The Partial differential coefficients for the second weghted Bartholdi zeta function of a graph

TL;DR

This paper extends the theory of the Bartholdi zeta function of a graph by introducing weighted derivatives and provides a formula relating the Kirchhoff index of regular coverings to that of the original graph.

Contribution

It introduces weighted partial derivatives for the second weighted Bartholdi zeta function and derives a formula for the Kirchhoff index of regular coverings in terms of the base graph.

Findings

Weighted derivatives of the Bartholdi zeta function are derived.

A formula relating Kirchhoff indices of regular coverings and base graphs is established.

Extensions of Li and Hou's results to weighted cases are provided.

Abstract

We consider the second weighted Bartholdi zeta function of a graph $G$, and present weighted versions for the result of Li and Hou's on the partial derivatives of the determinant part in the determinant expression of the Bartholdi zeta function of $G$. Furthermore, we give a formula for the weighted Kirchhoff index of a regular covering of $G$ in terms of that of $G$.

Equations

$$\mathbf{Z}(G,t)=\prod _{[C]}(1-t^{\mid C\mid }{)}^{-1},$$

$$\mathbf{Z}(G,t){}^{-1}=(1-t^2{)}^{m-n}\det ({\mathbf{I}}_n-t\mathbf{A}(G)+t^2\mathbf{Q}).$$

$$(1-t{)}^{-r}\mathbf{Z}(G,t{)}^{-1}\mid {}_{t=1}=2^r\chi (G)\kappa (G),$$

$$f_G(t)=\det (\mathbf{I}-t\mathbf{A}(G)+t^2\mathbf{Q}).$$

$$f_G^{\prime }(1)=2(m-n)\kappa (G),$$

$$r_{ij}=\frac{\det ({\mathbf{L}}^{(ij)})}{\kappa (G)},$$

$$r_{ij}=0.$$

$$Kf=\sum _{1\le i<j\le n}{r}_{ij}.$$

$$Kf=n\sum _{i=2}^n\frac{1}{\mu {}_i},$$

$$K{f}^{\ast }=\sum _{1\le i<j\le n}{d}_i{d}_j{r}_{ij}.$$

$$K{f}^{+}=\sum _{1\le i<j\le n}(d_i+d_j)r_{ij}.$$

$$K{f}^z=K{f}^z(G)=\sum _{1\le i<j\le n}(d_i-2)(d_j-2)r_{ij}.$$

$$f_G^{\prime \prime }(1)=2(K{f}^z+2mn-2n^2+n)\kappa (G).$$

$$w(T)=\prod _{e\in E(T)}w(e).$$

$$\kappa {}_w(G)=\sum _{T}w(T),$$

$$f_G(w,t)=\det ({\mathbf{I}}_n-t\mathbf{W}+({\mathbf{D}}_w-{\mathbf{I}}_n)t^2),$$

$$\kappa {}_w(G)=\frac{1}{2(w(G)-n)}{f}^{\prime }(w,1),$$

$$r_{pq}=\frac{\det ({\mathbf{L}}^{(pq)})}{\kappa {}_w(G)},$$

$$K{f}_w=\sum _{1\le i<j\le n}{r}_{ij},$$

$$K{f}_w^{\ast }=\sum _{1\le i<j\le n}{d}_i^w{d}_j^w{r}_{ij}$$

$$K{f}_w^{+}=\sum _{1\le i<j\le n}(d_i^w+d_j^w)r_{ij}.$$

$$K{f}_w^z=K{f}_w^z(G)=\sum _{1\le p<q\le n}(d_p^w-2)(d_q^w-2)r_{pq}.$$

$$f_G^{\prime \prime }(w,1)=2\{K{f}_w^z+2w(G)n-2n^2+n\}\kappa {}_w(G).$$

$$\zeta (G,u,t)=\prod _{[C]}(1-u^{cbc(C)}{t}^{\mid C\mid }{)}^{-1},$$

$$\zeta (G,u,t{)}^{-1}=(1-(1-u{)}^2{t}^2{)}^{m-n}\det ({\mathbf{I}}_n-t\mathbf{A}(G)+(1-u)(\mathbf{D}-(1-u){\mathbf{I}}_n)t^2).$$

