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\usepackage{amsthm, amsmath, amssymb}

\title{\bf A new determinant expression for \\ the weighted Bartholdi zeta function of a digraph}
\author{Iwao Sato\thanks{Supported by Grant-in-Aid for Science Research (C)} , 
\ Hideo Mitsuhasi , \and \ Hideaki Morita \\
\small Oyama National College of Technology \\
\small Oyama, Tochigi 323-0806, JAPAN \\
\small\tt isato@oyama-ct.ac.jp\\
 \\ 
\small Division of System Engineering for Mathematics \\
\small Muroran Institute of Technology\\ 
\small Muroran, Hokkaido 050-8585, JAPAN}

\date{\dateline{ }{ }\\
\small Mathematical Subject Classifications: 05C50, 15A15}

 \begin{document}
 \maketitle


\begin{abstract}
We consider the weighted Bartholdi zeta function of a digraph $D$, and 
give a new determinant expression of it. 
Furthermore, we treat a weighted $L$-function of $D$, and give a new 
determinant expression of it. 
As a corollary, we present determinant expressions for 
the Bartholdi edge zeta functions of a graph and a digraph. 


\bigskip\noindent \textbf{Key words:} zeta function, digraph covering, $L$-function
\end{abstract}




\section{Introduction}

Zeta functions of graphs started from zeta functions of regular graphs 
by Ihara [7]. 
In [7], 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 [12,13]. 
Hashimoto [6] 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 by a determinant containing the edge matrix. 
Bass [2] presented another determinant expression for the Ihara zeta function 
of an irregular graph by using its adjacency matrix. 

Stark and Terras [11] gave an elementary proof of Bass' Theorem, and 
discussed three different zeta functions of any graph. 
Furthermore, various proofs of Bass' Theorem were given by 
Foata and Zeilberger [4], Kotani and Sunada [8].

For two variable zeta function of a graph, Bartholdi [1] defined and gave 
a determinant expression of the Bartholdi zeta function of a graph. 
Mizuno and Sato [9] presented a decomposition formula for the Bartholdi 
zeta function of a regular covering of a graph. 

As a digraph version of the Bartholdi zeta function, Choe, Kwak, Park and Sato [3] 
defined the weighted Bartholdi zeta function of a digraph, and 
presented its determinant expression. 
 
As a multi-variable zeta function of a graph, Stark and Terras [11] defined 
the edge zeta function of a graph. 
Watanabe and Fukumizu [14] presented a determinant expression for the edge zeta function 
of a graph $G$ with $n$ vertices by $n \times n$ matrices.

In this paper, we present a new determinant expression of 
the weighted Bartholdi zeta function of a digraph $D$ by using 
the method of Watanabe and Fukumizu [14]: 

\vspace{1mm} 


{\bf Main Theorem}. 

Let $D$ be a connected digraph with $n$ vertices and $m$ arcs, and 
let ${\bf W} = {\bf W} (D)$ be a weighted matrix of $D$. 
Then the reciprocal of the weighted Bartholdi zeta function of $D$ 
is given by 
\[
\zeta (D,w,u,t )^{-1} = \det ( {\bf I}_n +(1-u) t^2 \tilde{{\bf D}} 
-t \tilde{{\bf A}}_1 -t \tilde{{\bf A}}_0 ) 
\prod^{m_1}_{i=1} ( 1- w(f_i ) w(f^{-1}_i ) (1-u )^2 t^2 ) , 
\]
where $\tilde{{\bf D}} $, $\tilde{{\bf A}}_1 $ and $ \tilde{{\bf A}}_0 $ are 
defined in Section 3, and $f^{\pm 1}_1 , \ldots , f^{\pm 1}_{m_1} $ are 
symmetric arcs of $D$. 


\vspace{1mm} 

Furthermore, we present a new decomposition formula for 
the weighted Bartholdi zeta function of a group covering of $D$, 
and a new determinant expression for the weighted Bartholdi $L$-function 
of $D$. 

\section{Preliminaries} 

Graphs and digraphs treated here are finite.
Let $G=(V(G),E(G))$ be a connected graph (possibly multiple edges and loops) 
with the set $V(G)$ of vertices and the set $E(G)$ of unoriented edges $uv$ 
joining two vertices $u$ and $v$. 
For $uv \in E(G)$, an arc $(u,v)$ is the oriented edge from $u$ to $v$. 
Set $D(G)= \{ (u,v),(v,u) \mid uv \in E(G) \} $. 
For $e=(u,v) \in D(G)$, set $u=o(e)$ and $v=t(e)$. 
Furthermore, let $e^{-1}=(v,u)$ be the {\em inverse} of $e=(u,v)$. 

A {\em path $P$ of length $n$} in $G$ is a sequence 
$P=(e_1, \cdots ,e_n )$ of $n$ arcs such that $e_i \in D(G)$,
$t( e_i )=o( e_{i+1} )(1 \leq i \leq n-1)$, 
where indices are treated $mod \  n$. 
Set $ \mid P \mid =n$, $o(P)=o( e_1 )$ and $t(P)=t( e_n )$. 
Also, $P$ is called an {\em $(o(P),t(P))$-path}. 
We say that a path $P=(e_1, \cdots ,e_n )$ has a {\em backtracking} 
or a {\em bump} at $t( e_i )$ if $ e^{-1}_{i+1} =e_i $ 
for some $i(1 \leq i \leq n-1)$. 
A $(v, w)$-path is called a {\em $v$-cycle} 
(or {\em $v$-closed path}) if $v=w$. 


We introduce an equivalence relation between cycles. 
Two cycles $C_1 =(e_1, \cdots ,e_m )$ and 
$C_2 =(f_1, \cdots ,f_m )$ are called {\em equivalent} if there exists 
$k$ such that $f_j =e_{j+k} $ for all $j$. 
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 {\em power} of $B$. 
A cycle $C$ is {\em reduced} if 
$C$ has no backtracking. 
Furthermore, a cycle $C$ is {\em prime} if it is not a power of 
a strictly smaller cycle. 


The {\em Ihara zeta function} of a graph $G$ is 
a function of $u \in {\bf C}$ with $|u|$ sufficiently small, defined by 
\[
{\bf Z} (G, t)= \prod_{[C]} (1- t^{ \mid C \mid } )^{-1} ,
\]
where $[C]$ runs over all equivalence classes of prime, reduced cycles 
of $G$(see [7]). 

Let $m$ be the number of edges of $G$. 
Furthermore, let two $m \times m$ matrices 
${\bf B} =( {\bf B}_{e,f} )_{e,f \in A(D)} $ and 
${\bf J}_0 =( {\bf J}_{e,f} )_{e,f \in A(D)} $ be defined as follows: 
\[
{\bf B}_{e,f} =\left\{
\begin{array}{ll}
1 & \mbox{if $t(e)=o(f)$, } \\
0 & \mbox{otherwise}
\end{array}
\right.
, 
{\bf J}_{e,f} =\left\{
\begin{array}{ll}
1 & \mbox{if $f= e^{-1} $, } \\
0 & \mbox{otherwise.}
\end{array}
\right.
\]
Then ${\bf B} -{\bf J}_0 $ is called the {\em edge matrix} of $G$. 
  

\newtheorem{theorem}{Theorem}
\begin{theorem}[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 
\[
{\bf Z} (G, t)^{-1} = \det ( {\bf I}_{2m} - t ( {\bf B} - {\bf J}_0 ))
=(1- t^2 )^{m-n} \det ( {\bf I} -t {\bf A} (G)+ t^2 ({\bf D} -{\bf I} )), 
\]
where ${\bf A} (G)$ is the adjacency matrix of $G$, and ${\bf D} =( d_{ij} )$ is the diagonal matrix 
with $d_{ii} = \deg v_i $ where $V(G)= \{ v_1 , \cdots , v_n \} $. 
\end{theorem}


Then the {\em Bartholdi zeta function} of $G$ is  defined by 
\[
\zeta {}_G (u,t)= \zeta (G,u,t)= 
\prod_{[C]} (1- u^{ cbc(C)} t^{ \mid C \mid } )^{-1} , 
\]
where $[C]$ runs over all equivalence classes of prime cycles 
of $G$(see [1]). 



\begin{theorem}[Bartholdi] 
Let $G$ be a connected graph with $n$ vertices and $m$ unoriented edges. 
Then the reciprocal of the Bartholdi zeta function of $G$ is given by 
\[
\begin{array}{rcl}
\  &  & \zeta (G,u,t )^{-1} = \det ( {\bf I}_{2m} -t ( {\bf B} -(1-u) {\bf J}_0 )) \\ 
\  &   &                \\ 
\  & = & (1-(1-u )^2 t^2 )^{m-n} 
\det ( {\bf I} -t {\bf A} (G)+(1-u)( {\bf D} -(1-u) {\bf I} ) t^2 ) . 
\end{array}
\]
\end{theorem}


In the case of $u=0$, Theorem 2 implies Theorem 1. 

