2 | = n, and Spec(A) =  has 4mn eigenvalues of the {±λ1 , . . . , ±λm , 0, . . . , 0}. Then the transition matrix P form 1. 4m eigenvalues:  !   λ2 − (c − 1) − (d − 1) ± (λ2 − (c − 1) − (d − 1))2 − 4(c − 1)(d − 1) j  j λ=± , 2(c − 1)(d − 1) where λj (1 ≤ j ≤ m) is an eigenvalue of the matrix A(G); 2. 2n − 2m eigenvalues: " λ = ±i

1 ; c−1

3. 2( − ν) eigenvalues: λ=±

1 . (c − 1)(d − 1)

Acknowledgements We would like to thank the referee for many useful suggestions and comments.

N. Konno et al. / Linear Algebra and its Applications 571 (2019) 154–179

179

The first author is partially supported by the Grant-in-Aid for Scientific Research (Challenging Exploratory Research) of Japan Society for the Promotion of Science (Grant No. 15K13443). The second author is partially supported by the Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of Science (Grant No. 16K05249). The third author is partially supported by the Grant-in-Aid for Young Scientists (B) of Japan Society for the Promotion of Science (Grant No. 26400001). The fourth author is partially supported by the Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of Science (Grant No. 15K04985). References [1] N. Alon, I. Benjamini, E. Lubetzky, S. Sodin, Non-backtracking random walks mix faster, Commun. Contemp. Math. 09 (2007) 585. [2] O. Angel, J. Friedman, S. Hoory, The non-backtracking spectrum of the universal cover of a graph, Trans. Amer. Math. Soc. 326 (6) (2015) 4287–4318. [3] H. Bass, The Ihara-Selberg zeta function of a tree lattice, Internat. J. Math. 3 (1992) 717–797. [4] R. Fitzner, R. van der Hofstad, Non-backtracking random walk, J. Stat. Phys. 150 (2) (2013) 264–284. [5] 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. [6] J.L. Gross, 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, Academic Press, New York, 1989, pp. 211–280. [8] K. Hashimoto, On the zeta- and L-functions of finite graphs, Internat. J. Math. 1 (1999) 381–396. [9] 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. [10] M. Kempton, Non-backtracking random walks and a weighted Ihara’s theorem, Open J. Discrete Math. 6 (2016) 207–226. [11] M. Kotani, T. Sunada, Zeta functions of finite graphs, J. Math. Sci. Univ. Tokyo 7 (2000) 7–25. [12] F. Krzakala, C. Moore, E. Mossel, J. Neeman, A. Sly, L. Zdeborova, P. Zhang, Spectral redemption in clustering sparse networks, Proc. Natl. Acad. Sci. 110 (52) (2013) 20935–20940. [13] M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, Mass, 1983. [14] H. Mizuno, I. Sato, Zeta functions of graph coverings, J. Combin. Theory Ser. B 80 (2000) 247–257. [15] H. Mizuno, I. Sato, Weighted zeta functions of graphs, J. Combin. Theory Ser. B 91 (2004) 169–183. [16] J.-P. Serre, Trees, Springer-Verlag, New York, 1980. [17] H.M. Stark, A.A. Terras, Zeta functions of finite graphs and coverings, Adv. Math. 121 (1996) 124–165. [18] H.M. Stark, A.A. Terras, Zeta functions of finite graphs and coverings. II, Adv. Math. 154 (2000) 132–195. [19] T. Sunada, L-functions in geometry and some applications, in: Lecture Notes in Math., vol. 1201, Springer-Verlag, New York, 1986, pp. 266–284. [20] T. Sunada, Fundamental Groups and Laplacians (in Japanese) Kinokuniya, Tokyo, 1988. [21] Y. Watanabe, K. Fukumizu, Loopy belief propagation, Bethe free energy and graph zeta function, Adv. Neural Inf. Process. Syst. 22 (2010) 2017–2025.