$$F(u,t)=\det ({\mathbf{I}}_n-t\mathbf{A}(G)+(1-u)t^2(\mathbf{D}-(1-u){\mathbf{I}}_n)).$$

$$\kappa (G)=\frac{1}{2t^{n-2}(mt-n)}\frac{\partial F(u,t)}{\partial t}{|}_{(1-1/t,t)},$$

$$\kappa (G)=-\frac{1}{2t^n(mt-n)}\frac{\partial F(u,t)}{\partial u}{|}_{(1-1/t,t)}.$$

$$\frac{\partial F(u,t)}{\partial t}{|}_{(0,1)}=-\frac{\partial F(u,t)}{\partial u}{|}_{(0,1)}=2(m-n)\kappa (G).$$

$$\frac{\partial {}^{2}F(u,t)}{\partial t^2}{|}_{(1-1/t,t)}=2t^{n-4}\kappa (G)(K{f}^z(t)+nt(2mt-2n+1)),$$

Full text

Running head title:

The partial derivative of the second weighted Bartholdi zeta function

The address for manuscript correspondence:

Iwao Sato

Oyama National College of Technology, Oyama, Tochigi 323-0806, JAPAN

Tel: +81-285-20-2176

Fax: +81-285-20-2880

E-mail: isato@oyama-ct.ac.jp

2000 Mathematical Subject Classification: 05C50, 05C05, 15A15.

Key words: complexity, Kirchhoff index, Laplacian matrix, Bartholdi zeta function, regular covering

1 Introduction

1.1 The Ihara zeta function, the complexity and the Kirchhoff index of a graph

Graphs and digraphs treated here are finite. Let $G$ be a connected graph and $D_{G}$ the symmetric digraph corresponding to $G$. 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 inverse of $e=(u,v)$.

The Ihara(-Selberg) zeta function of $G$ is defined by

[TABLE]

where $[C]$ runs over all equivalence classes of prime, reduced cycles of $G$ and $|C|$ is the length of a prime, reduced cycle $C$. Ihara [14] defined Ihara zeta functions of graphs, and showed that the reciprocals of Ihara zeta functions of regular graphs are explicit polynomials. The Ihara zeta function of a regular graph $G$ associated with a unitary representation of the fundamental group of $G$ was developed by Sunada [23,24]. Hashimoto [12] 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.

Let $G$ be a connected graph with $n$ vertices $v_{1},\cdots,v_{n}$ and $m$ edges. Then the adjacency matrix ${\bf A}(G)=(a_{ij})$ is the square matrix such that $a_{ij}=1$ if $v_{i}$ and $v_{j}$ are adjacent, and $a_{ij}=0$ otherwise. The degree $\deg{}_{G}v_{i}$ of a vertex $v_{i}$ of $G$ is the number of vertices adjacent to $v_{i}$ in $G$. Let ${\bf D}=(d_{ij})$ be the diagonal matrix with $d_{ii}=deg{}_{G}\ v_{i}$, and ${\bf Q}={\bf D}-{\bf I}$.

Bass [3] proved the following result for the Ihara zeta function.

Theorem 1** (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

[TABLE]

The complexity $\kappa(G)$($=$ the number of spanning trees in $G$) of a connected graph $G$ is closely related to the Ihara zeta function of $G$. The complexities for various graphs were given in [4,6,8]. Hashimoto expressed the complexity of a regular graph as a limit involving its zeta function in [12]. For an irregular graph $G$, Hashimoto [13] and Northshield [21] gave the value of $(1-t)^{-r}{\bf Z}(G,t)^{-1}$ at $t=1$ in term of the complexity of $G$, where $r$ is the Betti number of $G$.

Theorem 2** (Hashimoto; Northshield)**

For any finite graph $G$ such that $r>1$, we have

[TABLE]

where $\chi(G)=1-r$ is the Euler number of $G$.

For a connected graph $G$, let

[TABLE]

For a connected graph $G$, Northshield [21] showed that the complexity of $G$ is given by the derivative of the above function.

Theorem 3** (Northshield)**

For a connected graph $G$,

[TABLE]

where $n=\mid V(G)\mid$ and $m=\mid E(G)\mid$.