Next, we state the weighted Bartholdi zeta function of a digraph.  
Let $D=(V(D),A(D))$ be a connected digraph with the set $V(D)$ of vertices 
and the set $A(D)$ of arcs. 
Furthermore, let $D$ have $n$ vertices $v_1 , \cdots , v_n $ 
and $m$ arcs. 
Then we consider an $n \times n$ matrix 
${\bf W} ={\bf W} (D)=( w_{ij} )_{1 \leq i,j \leq n }$ with $ij$ entry 
nonzero complex number $w_{ij}$ if $( v_i , v_j ) \in A(D)$, 
and $w_{ij} =0$ otherwise. 
The matrix ${\bf W} = {\bf W} (D)$ is called the 
{\em weighted matrix} of $D$.
Furthermore, let $w( v_i , v_j )= w_{ij}, \  v_i , v_j \in V(D)$ and 
$w(e)= w_{ij}, e=( v_i , v_j ) \in A(D)$. 
For each path $P=( e_1 , \cdots , e_r )$ of $G$, the {\em norm} 
$w(P)$ of $P$ is defined as follows: 
$w(P)= w (e_1 ) \cdots w ( e_r ) $. 


The {\em cyclic bump count} $cbc(C)$ of a cycle 
$C=( e_1 , \cdots , e_n )$ of $G$ is 
\[
cbc(C)= \mid \{  i=1, \cdots , n \mid e_i = e^{-1}_{i+1} \} \mid , 
\]
where $e_{n+1} = e_1 $. 
Then the {\em weighted Bartholdi zeta function} of $D$ is 
a function of $u,t \in {\bf C}$ with $|u|,|t|$ sufficiently small, defined by 
\[
\zeta (D,w,u,t)= \prod_{[C]} (1-w(C) u^{cbc(C)} t^{ \mid C \mid } ) {}^{-1} ,
\]
where $[C]$ runs over all equivalence classes of prime cycles of $D$. 

If $w= {\bf 1} $, i.e., $w(v_i,v_j)=1$ for each $( v_i , v_j ) \in A(D)$, then the weighted 
Bartholdi zeta function of $D$ is the Bartholdi zeta function of $D$. 
If $D= D_G$ is the symmetric digraph corresponding to a graph $G$, and $w= {\bf 1} $, 
then the weighted Bartholdi zeta function of $D_G$ is the Bartholdi zeta 
function of $G$. 
If $D= D_G$, $w= {\bf 1} $ and $u=0$, then the weighted Bartholdi zeta function of $G$ is 
the Ihara zeta function of $G$. 

Two $m \times m$ matrices 
${\bf B}_w =( {\bf B}^w_{e,f} )_{e,f \in A(D)} $ and 
${\bf J}_w =( {\bf J}^w_{e,f} )_{e,f \in A(D)} $ are defined as follows: 
\[
{\bf B}^w_{e,f} =\left\{
\begin{array}{ll}
w(e) & \mbox{if $t(e)=o(f)$, } \\
0 & \mbox{otherwise}
\end{array}
\right.
, 
{\bf J}^w_{e,f} =\left\{
\begin{array}{ll}
w(e) & \mbox{if $f= e^{-1} $, } \\
0 & \mbox{otherwise.}
\end{array}
\right.
\]
Furthermore, we define two $n \times n$ matrices 
${\bf W}_1 ={\bf W}_1 (D)=(a_{uv} )$ and ${\bf W}_0$ as follows: 
\[
a_{uv} =\left\{
\begin{array}{ll}
w(u,v) & \mbox{if both $(u,v)$ and $(v,u) \in A(D)$, } \\
0 & \mbox{otherwise }
\end{array}
\right. 
\]
and 
\[
{\bf W}_0 = {\bf W}_0 (D)= {\bf W} (D)- {\bf W}_1 . 
\]
Let an $n \times n$ matrix ${\bf S} =(s_{xy} )$ is the diagonal matrix 
defined by 
\[
s_{xx} = \mid \{ e \in A(D) \mid o(e)=x, e^{-1} \in A(D) \} \mid . 
\] 


\begin{theorem}[Choe, Kwak, Park and Sato]
Let $D$ be a connected digraph, and 
let ${\bf W} = {\bf W} (D)$ be a weighted matrix of $D$. 
Furthermore, let $m_1 =\mid \{ e \in A(D) \mid e^{-1} \in A(D) \} \mid /2$. 
Then the reciprocal of the weighted Bartholdi zeta function of $D$ 
is given by 
\[
\zeta (D,w,u,t )^{-1} 
= \det ({\bf I}_{m} -( {\bf B}_w -(1-u) {\bf J}_w )t) ,  
\]
where $n= \mid V(D) \mid $ and $m= \mid A(D) \mid $. 

Furthermore, if $w( e^{-1} )=w(e )^{-1}$ for each $e \in A(D)$ such that 
$e^{-1} \in A(D)$, then 
\[
\begin{array}{rcl}
\  &  & \zeta (D,w,u,t )^{-1} = (1-(1-u )^2 t^2 )^{m_1 -n} \\ 
\  &   &                \\ 
\  & \times & 
\det ({\bf I}_n -t {\bf W}_1 (D) -(1-(1-u )^2 t^2 )t {\bf W}_0 (D) 
+(1-u) t^2 ( {\bf S} -(1-u) {\bf I}_n )) . 
\end{array}
\]
\end{theorem}


If $D=D_G$, $w= {\bf 1} $ and $u=0$, then Theorem 2 implies Theorem 1.

Now, we proceed to the edge zeta function of a graph $G$ with $m$ edges. 
Let $G$ be a connected graph and $D(G)= \{ e_1, \ldots, e_m, e_{m+1} , \ldots , 
e_{2m} \} (e_{m+i} = e^{-1}_i (1 \leq i \leq m ))$. 
We introduce $2m$ variables $z_1 , \ldots , z_{2m} $, and set $g(C)=  z_{i_1 } \cdots z_{i_k} $ 
for each cycle $C=( e_{i_1 }, \ldots, e_{i_k} )$ of $G$. 
Set $z_{e_i} =z_i (1 \leq i \leq 2m)$ and ${\bf z} =(z_1 , \ldots , z_{2m} )$. 
Then the {\em edge zeta function} $\zeta {}_G ( {\bf z} )$ of $G$ is defined by 
\[
\zeta {}_G ( {\bf z} )= \prod_{[C]} (1-g(C) )^{-1} , 
\]
where $[C]$ runs over all equivalence classes of prime, reduced cycles 
of $G$. 


\begin{theorem}[Stark and Terras]
Let $G$ be a connected graph with $m$ edges. 
Then 
\[
\zeta {}_G ( {\bf z} ) {}^{-1} = 
\det ( {\bf I}_{2m} - ( {\bf B} - {\bf J} {}_0) {\bf U} ) ,  
\]
where 
\[
{\bf U} =
\left[ 
\begin{array}{cccccc}
z_1 &   &   &   &   & 0 \\
   & \ddots &   &   &   &   \\
   &    & z_m  &   &   &  \\
   &    &   & z_{m+1}  &   &  \\ 
   &    &   &   & \ddots &  \\
0  &    &   &   &   & z_{2m} 
\end{array} 
\right] 
. 
\]
\end{theorem}


Let $G$ be a graph with $n$ vertices.  
Then we define an $n \times n$ matrix $\widehat{{\bf A}} =(a_{xy} )$ as follows: 
\[
a_{xy} =\left\{
\begin{array}{ll}
z_{(x,y)}/(1- z_{(x,y)} z_{(y,x)} ) & \mbox{if $(x,y) \in D(G)$, } \\
0 & \mbox{otherwise. }
\end{array}
\right. 
\]
Furthermore, an $n \times n$ matrix $\widehat{{\bf D}} =(d_{xy} )$ is the diagonal matrix 
defined by 
\[
d_{xx} = \sum_{o(e)=x} 
\frac{z_e z_{e^{-1}} }{1- z_e z_{ e^{-1}} } . 
\]


\begin{theorem}[Watanabe and Fukumizu] 
Let $G$ be a connected graph with $n$ vertices and $m$ edges.  
Then    
\[
\zeta {}_G ( {\bf z} ) {}^{-1} = \det ( {\bf I}_{n} + \widehat{{\bf D}} 
- \widehat{{\bf A}} ) \prod^{m}_{i=1} ( 1- z_{f_i } z_{f^{-1}_i } ) ,  
\]
where $D(G)= \{ f_1 ,f^{-1}_1 , \ldots , f_m f^{-1}_m \} $. 
\end{theorem}

In Section 2, we present a new determinant expression of 
the weighted Bartholdi zeta function of a digraph $D$ by using 
the method of Watanabe and Fukumizu [14]. 
In Section 3, we present a new decomposition formula for 
the weighted Bartholdi zeta function of a group covering of $D$. 
In Section 4, we present a new determinant expression for 
the weighted Bartholdi $L$-function of $D$. 
In Section 5, we define the Bartholdi edge zeta functions of graphs and digraphs, 
and present their determinant expressions as corollaries of Theorem 6. 


\section{Weighted Bartholdi zeta functions of digraphs}

We present a new determinant expression of 
the weighted Bartholdi zeta function of a digraph. 

Let $D$ be a connected digraph with $n$ vertices $v_1 , \cdots , v_n $ 
and $m$ arcs, and ${\bf W} = {\bf W} (D)$ a weighted matrix of $D$. 
Then we define two $n \times n$ matrices 
$\tilde{{\bf A}}_1 =\tilde{{\bf A}}_1 (D)=(a_{xy} )$ and 
$\tilde{{\bf A}}_0 =\tilde{{\bf A}}_0 (D)= (b_{xy} )$ as follows: 
\[
a_{xy} =\left\{
\begin{array}{ll}
w(x,y)/(1- w(x,y) w(y,x) (1-u )^2 t^2 ) & \mbox{if both $(x,y)$ and $(y,x) \in A(D)$, } \\
0 & \mbox{otherwise }
\end{array}
\right. 
\]
and 
\[
b_{xy} =\left\{
\begin{array}{ll}
w(x,y) & \mbox{if $(x,y) \in A(D)$ and $(y,x) \not\in A(D)$, } \\
0 & \mbox{otherwise }
\end{array}
\right. 
\]
Furthermore, an $n \times n$ matrix $\tilde{{\bf D}} = \tilde{{\bf D}} (D)= (d_{xy} )$ 
is the diagonal matrix defined by 
\[
d_{xx} = \sum_{o(e)=x, e^{-1} \in A(D)} 
\frac{w(e) w(e^{-1} )}{1- w(e) w( e^{-1} ) (1-u )^2 t^2 } . 
\] 

Let \( {\bf M}_{1} \oplus \cdots \oplus {\bf M}_{s} \) be the 
block diagonal sum of square matrices 
${\bf M}_{1}, \cdots , {\bf M}_{s}$. 
A new determinant expression for $ \zeta (D,w,u,t )$ is given as follows: 


\begin{theorem}
Let $D$ be a connected digraph, and 
let ${\bf W} = {\bf W} (D)$ be a weighted matrix of $D$. 
Then the reciprocal of the weighted Bartholdi zeta function of $D$ 
is given by 
\[
\zeta (D,w,u,t )^{-1} = \det ( {\bf I}_n +(1-u) t^2 \tilde{{\bf D}} 
-t \tilde{{\bf A}}_1 -t \tilde{{\bf A}}_0 ) 
\prod^{m_1}_{i=1} ( 1- w(f_i ) w(f^{-1}_i ) (1-u )^2 t^2 ) , 
\]
where $n= \mid V(D) \mid $, $m= \mid A(D) \mid $ and 
$f^{\pm 1}_1 , \ldots , f^{\pm 1}_{m_1} $ are symmetric arcs of $D$. 
\end{theorem}


{\bf Proof}.  Let $V(D)= \{ v_1, \cdots , v_n \} $ and, 
let $A(D)= \{ e_1, \cdots , e_{m_0 } , f_1, 
\cdots , f_{m_1}, f^{-1}_{1}, \cdots $, $ f^{-1}_{m_1} \} $ 
such that $e^{-1}_i \not\in A(D) (1 \leq i \leq m_0 )$. 
Note that  $m= m_0 +2 m_1 $. 

Arrange arcs of $D$ as follows: 
\[
e_1, \cdots , e_{m_0 } , f_{1}, f^{-1}_1 , 
\cdots , f_{ m_1}, f^{-1}_{m_1 } . 
\]
Let 
\[
{\bf U} =
\left[ 
\begin{array}{cccccc}
w(e_1 ) &   &   &   &   & 0 \\
   & \ddots &   &   &   &   \\
   &    & w(e_{m_0} ) &   &   &  \\
   &    &   & w(f_1)  &   &  \\ 
   &    &   &   & w(f^{-1}_1 ) &  \\
0  &    &   &   &   &  \ddots 
\end{array} 
\right] 
. 
\]
Then we have 
\[
{\bf U} {\bf B} = {\bf B}_w \  and \  
{\bf U} {\bf J}_0 = {\bf J}_w . 
\]
Thus, 
\[
{\bf B}_w -(1-u) {\bf J}_w = {\bf U} ({\bf B} -(1-u) {\bf J}_0 ) . 
\]
By Theorem 2, it follows that 
\[
 \zeta (D,w,u,t )^{-1} = \det ( {\bf I}_m 
-t {\bf U} ({\bf B} -(1-u) {\bf J}_0 )) . 
\]


Now, let ${\bf K} =( {\bf K}_{ev} )$ ${}_{e \in A(D); v \in V(D)} $ be the $m \times n$ matrix defined 
as follows: 
\[
{\bf K}_{ev} :=\left\{
\begin{array}{ll}
1 & \mbox{if $o(e)=v$, } \\
0 & \mbox{otherwise. } 
\end{array}
\right.
\]
Furthermore, we define the $m \times n$ matrix 
${\bf L} =( {\bf L}_{ev} )_{e \in A(D); v \in V(D)} $ as follows: 
\[
{\bf L}_{ev} :=\left\{
\begin{array}{ll}
1 & \mbox{if $t(e)=v$, } \\
0 & \mbox{otherwise. } 
\end{array}
\right.
\] 
Then we have 
\[
{\bf L} {}^t {\bf K} = {\bf B} . 
\]
Thus, 
\[
\begin{array}{rcl}
\  &   & \det ( {\bf I}_m -t {\bf U} ({\bf B} -(1-u) {\bf J}_0 )) \\  
\  &   &                \\ 
\  & = & \det ( {\bf I}_m -t {\bf U} 
({\bf L} {}^t {\bf K} -(1-u) {\bf J}_0 )) 
= \det ( {\bf I}_m -t {\bf U} {\bf L} {}^t {\bf K} 
+(1-u)t {\bf U} {\bf J}_0 ) . 
\end{array}
\]


But, we have 
\begin{equation}
{\bf I}_m +(1-u)t {\bf U} {\bf J}_0 
= {\bf I}_{m_0 } \oplus ( \oplus {}^{m_1 }_{j=1} 
\left[ 
\begin{array}{cc}
1  &  (1-u)t w(f_j ) \\
(1-u)t w(f^{-1}_j ) & 1 
\end{array} 
\right] 
) 
. 
\end{equation}
Since  $|u|,|t|$ are sufficiently small, we have 
\[
\det ( 
\left[ 
\begin{array}{cc}
1  &  (1-u)t w(f_j ) \\
(1-u)t w(f^{-1}_j ) & 1 
\end{array} 
\right] 
) 
=1-(1-u )^2 t^2 w(f_j ) w(f^{-1}_j ) \neq 0\  (1 \leq j \leq m_1 ). 
\]
Thus, ${\bf I}_m +(1-u)t {\bf U} {\bf J}_0 $ is invertible. 
Therefore,  
\[
\begin{array}{rcl}
\  &   & \det ( {\bf I}_m -t {\bf U} ({\bf B} -(1-u) {\bf J}_0 )) \\  
\  &   &                \\ 
\  & = & \det ( {\bf I}_m -t {\bf U} {\bf L} {}^t {\bf K} 
({\bf I}_m +(1-u)t {\bf U} {\bf J}_0 )^{-1} ) 
\det ( {\bf I}_m +(1-u)t {\bf U} {\bf J}_0 ) . 
\end{array}
\]

But, if ${\bf A}$ and ${\bf B}$ are a $m \times n $ and $n \times m$ 
matrices, respectively, then we have 
\begin{equation}
\det ( {\bf I}_m - {\bf A} {\bf B} )= 
\det ( {\bf I}_n - {\bf B} {\bf A} ) . 
\end{equation}
Thus, we have 
\[
\begin{array}{rcl}
\  &   & \det ( {\bf I}_m -t {\bf U} ({\bf B} -(1-u) {\bf J}_0 )) \\
\  &   &                \\ 
\  & = & \det ( {\bf I}_n -t \  {}^t {\bf K} 
({\bf I}_m +(1-u)t {\bf U} {\bf J}_0 )^{-1} {\bf U} {\bf L} ) 
\det ( {\bf I}_m +(1-u)t {\bf U} {\bf J}_0 ) . 
\end{array}
\]

Next, we have 
\[
\det ( {\bf I}_m +(1-u)t {\bf U} {\bf J}_0 ) 
= \prod^{m_1}_{i=1} ( 1- w(f_i ) w(f^{-1}_i ) (1-u )^2 t^2 ) . 
\]
Furthermore, the $m \times n$ matrix 
${\bf U} {\bf L} =( c_{ev} )_{e \in A(D); v \in V(D)} $ is given as follows: 
\[
c_{ev} :=\left\{
\begin{array}{ll}
w(e) & \mbox{if $t(e)=v$, } \\
0 & \mbox{otherwise. } 
\end{array}
\right.
\] 
But, we have 
\[
( {\bf I}_m +(1-u)t {\bf U} {\bf J}_0 )^{-1} =
{\bf I}_{m_0 } \oplus ( \oplus {}^{m_1 }_{j=1} 
\left[ 
\begin{array}{cc}
1/x_j & -(1-u)t  w(f_j )/x_j \\ 
-(1-u)t w(f^{-1}_j )/x_j & 1/x_j 
\end{array} 
\right] 
) 
,   
\]
where $x_i =1-w(f_i) w(f^{-1}_i ) (1-u )^2 t^2 \  (1 \leq i \leq m_1 )$.  