Let $G$ be a connected graph with $n$ vertices $v_{1},\ldots,v_{n}$ and $m$ edges. Furthermore, let $d_{i}=d_{v_{i}}=\deg{}_{G}v_{i}(1\leq i\leq n)$, and let ${\bf L}={\bf D}-{\bf A}(G)$ be the Laplacian of $G$. Klein and Randic [16] defined the resistance distance between $v_{i}$ and $v_{j}$ in $G$ as follows:

[TABLE]

where ${\bf L}^{(ij)}$ is the matrix obtained from ${\bf L}$ by deleting its $i$ th and $j$ th rows and columns. If $i=j$, then we set

[TABLE]

Kline and Randic [16] introduced the resistance distance between two vertices as the effective resistance between two vertices when $G$ is regarded as an electric network with a resistor of 1 ohm placed on each edge. In fact, Bapat et al [1] proved that the resistance distance for simple connected graphs can be calculated using the Laplacian.

The Kirchhoff index $Kf$ of $G$ is defined by using the resistance distances as follows(see [5,11,17]):

[TABLE]

The Kirchhoff index of a graph is expressed by the spectra of its Laplacian(see [5,6,16]):

[TABLE]

where $Spec({\bf L})=\{\mu{}_{1}=0,\mu{}_{2},\ldots,\mu{}_{n}\}$ is the set of all eigenvalues of ${\bf L}$. Furthermore, Chen and Zhang [7] defined the multiplicative Kirchhoff index $Kf^{*}$ of $G$ as follows:

[TABLE]

In [10], the additive Kirchhoff index $Kf^{+}$ of $G$ was defined as follows:

[TABLE]

Somodi [22] introduced a new Kirchhoff index $Kf^{z}$ of a graph by using its resistance distances:

[TABLE]

Note that $Kf^{z}=Kf^{*}-2Kf^{+}+4Kf$.

Somodi [22] showed that the new Kirchhoff index $Kf^{z}(G)$ of $G$ is given by the second derivative of $f_{G}(u)$.

Theorem 4** (Somodi)**

Let $G$ be a connected graph with $n$ vertices and $m$ edges. Suppose that the minimum degree $\delta(G)$ is not less than two. Then

[TABLE]

1.2 The weighted complexity and the weighted Kirchhoff index of a graph

Let $G$ be a connected graph and $V(G)=\{v_{1},\cdots,v_{n}\}$. Then we consider an $n\times n$ matrix ${\bf W}=(w_{ij})_{1\leq i,j\leq n}$ with $ij$ entry the complex variable $w_{ij}$ if $(v_{i},v_{j})\in D(G)$, and $w_{ij}=0$ otherwise. The matrix ${\bf W}={\bf W}(G)$ is called the weighted matrix of $G$. Furthermore, let $w(v_{i},v_{j})=w_{ij},\ v_{i},v_{j}\in V(G)$ and $w(e)=w_{ij},e=(v_{i},v_{j})\in D(G)$.

In the case that $w$ is symmetric, i.e., $w(e^{-1})=w(e)$ for each $e\in D(G)$, $w:D(G)\longrightarrow{\bf C}$ is considered as a symmetric function from $E(G)$ to ${\bf C}$. Then, for a spanning tree $T$ of $G$, let

[TABLE]

Furthermore, let

[TABLE]

where $T$ runs over all spanning tress of $G$. Then this sum is called the weighted complexity of $G$.

Now, let

[TABLE]

where $n=\mid V(G)\mid$ and ${\bf D}_{w}=(d_{ij})$ is the diagonal matrix with $d_{ii}=\sum_{o(e)=v_{i}}w(e)$, $V(G)=\{v_{1},\cdots,v_{n}\}$.

Mizuno and Sato [19] showed the following result for the weighted complexity of a graph.

Theorem 5** (Mizuno and Sato)**

[TABLE]

where $w(G)=\sum_{vw\in E(G)}w(v,w)$.

When $w=1$, i.e., $w(e)=1$ for each $e\in D(G)$, we obtain Theorem 3.