Now, for a symmetric arc $(x,y) \in A(D)$, 
\[
( {}^t {\bf K} ({\bf I}_m +(1-u)t {\bf U} {\bf J}_0 )^{-1} 
{\bf U} {\bf L} )_{xy} = w(x,y)/(1- w(x,y) w(y,x) (1-u )^2 t^2 ) . 
\]
For a nonsymmetric arc $(x,y) \in A(D)$, 
\[
( {}^t {\bf K} ({\bf I}_m +(1-u)t {\bf U} {\bf J}_0 )^{-1} 
{\bf U} {\bf L} )_{xy} = w(x,y) .  
\]
Furthermore, if $x=y$, then 
\[
( {}^t {\bf K} ({\bf I}_m +(1-u)t {\bf U} {\bf J}_0 )^{-1} 
{\bf U} {\bf L} )_{xx} 
=- \sum_{o(e)=x, e^{-1} \in A(D)} 
\frac{(1-u)t w(e) w(e^{-1} )}{1- w(e) w( e^{-1} ) (1-u )^2 t^2 } . 
\]

Thus, 
\[
\det ( {\bf I}_n -t \  {}^t {\bf K} 
({\bf I}_m +(1-u)t {\bf U} {\bf J}_0 )^{-1} {\bf U} {\bf L} ) 
= \det ( {\bf I}_n +(1-u) t^2 \tilde{{\bf D}} 
-t \tilde{{\bf A}}_1 -t \tilde{{\bf A}}_0 ) . 
\]
Therefore, it follows that 
\[
\zeta (D,w,u,t )^{-1} = \det ( {\bf I}_n +(1-u) t^2 \tilde{{\bf D}} 
-t \tilde{{\bf A}}_1 -t \tilde{{\bf A}}_0 ) 
\prod^{m_1}_{i=1} ( 1- w(f_i ) w(f^{-1}_i ) (1-u )^2 t^2 ) . 
\]
$\Box$

By Theorem 5, we obtain the second identity of Theorem 2. 


\newtheorem{corollary}{Corollary}
\begin{corollary}[Choe, Kwak, Park and Sato]
Let $D$ be a connected digraph, and 
let ${\bf W} = {\bf W} (D)$ be a weighted matrix of $D$. 
Furthermore, assume that $w( e^{-1} )=w(e )^{-1}$ for each $e \in A(D)$ such that 
$e^{-1} \in A(D)$. 
Then the reciprocal of the weighted Bartholdi zeta function of $D$ 
is given by 
\[
\begin{array}{rcl}
\  &  & \zeta (D,w,u,t )^{-1} 
= (1-(1-u )^2 t^2 )^{m_1 -n} \\
\  &   &                \\ 
\  & \times & 
\det ({\bf I}_n -t {\bf W}_1 (D) -(1-(1-u )^2 t^2 )t {\bf W}_0 (D) 
+(1-u) t^2 ( {\bf S} -(1-u) {\bf I}_n )) . 
\end{array}
\]
where $n= \mid V(D) \mid $ and $m= \mid A(D) \mid $. 
\end{corollary}


{\bf Proof}.  Since $w( e^{-1} )=w(e )^{-1}$ for each symmetric arc $e \in A(D)$, 
we have $w( e^{-1} )$ $w(e )^{-1} =1$. 
Then we have 
\[
\tilde{{\bf D}} = \frac{1}{1-(1-u )^2 t^2} {\bf S} , \  
\tilde{{\bf A}}_1 = \frac{1}{1-(1-u )^2 t^2} {\bf W}_1 (D). 
\]
Furthermore, $\tilde{{\bf A}}_0 = {\bf W}_0 (D)$. 
Thus, 
\[
\begin{array}{rcl}
\  &  & \zeta (D,w,u,t )^{-1} 
= (1-(1-u )^2 t^2 )^{m_1 } \\
\  &   &                \\ 
\  & \times & 
\det ({\bf I}_n -t/(1-(1-u )^2 t^2 ) {\bf W}_1 (D) -t {\bf W}_0 (D) 
+(1-u) t^2 /(1-(1-u )^2 t^2 ) {\bf S} ) \\
\  &   &                \\ 
\  & = & (1-(1-u )^2 t^2 )^{m_1 -n} 
\det ({\bf I}_n -t {\bf W}_1 (D) -(1-(1-u )^2 t^2 )t {\bf W}_0 (D) \\
\  &   &                \\ 
\  & + & (1-u) t^2 ( {\bf S} -(1-u) {\bf I}_n )) .  
\end{array}
\] 
$\Box$ 


\section{Weighted Bartholdi zeta functions of group coverings of digraphs}

We can generalize the notion of a $\Gamma$-covering of a graph to a simple 
digraph. 
Let $D$ be a connected digraph and $ \Gamma $ a finite group.
Then a mapping $ \alpha : A(D) \longrightarrow \Gamma $
is called a {\em pseudo ordinary voltage} {\em assignment}
if $ \alpha (v,u)= \alpha (u,v)^{-1} $ for each $(u,v) \in A(D)$ 
such that $(v,u) \in A(D)$.
The pair $(D, \alpha )$ is called an 
{\em ordinary voltage digraph}.
The {\em derived digraph} $D^{ \alpha } $ of the ordinary
voltage digraph $(D, \alpha )$ is defined as follows:
$V(D^{ \alpha } )=V(D) \times \Gamma $ and $((u,h),(v,k)) \in 
A(D^{ \alpha })$ if and only if $(u,v) \in A(D)$ and $k=h \alpha (u,v) $. 
The digraph $D^{ \alpha }$ is called a {\em $ \Gamma $-covering} of $D$.
Note that a $\Gamma$-covering of the symmetric digraph corresponding 
to a graph $G$ is a $\Gamma$-covering of $G$(see [5]). 


Let $D$ be a connected digraph, $ \Gamma $ a finite group and 
$ \alpha : A(D) \longrightarrow \Gamma $ a pseudo ordinary voltage assignment. 
In the $\Gamma $-covering $D^{ \alpha } $, set $v_g =(v,g)$ and $e_g =(e,g)$, 
where $v \in V(D), e \in A(D), g\in \Gamma $. 
For $e=(u,v) \in A(D)$, the arc $e_g$ emanates from $u_g$ and 
terminates at $v_{g \alpha (e)}$. 

Let ${\bf W} = {\bf W} (D)$ be a weighted matrix of $D$. 
Then we define the {\em weighted matrix} 
$\tilde{{\bf W}} = {\bf W} ( D^{ \alpha } )
=( \tilde{w} ( u_g , v_h ))$ of $ D^{ \alpha } $ 
{\em derived from} ${\bf W}$ as follows: 
\[
\tilde{w} ( u_g , v_h ) :=\left\{
\begin{array}{ll}
w(u,v) & \mbox{if $(u,v) \in A(D)$ and $h=g \alpha (u,v)$, } \\
0 & \mbox{otherwise.}
\end{array}
\right.
\]


If \( {\bf M}_{1} = {\bf M}_{2} = \cdots = {\bf M}_{s} = {\bf M} \),
then we write 
\( s \circ {\bf M} = {\bf M}_{1} \oplus \cdots \oplus {\bf M}_{s} \).
The {\em Kronecker product} $ {\bf A} \bigotimes {\bf B} $
of matrices {\bf A} and {\bf B} is considered as the matrix 
{\bf A} having the element $a_{ij}$ replaced by the matrix $a_{ij} {\bf B}$.


\begin{theorem} 
Let $D$ be a connected digraph with $n$ vertices and $m$ arcs, 
$ \Gamma $ a finite group, $ \alpha : A(D) \longrightarrow \Gamma  $ 
a pseudo ordinary voltage assignment and ${\bf W} = {\bf W} (D)$ a weighted 
matrix of $D$. 
Set $m_1 = \mid \{ e \in A(D) \mid e^{-1} \in A(D) \} \mid /2$ and 
$\mid \Gamma \mid =r$. 
Furthermore, let $ {\rho}_{1} =1, {\rho}_{2} , \cdots , {\rho}_{k} $
be the irreducible representations of $ \Gamma $, and 
$d_i$ the degree of $ {\rho}_{i} $ for each $i$, where 
$d_1=1$.
For $g \in \Gamma $, the matrix ${\bf A} {}_{1,g} =(a^{(g)}_{xy} )$ 
is defined as follows:
\[
a^{(g)}_{xy} :=\left\{
\begin{array}{ll}
w(x,y)/(1-w(x,y) w(y,x) (1-u )^2 t^2 ) & \mbox{if $(x,y),(y,x) \in A(D)$ 
and $ \alpha (x,y)= g $, } \\
0 & \mbox{otherwise.}
\end{array}
\right.
\]
Furthermore, the matrix ${\bf A} {}_{0,g} =(b^{(g)}_{xy} )$ 
is defined as follows:
\[
b^{(g)}_{xy} :=\left\{
\begin{array}{ll}
w(x,y) & \mbox{if $(x,y) \in A(D)$,$(y,x) \not\in A(D)$ and 
$ \alpha (x,y)= g $, } \\
0 & \mbox{otherwise.}
\end{array}
\right.
\]

Suppose that the $ \Gamma $-covering $D^{ \alpha } $ of $D$ is connected. 
Then the reciprocal of the weighted Bartholdi zeta function of $D {}^{ \alpha } $ is
\[
\zeta (D {}^{ \alpha } , \tilde{w} ,u,t )^{-1} 
= \prod^{m_1}_{i=1} ( 1- w(f_i ) w(f^{-1}_i )(1-u )^2 t^2 )^r  
\]
\[
\times 
\prod^{k}_{i=1} 
\{ \det ({\bf I}_{n d_i } -t \sum_{h \in \Gamma } 
{\rho}_{i} (h) \bigotimes {\bf A} {}_{1,h} 
-t \sum_{h \in \Gamma } 
{\rho}_{i} (h) \bigotimes {\bf A} {}_{0,h} 
+(1-u) t^2 
( {\bf I}_{d_i} \bigotimes \tilde{{\bf D}} (D))) \} {}^{d_i} , 
\] 
where $f^{\pm 1}_1 , \ldots , f^{\pm 1}_{m_1} $ are symmetric arcs of $D$. 
\end{theorem}


{\bf Proof }. 
Let $V(D)= \{ v_1, \cdots , v_{n} \} $ and 
$ \Gamma = \{ 1=g_1, g_2, \cdots ,g_r \} $.
Arrange vertices of $D^{ \alpha } $ in $n$ blocks:
$(v_1,1), \cdots , (v_{n},1);(v_1,g_2), \cdots , (v_{n},g_2); 
\cdots ; (v_1,g_r), \cdots ,(v_{n},g_r). $
We consider the three matrices $\tilde{{\bf A}}_1 ( D {}^{ \alpha } ) $, 
$\tilde{{\bf W}}_0 ( D {}^{ \alpha } ) $ and $\tilde{{\bf D}} ( D {}^{ \alpha } ) $
under this order.
By Theorem 5, we have 
\[
\zeta (D {}^{ \alpha } , \tilde{w} ,u,t )^{-1} =
\det ( {\bf I}_{\nu m} -t \tilde{{\bf A}} {}_1 ( D^{\alpha} )
-t \tilde{{\bf A}} {}_0 ( D^{\alpha} )+
(1-u) t^2 \tilde{{\bf D}} ( D^{\alpha } )) 
\]
\[
\cdot \prod^{m_1}_{i=1} ( 1- w(f_i ) w(f^{-1}_i )(1-u )^2 t^2 )^r . 
\]

For $h \in \Gamma $, the matrix ${\bf P}_{h}=(p^{(h)}_{ij} )$ 
is defined as follows:
\[
p^{(h)}_{ij} = \left\{
\begin{array}{ll}
1 & \mbox{if $g_i h=g_j$,} \\
0 & \mbox{otherwise.}
\end{array}
\right.
\]
Suppose that $p^{(h)}_{ij} =1 $, i.e., $g_j=g_ih$.
Then $((u,g_i),(v,g_j)) \in A(D {}^{ \alpha } ) $
if and only if $(u,v) \in A(D)$ and 
$g_{j} = g_{i} \alpha (u,v)$,
i.e., $ \alpha (u,v)=g^{-1}_{i} g_j =g^{-1}_{i} g_i h=h$.
Thus we have
\[
\tilde{{\bf A}} {}_0 (D {}^{ \alpha } )= \sum_{h \in \Gamma } {\bf P}_{h} 
\bigotimes {\bf A} {}_{0,h} \  and \  
\tilde{{\bf A}} {}_1 (D {}^{ \alpha } )= \sum_{h \in \Gamma } {\bf P}_{h} 
\bigotimes {\bf A} {}_{1,h} . 
\]

Let $\rho$ be the right regular representation of $ \Gamma $.
Furthermore, let $ {\rho}_{1} =1, {\rho}_{2} , \cdots , {\rho}_{k} $
be the irreducible representations of $ \Gamma $, and 
$d_i$ the degree of $ {\rho}_{i} $ for each $i$, where 
$d_1=1$.
Then we have $\rho (h)= {\bf P}_{h} $ for $h \in \Gamma $.
Furthermore, there exists a nonsingular matrix ${\bf P}$ such that 
${\bf P}^{-1} \rho (h) {\bf P} = (1) \oplus d_2 \circ {\rho}_{2} (h) 
\oplus \cdots \oplus d_k \circ {\rho}_{k} (h)$ 
for each $h \in \Gamma $(see [10]). 
Putting 
${\bf B} =( {\bf P}^{-1} \bigotimes {\bf I}_{n} ) 
( \tilde{{\bf A}}_1 (D {}^{ \alpha } )+ \tilde{{\bf A}}_0 (D {}^{ \alpha } )) 
( {\bf P} \bigotimes {\bf I}_{n} )$,
we have 
\[
{\bf B}= \sum_{h \in \Gamma } 
\{ (1) \oplus d_2 \circ {\rho}_{2} (h) \oplus \cdots \oplus 
d_k \circ {\rho}_{k} (h) \} \bigotimes 
( {\bf A}_{1,h} + {\bf A}_{0,h} ) . 
\]
Note that $\tilde{{\bf A}}_i (D) = \sum_{h \in \Gamma } {\bf A} {}_{i,h}\  (i=0,1)$ 
and $1+ d^2_2 + \cdots + d^2_k =r$.
Therefore it follows that 
\[
\zeta (D {}^{ \alpha } , \tilde{w} ,u,t )^{-1} 
= \prod^{m_1}_{j=1} ( 1- w(f_j ) w(f^{-1}_j )(1-u )^2 t^2 )^{r} 
\]
\[
\times 
\prod^{k}_{i=1} \det ({\bf I}_{n d_i } -t \sum_{h \in \Gamma } 
{\rho}_{i} (h) \bigotimes {\bf A} {}_{1,h} 
-t \sum_{h \in \Gamma } 
{\rho}_{i} (h) \bigotimes {\bf A} {}_{0,h} 
+(1-u) t^2 
( {\bf I}_{d_i} \bigotimes \tilde{{\bf D}} (D))) {}^{d_i} .
\]
$\Box$  


\section{$L$-functions of digraphs}

Let $D$ be a connected digraph with $m$ arcs, $ \Gamma $ a finite group, 
$ \alpha : A(D) \longrightarrow \Gamma $ a pseudo ordinary voltage 
assignment and ${\bf W} = {\bf W} (D)$ a weighted matrix of $D$. 
For each path $P=( e_1, \cdots , e_l)$ of $D$, set 
$ \alpha (P)= \alpha ( e_1 ) \cdots \alpha ( e_l )$ and 
$w(P)=w( e_1 ) \cdots w( e_l )$. . 
Furthermore, let $ \rho $ be a representation of $ \Gamma $ 
and $d$ its degree. 

The {\em weighted Bartholdi $L$-function} of $D$ associated with 
$ \rho $ and $ \alpha $ is defined by 
\[
\zeta {}_D (w, u,t, \rho , \alpha )= 
\prod_{[C]} \det ( {\bf I}_d -w(C) \rho ( \alpha (C)) 
u^{ cbc(C) } t^{ \mid C \mid } )^{-1} , 
\]
where $[C]$ runs over all equivalence classes of prime cycles of $D$. 