Let $G$ be a connected graph with $n$ vertices $v_{1},\cdots,v_{n}$, $w:D(G)\longrightarrow{\bf C}$ a symmetric weight function and ${\bf W}$ the weighted matrix of $G$ corresponding to $w$. Furthermore, let ${\bf L}={\bf D}_{w}-{\bf W}$ be the weighted Laplacian of $G$. Set $d^{w}_{i}=\sum_{o(e)=v_{i}}w(e)$ for each $i=1,\ldots,n$. For $1\leq p\neq q\leq n$, let

[TABLE]

where ${\bf L}^{(pq)}={\bf L}(pq)$ is the submatrix of ${\bf L}$ obtained from deleting the $p,q$ th rows and the $p,q$ th columns. Then we define three weighted Kirchhoff indices of $G$ as follows:

[TABLE]

[TABLE]

and

[TABLE]

Furthermore, another weighted Kirchhoff index $Kf^{z}_{w}=Kf^{z}_{w}(G)$ of $G$ is defined as follows:

[TABLE]

Note that $Kf^{z}_{w}=Kf^{*}_{w}-2Kf^{+}_{w}+4Kf_{w}$.

The following theorem is a generalization of Theorem 4 (see [18]).

Theorem 6** (Mitsuhashi, Morita and Sato)**

Let $G$ be a connected graph with $n$ vertices $v_{1},\cdots,v_{n}$, $w:D(G)\longrightarrow{\bf C}$ a symmetric weight function and ${\bf W}$ the weighted matrix of $G$ corresponding to $w$. Then

[TABLE]

1.3 Barthlodi zeta function and the Kirchhoff index of a graph

Let $G$ be a connected graph. Then the Bartholdi zeta function ${\zeta}(G,u,t)$ of a graph $G$ is defined as follows(see [2]):

[TABLE]

where $[C]$ runs over all equivalence classes of prime cycles of $G$ and $cbc(C)$ is the cyclic bump count of a cycle $C$. Bartholdi [2] gave a determinant expression of the Bartholdi zeta function of a graph.

Theorem 7** (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

[TABLE]

Let $G$ be a connected graph with $n$ vertices $v_{1},\cdots,v_{n}$. Then Li and Hou [17] introduced the following function:

[TABLE]

Li and Hou [17] gave a generalization of Northshield Theorem [21].

Theorem 8** (Li and Hou)**

Let $G$ be a connected graph with $n$ vertices and $m$ edges. Then

[TABLE]

[TABLE]

For $t=1$, the following result holds(see [15]).

Corollary 1** (Kim, Kwon and Lee)**

[TABLE]

Furthermore, Li and Hou [17] gave a result for the second differential coefficients of $F(u,t)$. Set $Kf^{z}(t)=Kf^{*}t^{2}-2Kf^{+}t+4Kf$.

Theorem 9** (Li and Hou)**

Let $G$ be a connected graph with $n$ vertices and $m$ edges. Then

[TABLE]

[TABLE]

[TABLE]

In this paper, we treat the second weighted Bartholdi zeta function of a graph $G$, and present weighted versions for the results of Li and Hou’s on the partial derivatives of the determinant part in the determinant expression of the Bartholdi zeta function of $G$.

In Section 2, we consider the second weighted Bartholdi zeta function of a graph $G$, and express its first partial derivatives by using the weighted complexity of $G$. In Section 3, we express the second partial derivatives of the second weighted Bartholdi zeta function of $G$ by using the weighted complexity and the weighted Kirchhoff index of $G$. As an application, we give a generalization of a theorem by Hashimoto and Northshield on the complexity of a graph. In Section 4, we give formulas for the weighted Kirchhoff index of a regular covering of $G$ by using the weighted Kirchhoff index of $G$.

For a general theory of graph coverings, the reader is referred to [9].

2 A partial differential coefficient for the second weighted Bartholdi zeta function of a graph

Mizuno and Sato [19] defined a new zeta function of a graph by using not an infinite product but a determinant.

Let $G$ be a connected graph with $n$ vertices and $m$ unoriented edges, and ${\bf W}={\bf W}(G)$ a weighted matrix of $G$. Two $2m\times 2m$ matrices ${\bf B}_{w}={\bf B}_{w}(G)=({\bf B}_{e,f})_{e,f\in D(G)}$ and ${\bf J}_{0}={\bf J}_{0}(G)=({\bf J}_{e,f})_{e,f\in D(G)}$ are defined as follows:

[TABLE]

Then the second weighted Bartholdi zeta function of $G$ is defined by

[TABLE]

If $w(e)=1$ for any $e\in D(G)$, then the second weighted Bartholdi zeta function of $G$ is the Bartholdi zeta function of $G$(see [2]).