Two $md \times md $ matrices 
${\bf B}^{\rho }_w =( {\bf B}_{e,f} )_{e,f \in A(D)} $ and 
${\bf J}^{\rho }_w =( {\bf J}_{e,f} )_{e,f \in A(D)} $ are defined as follows: 
\[
{\bf B}_{e,f} =\left\{
\begin{array}{ll}
w(e) \rho ( \alpha (e)) & \mbox{if $t(e)=o(f)$, } \\
{\bf 0}_d & \mbox{otherwise}
\end{array}
\right.
, 
{\bf J}_{e,f} =\left\{
\begin{array}{ll}
w(e) \rho ( \alpha (e)) & \mbox{if $f= e^{-1} $, } \\
{\bf 0}_d & \mbox{otherwise.}
\end{array}
\right.
\]

A determinant expression for the weighted Bartholdi $L$-function of $D$ associated with 
$ \rho $ and $ \alpha $ was given by Choe, Kwak, Park and Sato [3]. 
Let $1 \leq i,j \leq n$. 
Then the {\em $(i,j)$-block} ${\bf F}_{ij} $ of a $dn \times dn$ 
matrix ${\bf F} $ is the submatrix of ${\bf F} $ consisting of 
$d(i-1)+1, \ldots , di$ rows and $d(j-1)+1, \ldots , dj$ columns. 


\begin{theorem}[Choe, Kwak, Park and Sato]
Let $D$ be a connected digraph with $m$ arcs, 
$ \Gamma $ a finite group, $ \alpha : A(D) \longrightarrow \Gamma $ 
a pseudo ordinary voltage assignment and 
${\bf W} = {\bf W} (D)$ a weighted matrix of $D$. 
Furthermore, let $ \rho $ be a representation of $ \Gamma $, 
and $d$ the degree of $ \rho $. 
Then the reciprocal of the weighted Bartholdi $L$-function of $D$ associated with 
$ \rho $ and $ \alpha $ is
\[
\zeta {}_D (w,u,t, \rho , \alpha )^{-1} 
= \det ({\bf I}_{md} -( {\bf B}^{\rho }_w -(1-u) {\bf J}^{\rho }_w )t) . 
\]
\end{theorem}


A new determinant expression for the weighted Bartholdi $L$-function of $D$ associated with 
$ \rho $ and $ \alpha $ is given as follows: 


\begin{theorem}
Let $D$ be a connected digraph, and 
let ${\bf W} = {\bf W} (D)$ be a weighted matrix of $D$. 
Then the reciprocal of the weighted Bartholdi $L$-function of $D$ 
is given by 
\[
\zeta {}_D (w,u,t, \rho , \alpha )^{-1} 
= \prod^{m_1}_{i=1} ( 1- w(f_i ) w(f^{-1}_i ) (1-u )^2 t^2 )^d 
\]
\[
\times 
\det ( {\bf I}_{nd} +(1-u) t^2 {\bf I}_d \bigotimes \tilde{{\bf D}} (D) 
-t \sum_{g \in \Gamma } \rho (g) \bigotimes {\bf A}_{1,g}  
-t \sum_{g \in \Gamma } \rho (g) \bigotimes {\bf A}_{0,g} ) , 
\]
where $n= \mid V(D) \mid $, $m= \mid A(D) \mid $ and 
 $f^{\pm 1}_1 , \ldots , f^{\pm 1}_{m_1} $ are symmetric arcs of $D$.  
\end{theorem}


{\bf Proof}.  Let $V(D)= \{ v_1, \cdots , v_{n} \} $ and, 
let $A(D)= \{ e_1, \cdots , e_{m {}_0 } , f_{1}, 
\cdots , f_{m_1}, f^{-1}_{1}, \cdots $, $f^{-1}_{m_1} \} $ 
such that $e^{-1}_i \not\in A(D) (1 \leq i \leq m_0 )$. 
Note that  $m= m_0 +2 m_1 $. 

Arrange arcs of $D$ as follows: 
\[
e_1, \cdots , e_{m_0 } , f_{1}, f^{-1}_1 , 
\cdots , f_{ m_1}, f^{-1}_{m_1 } . 
\]
Let 
\[
{\bf U} =
\left[ 
\begin{array}{cccccc}
w(e_1 ) &   &   &   &   & 0 \\
   & \ddots &   &   &   &   \\
   &    & w(e_{m_0} ) &   &   &  \\
   &    &   & w(f_1)  &   &  \\ 
   &    &   &   & w(f^{-1}_1 ) &  \\
   &    &   &   &   &  \ddots 
\end{array} 
\right] 
. 
\]
Furthermore, let two $md \times md$ matrices 
${\bf B}_{\rho } =( {\bf B}^{\rho }_{e,f} )_{e,f \in A(D)} $ and 
${\bf J}_{\rho } =( {\bf J}^{\rho }_{e,f} )_{e,f \in A(D)} $ be defined as follows: 
\[
{\bf B}^{\rho }_{e,f} =\left\{
\begin{array}{ll}
\rho ( \alpha (e)) & \mbox{if $t(e)=o(f)$, } \\
{\bf 0}_d & \mbox{otherwise}
\end{array}
\right.
, 
{\bf J}^{\rho }_{e,f} =\left\{
\begin{array}{ll}
\rho ( \alpha (e)) & \mbox{if $f= e^{-1} $, } \\
{\bf 0}_d & \mbox{otherwise.}
\end{array}
\right.
\]
Then we have 
\[
( {\bf U} \bigotimes {\bf I}_d ) {\bf B}_{\rho } = {\bf B}^{\rho }_w \  and \  
( {\bf U} \bigotimes {\bf I}_d ) {\bf J}_{\rho } = {\bf J}^{\rho }_w . 
\]
Thus, 
\[
{\bf B}^{\rho }_w -(1-u) {\bf J}^{\rho }_w =( {\bf U} \bigotimes {\bf I}_d )
({\bf B}_{\rho } -(1-u) {\bf J}_{\rho } ) . 
\]
By Theorem 7, it follows that 
\[
 \zeta {}_D (w,u,t, \rho , \alpha )^{-1} = \det ( {\bf I}_{md}  
-t ( {\bf U} \bigotimes {\bf I}_d )({\bf B}_{\rho } -(1-u) {\bf J}_{\rho } )) . 
\]


Now, let ${\bf K} =( {\bf K}_{ev} )$ ${}_{e \in A(D); v \in V(D)} $ be the $md \times nd$ matrix defined 
as follows: 
\[
{\bf K}_{ev} :=\left\{
\begin{array}{ll}
{\bf I}_d & \mbox{if $o(e)=v$, } \\
{\bf 0}_d & \mbox{otherwise. } 
\end{array}
\right.
\]
Furthermore, we define the $md \times nd$ matrix 
${\bf L} =( {\bf L}_{ev} )_{e \in A(D); v \in V(D)} $ as follows: 
\[
{\bf L}_{ev} :=\left\{
\begin{array}{ll}
\rho ( \alpha (e)) & \mbox{if $t(e)=v$, } \\
{\bf 0}_d & \mbox{otherwise. } 
\end{array}
\right.
\] 

Set ${\bf U}_d = {\bf U} \bigotimes {\bf I}_d $. 
Then we have 
\[
{\bf L} {}^t {\bf K} = {\bf B}_{\rho } . 
\]
Thus, 
\[
\begin{array}{rcl}
\  &   & \det ( {\bf I}_{md} -t {\bf U}_d ({\bf B}_{\rho } -(1-u) {\bf J}_{\rho } )) \\  
\  &   &                \\ 
\  & = & \det ( {\bf I}_{md} -t {\bf U}_d 
({\bf L} {}^t {\bf K} -(1-u) {\bf J}_{\rho } )) 
= \det ( {\bf I}_{md} -t {\bf U}_d {\bf L} {}^t {\bf K} 
+(1-u)t {\bf U}_d {\bf J}_{\rho } ) . 
\end{array}
\]


But, we have 
\[
{\bf I}_{md} +(1-u)t {\bf U}_d {\bf J}_{\rho } 
\]
\begin{equation}
= {\bf I}_{m_0 d} \oplus ( \oplus {}^{m_1 }_{j=1}
\left[ 
\begin{array}{cc}
{\bf I}_d &  (1-u)t w(f_j ) \rho ( \alpha (f_j )) \\
(1-u)t w(f^{-1}_j ) \rho ( \alpha (f^{-1}_j )) & {\bf I}_d  
\end{array} 
\right] 
) 
. 
\end{equation}
Since  $|u|,|t|$ are sufficiently small, we have 
\[
\det ( 
\left[ 
\begin{array}{cc}
{\bf I}_d &  (1-u)t w(f_j ) \rho ( \alpha (f_j )) \\
(1-u)t w(f^{-1}_j ) \rho ( \alpha (f^{-1}_j )) & {\bf I}_d 
\end{array} 
\right] 
) 
\]
\[
=(1-(1-u )^2 t^2 w(f_j ) w(f^{-1}_j ) )^d \neq 0\  (1 \leq j \leq m_1 ). 
\]
Thus, ${\bf I}_{md} +(1-u)t {\bf U}_d {\bf J}_{\rho } $ is invertible. 
Therefore,  
\[
\begin{array}{rcl}
\  &   & \det ( {\bf I}_{md} -t {\bf U}_d ({\bf B}_{\rho } -(1-u) {\bf J}_{\rho } )) \\  
\  &   &                \\ 
\  & = & \det ( {\bf I}_{md} -t {\bf U}_d {\bf L} {}^t {\bf K} 
({\bf I}_{md} +(1-u)t {\bf U}_d {\bf J}_{\rho } )^{-1} ) 
\det ( {\bf I}_{md} +(1-u)t {\bf U}_d {\bf J}_{\rho } ) . 
\end{array}
\]

By (2), we have 
\[
\begin{array}{rcl}
\  &   & \det ( {\bf I}_{md} -t {\bf U}_d ({\bf B}_{\rho } -(1-u) {\bf J}_{\rho } )) \\
\  &   &                \\ 
\  & = & \det ( {\bf I}_{nd} -t \  {}^t {\bf K} 
({\bf I}_{nd} +(1-u)t {\bf U}_d {\bf J}_{\rho } )^{-1} {\bf U}_d {\bf L} ) 
\det ( {\bf I}_{md} +(1-u)t {\bf U}_d {\bf J}_{\rho } ) . 
\end{array}
\]

Next, we have 
\[
\det ( {\bf I}_{md} +(1-u)t {\bf U}_d {\bf J}_{\rho } ) 
= \prod^{m_1}_{i=1} ( 1- w(f_i ) w(f^{-1}_i ) (1-u )^2 t^2 )^d . 
\]
Furthermore, the $md \times nd$ matrix 
${\bf U}_d {\bf L} =( c_{ev} )_{e \in A(D); v \in V(D)} $ is given as follows: 
\[
c_{ev} :=\left\{
\begin{array}{ll}
w(e) \rho ( \alpha (e)) & \mbox{if $t(e)=v$, } \\
0 & \mbox{otherwise. } 
\end{array}
\right.
\] 

But, we have 
\[
( {\bf I}_{md} +(1-u)t {\bf U}_d {\bf J}_{\rho } )^{-1} 
\]
\[
= {\bf I}_{m_0 d} \oplus ( \oplus {}^{m_1 }_{j=1} 
\left[ 
\begin{array}{cc}
1/ x_j {\bf I}_d &  -(1-u)t w(f_j )/ x_j  \rho ( \alpha (f_j )) \\
-(1-u)t w(f^{-1}_j )/ x_j  \rho ( \alpha (f^{-1}_j )) & 1/ x_j {\bf I}_d  
\end{array} 
\right] 
) 
. 
\]
where $x_i =1-w(f_i) w(f^{-1}_i ) (1-u )^2 t^2 \  (1 \leq i \leq m_1 )$.  

But, for a symmetric arc $(x,y) \in A(D)$, 
\[
( {}^t {\bf K} ({\bf I}_{md} +(1-u)t {\bf U}_d {\bf J}_{\rho } )^{-1} 
{\bf U}_d {\bf L} )_{xy} = w(x,y)/(1- w(x,y) w(y,x) (1-u )^2 t^2 ) \rho ( \alpha (x,y)) . 
\]
For a nonsymmetric arc $(x,y) \in A(D)$, 
\[
( {}^t {\bf K} ({\bf I}_{md} +(1-u)t {\bf U}_d {\bf J}_{\rho } )^{-1} 
{\bf U}_d {\bf L} )_{xy} = w(x,y) \rho ( \alpha (x,y)) .  
\]
Furthermore, if $x=y$, then 
\[
( {}^t {\bf K} ({\bf I}_{md} +(1-u)t {\bf U}_d {\bf J}_{\rho } )^{-1} 
{\bf U}_d {\bf L} )_{xx} 
=- \sum_{o(e)=x, e^{-1} \in A(D)} 
\frac{(1-u)t w(e) w(e^{-1} )}{1- w(e) w( e^{-1} ) (1-u )^2 t^2 } {\bf I}_d . 
\]

Thus, 
\[
\det ( {\bf I}_{nd} -t \  {}^t {\bf K} 
({\bf I}_{nd} +(1-u)t {\bf U}_d {\bf J}_{\rho } )^{-1} {\bf U}_d {\bf L} ) 
\]
\[
= \det ( {\bf I}_{nd} +(1-u) t^2 \tilde{{\bf D}} (D) \bigotimes {\bf I}_d 
-t \sum_{g \in \Gamma } {\bf A}_{1,g} \bigotimes \rho (g) 
-t \sum_{g \in \Gamma } {\bf A}_{0,g} \bigotimes \rho (g)) , 
\]
Therefore, it follows that 
\[
\zeta {}_D (w,u,t, \rho , \alpha  )^{-1}  
= \prod^{m_1}_{i=1} ( 1- w(f_i ) w(f^{-1}_i ) (1-u )^2 t^2 )^d 
\]
\[
\times 
\det ( {\bf I}_{nd} +(1-u) t^2 {\bf I}_d \bigotimes \tilde{{\bf D}} (D) 
-t \sum_{g \in \Gamma } \rho (g) \bigotimes {\bf A}_{1,g} 
-t \sum_{g \in \Gamma } \rho (g) \bigotimes {\bf A}_{0,g} ) , 
\] 
$\Box$


By Theorems 6,8, the following result holds. 


\begin{corollary}[Choe, Kwak, Park and Sato]
Let $D$ be a connected digraph, $ \Gamma $ a finite group, 
$ \alpha : A(D) \longrightarrow \Gamma $ a pseudo ordinary voltage 
assignment and ${\bf W} = {\bf W} (D)$ a weighted matrix of $D$. 
Then we have 
\[
\zeta (D {}^{ \alpha  } , \tilde{w} ,u,t) = \prod_{ \rho } 
\zeta {}_D (w,u,t, \rho, \alpha )^{ \deg \rho } , 
\]
where $ \rho $ runs over all inequivalent irreducible representations 
of $ \Gamma $. 
\end{corollary}


\section{Bartholdi edge zeta function of a digraph} 

Let $D$ be a connected digraph with $m$ arcs $e_1 , \ldots , e_m $. 
Furthermore, let $z_1 , \ldots z_{m} $ be $m$ variables. 
Set $z_{e_i } =z_i (1 \leq i \leq m)$ and ${\bf z} =( z_1 , \ldots , z_m )$. 
Then the {\em Bartholdi edge zeta function} $\zeta (D,w,u)$ of $D$ is defined by 
\[
\zeta (D,{\bf z} ,u)= \prod_{[C]} (1-g(C) u^{cbc(C)} )^{-1} , 
\]
where $[C]$ runs over all equivalence classes of prime cycles of $D$. 
If $D=D_G$ is the symmetric digraph of a graph $G$, then 
the Bartholdi edge zeta function $\zeta (D_G,{\bf z} ,u)$ of $D_G$ is called the 
{\em Bartholdi edge zeta function} $\zeta (G,{\bf z} ,u)$ of $G$. 

Now, set $|V(D)|=n$. 
Then we define an $n \times n$ matrix  
${\bf A}^{\prime }_1 ={\bf A}^{\prime }_1 (D)=(a_{xy} )$ as follows: 
\[
a_{xy} =\left\{
\begin{array}{ll}
z_{(x,y)} /(1- z_{(x,y)} z_{(y,x)} (1-u )^2 ) & \mbox{if both $(x,y)$ and $(y,x) \in A(D)$, } \\
0 & \mbox{otherwise.  }
\end{array}
\right. 
\]
Furthermore, an $n \times n$ matrix ${\bf D}^{\prime } = {\bf D}^{\prime } (D)= (d_{xy} )$ 
is the diagonal matrix defined by 
\[
d_{xx} = \sum_{o(e)=x, e^{-1} \in A(D)} 
\frac{z_e z_{e^{-1} }}{1- z_e z_{ e^{-1} } (1-u )^2 } . 
\] 

Substituting $t=1$ in Theorem 5, we obtain the following result. 


\begin{corollary}
Let $D$ be a connected digraph with $m$ arcs and 
let ${\bf z} =( z_1 , \ldots , z_m )$ be $m$ variables.
Then the reciprocal of the Bartholdi edge zeta function of $D$ 
is given by 
\[
\zeta (D,{\bf z} ,u )^{-1} = \det ( {\bf I}_n +(1-u) {\bf D}^{\prime }  
- {\bf A}^{\prime }_1 (D) - \tilde{{\bf A}}_0 )  
\prod^{m_1}_{i=1} ( 1- z_{f_i } z_{f^{-1}_i } (1-u )^2 ) ,  
\]
where  $n= \mid V(D) \mid $ and 
$f^{\pm 1}_1 , \ldots , f^{\pm 1}_{m_1} $ are symmetric arcs of $D$. 
\end{corollary}


If $D=D_G$, then 

\begin{corollary}
Let $G$ be a connected graph with $m$ edges and 
let ${\bf z} =( z_1 , \ldots , z_{2m} )$ be $2m$ variables.and 
let ${\bf W} = {\bf W} (G)$ be a weighted matrix of $G$. 
Then the reciprocal of the Bartholdi edge zeta function of $G$ 
is given by 
\[
\zeta (G,{\bf z} ,u )^{-1} = \det ( {\bf I}_n +(1-u) {\bf D}^{\prime }  
- {\bf A}^{\prime }_1 (G) - \tilde{{\bf A}}_0 )  
\prod^{m}_{i=1} ( 1- z_{f_i } z_{f^{-1}_i } (1-u )^2 ) ,  
\]
where  $n= \mid V(G) \mid $ and 
$D(G)= \{ f^{\pm 1}_1 , \ldots , f^{\pm 1}_{m} \}$.  
\end{corollary}


\section{Example}

Finally, we give an example.
Let $D$ be the digraph with three vertices $v_1 , v_2 , v_3 $ and 
five arcs $( v_1 ,v_2 )$, $( v_2 ,v_1 ),( v_2 ,v_3 ),( v_3 ,v_2 ),
( v_3 ,v_1 )$. 
Furthermore, let 
\[
{\bf W} (D)=
\left[ 
\begin{array}{ccc}
0 & a & 0 \\
b & 0 & c \\
d & e & 0 
\end{array} 
\right] 
. 
\]
Then we have $n=3, m=5, m_1 =2$. 
By Theorem 5, we have 
\[
\begin{array}{rcl}
\  &  & {\zeta} (D,w,u,t )^{-1} \\
\  &   &                \\ 
\  & = & (1-ab(1-u )^2 t^2 )(1-ce(1-u )^2 t^2 ) 
\det ({\bf I}_3 -t \tilde{{\bf A}}_1 -t \tilde{{\bf A}}_0 
+(1- u) t^2 \tilde{{\bf D}} ) \\ 
\  &   &                \\ 
\  & = & AB 
\det \left(
\left[ 
\begin{array}{ccc}
1+abF/A & -at/A & 0 \\
-bt/A & 1+abF/A+ceF/B & -ct/B \\
-dt & -et/B & 1+ceF/B 
\end{array} 
\right] 
\right)
\\
\  &   &                \\ 
\  & = & 1-(ab+ce) u^2 t^2 +abce( u^4 -u^2 ) t^4 -acd t^3 ,  
\end{array}
\]
where $A=1-ab(1-u )^2 t^2 $, $B=1-ce(1-u )^2 t^2 $ and $F=(1-u) t^2 $. 

Let $\Gamma = Z_3 = \{ 1, \tau , { \tau }^2 \} ( { \tau }^3 =1)$ be 
the cyclic group of order 3, and let 
$ \alpha : A(D) \longrightarrow Z_3$ be the pseudo ordinary voltage 
assignment such that $ \alpha ( v_1, v_2 )= \tau $,
$ \alpha ( v_2, v_1 )= \tau {}^2 $ and 
$ \alpha ( v_2 , v_3 )= \alpha ( v_3 , v_2 )= \alpha ( v_3 , v_1 )=1$. 
The characters of ${\bf Z}_3$ are given as follows: 
$\chi {}_i ( \tau {}^j )=( \xi {}^i )^j $, $ 0 \leq i, j \leq 2$, 
where $ \xi = \frac{-1+ \sqrt{-3} }{2} $. 

Now, we present the weighted Bartholdi $L$-function 
${\zeta}_D (w,u,t, \chi {}_1 , \alpha )$ of $D$ 
associated with $ \chi {}_1 $ and $ \alpha $. 
Theorem 8 implies that 
\[
\begin{array}{rcl}
\  &  & {\zeta}_D (w,u,t, \chi {}_1 , \alpha )^{-1} \\
\  &   &                \\ 
\  & = & AB  
\det ({\bf I}_3 -t \sum^2_{i=0} \chi {}_1 ( \tau {}^i ) 
{\bf A}_{1, \tau {}^i } -t \sum^2_{i=0} \chi {}_1 ( \tau {}^i ) 
{\bf A}_{0, \tau {}^i } 
+(1-u) t^2 \tilde{{\bf D}} ) \\ 
\  &   &                \\ 
\  & = & AB  
\det \left(
\left[ 
\begin{array}{ccc}
1+abF/A & -at \xi /A & 0 \\
-bt \xi {}^2 /A & 1+abF/A+ceF/B & -ct/B \\
-dt & -et/B & 1+ceF/B 
\end{array} 
\right] 
\right)
\\
\  &   &                \\ 
\  & = & 1-(ab+ce) u^2 t^2 +abce( u^4 -u^2 ) t^4 -acd t^3 \xi . 
\end{array}
\]

Similarly, we have 
\[
{\zeta}_D (w,u,t, \chi {}_2 , \alpha )^{-1} = 
1-(ab+ce) u^2 t^2 +abce( u^4 -u^2 ) t^4 -acd t^3 \xi {}^2 . 
\]
By Corollary 2, it follows that 
\[
{\zeta} ( D^{\alpha} , \tilde{w} , u,t )^{-1} = 
{\zeta} (D,w, u,t )^{-1} {\zeta}_D (w,u,t, \chi {}_1 , \alpha )^{-1} 
{\zeta}_D (w,u,t, \chi {}_2 , \alpha )^{-1} 
\]
\[
=(1-(ab+ce) u^2 t^2 +abce( u^4 -u^2 ) t^4 )^3 - a^3 c^3 d^3 t^9 . 
\]

If $w( e^{-1} )=w(e )^{-1}$ for each symmetric arc $e \in A(D)$, then 
\[
{\zeta} (D,w,u,t )^{-1} =1-2 u^2 t^2 +( u^4 -u^2 ) t^4 -acd t^3 ,  
\]
\[
{\zeta}_D (w,u,t, \chi {}_i , \alpha )^{-1} 
=1-2 u^2 t^2 +( u^4 -u^2 ) t^4 -acd t^3 \xi {}^i \  (i=1,2) 
\]
and 
\[
{\zeta} ( D^{\alpha} , \tilde{w} , u,t )^{-1} 
=(1-2 u^2 t^2 +( u^4 -u^2 ) t^4 )^3 - a^3 c^3 d^3 t^9 . 
\]

 
\vspace{5mm}

\begin{center}
{\bf Acknowledgment} 
\end{center}

We would like to thank the referee for many valuable comments and 
many helpful suggestions. 

\vspace{5mm}


\begin{thebibliography}{99}


\item
L. Bartholdi,
Counting paths in graphs, 
Enseign. Math. 45 (1999) 83-131.

\item
H. Bass,
The Ihara-Selberg zeta function of a tree lattice,
Internat. J. Math. 3 (1992) 717-797.

\item
Y. Choe, J. H. Kwak, Y. S. Park and I. Sato, 
Bartholdi zeta and $L$-functions of weighted digraphs, their coverings and products, 
Adv. Math. {\bf 213} (2007), 865-886. 

\item
D. Foata and D. Zeilberger,
A combinatorial proof of Bass's evaluations of the Ihara-Selberg zeta 
function for graphs, 
Trans. Amer. Math. Soc. 351 (1999) 2257-2274.

\item
J. L. Gross and T. W. Tucker,
Topological Graph Theory,
(Wiley-Interscience, New York, 1987).

\item
K. Hashimoto,
Zeta Functions of Finite Graphs and Representations 
of $p$-Adic Groups,
in Adv. Stud. Pure Math. Vol. 15 (Academic Press, New York, 1989) 
211-280. 

\item
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.

\item
M. Kotani and T. Sunada,
Zeta functions of finite graphs,
J. Math. Sci. U. Tokyo 7 (2000) 7-25.

\item
H. Mizuno and I. Sato,
Bartholdi zeta functions of graph coverings,
J. Combin. Theory Ser. B. 89 (2003) 27-41.

\item
J. -P. Serre, 
Linear Representations of Finite Group,
(Springer-Verlag, New York, 1977).

\item
H. M. Stark and A. A. Terras,
Zeta functions of finite graphs and coverings,
Adv. Math. 121 (1996) 124-165.

\item
T. Sunada,
$L$-Functions in Geometry and Some Applications, 
in Lecture Notes in Math., Vol. 1201 
(Springer-Verlag, New York, 1986) 266-284. 

\item
T. Sunada,
Fundamental Groups and Laplacians(in Japanese),
(Kinokuniya, Tokyo, 1988).


\item 
Y. Watanabe and K. Fukumizu, 
Graph zeta function in the Bethe free energy and loopy belief propagation,
Advances in Neural Information Processing Systems 22 (2010), 2017-2025. 

\end{thebibliography}




\end{document}