The determinant expression for the second weighted Bartholdi zeta function of a graph was given by Mizuno and Sato(see [20]):

Theorem 10** (Mizuno and Sato)**

Let $G$ be a connected graph, and let ${\bf W}={\bf W}(G)$ be a weighted matrix of $G$. Then the reciprocal of the second weighted Bartholdi zeta function of $G$ is given by

[TABLE]

where $n=\mid V(G)\mid$, $m=\mid E(G)\mid$.

Let $G$ be a connected graph with $n$ vertices $v_{1},\cdots,v_{n}$, and $w:D(G)\longrightarrow{\bf C}$ be a symmetric weight function. Then we introduce the following function:

[TABLE]

The following theorem is a generalization of Theorem 8.

Theorem 11

Let $G$ be a connected graph with $n$ vertices $v_{1},\cdots,v_{n}$, $w:D(G)\longrightarrow{\bf C}$ a symmetric weight function and ${\bf W}$ the weighted matrix of $G$ corresponding to $w$. Then

[TABLE]

[TABLE]

Proof. The argument is an analogue of Somodi’s method [22]. At first,

[TABLE]

where ${\bf L}={\bf D}_{w}-{\bf W}$.

Let $V(G)=\{v_{1},\cdots,v_{n}\}$, and

[TABLE]

Furthermore, let ${\bf M}{}^{t}_{k}=(m^{t}_{k;ij})$ denote the matrix ${\bf M}_{w}(u,t)$ with each entry of the $k$ th row replaced by its corresponding partial derivative with respect to $t$. Then

[TABLE]

Here, the $(i,j)$ entry of ${\bf M}{}^{t}_{k}$ is

[TABLE]

where $\delta{}_{ij}$ is the Kronecker delta and ${\bf L}=(l_{ij})$.

Thus, the $k$ th row of ${\bf M}^{t}_{k}|_{(1-1/t,t)}$ is

[TABLE]

where ${\bf L}_{i}$ is the $i$ th row of ${\bf L}$. Furthermore, the $i(i\neq k)$ th row of ${\bf M}^{t}_{k}|_{(1-1/t,t)}$ is ${\bf L}_{i}$. Therefore,

[TABLE]

where ${\bf L}^{(k)}={\bf L}(k)$ is the submatrix from ${\bf L}^{(k)}$ obtained by deleting the $k$ th row and the $k$ th column. Moreover, by the Matrix-Tree Theorem, we have

[TABLE]

Furthermore, we have

[TABLE]

Therefore, it follows that

[TABLE]

Hence,

[TABLE]

Next, let ${\bf M}{}^{u}_{k}=(m^{u}_{k;ij})$ denote the matrix ${\bf M}_{w}(u,t)$ with each entry of the $k$ th row replaced by its corresponding partial derivative with respect to $u$. Then

[TABLE]

Here, the $(i,j)$ entry of ${\bf M}{}^{u}_{k}$ is

[TABLE]

Thus, the $k$ th row of ${\bf M}^{u}_{k}|_{(1-1/t,t)}$ is

[TABLE]

Furthermore, the $i(i\neq k)$ th row of ${\bf M}^{u}_{k}|_{(1-1/t,t)}$ is ${\bf L}_{i}$. Therefore,

[TABLE]

Hence,

[TABLE]

Q.E.D.

For $w={\bf 1}$, Theorem 11 implies the result of Li and Hou(Theorem 8). Furthermore, if $t=1$, then Theorem 11 implies Theorem 5.

3 A second partial differential coefficient for the second weighted Bartholdi zeta function of a graph

Let $G$ be a connected graph with $n$ vertices $v_{1},\cdots,v_{n}$, and $w:D(G)\longrightarrow{\bf C}$ be a symmetric weight function. Then the weighted Kirchhoff index function $Kf^{z}_{w}(t)=Kf^{z}_{w}(G,t)$ of $G$ is defined as follows:

[TABLE]

The following theorem is a generalization of Theorem 9.

Theorem 12

Let $G$ be a connected graph with $n$ vertices $v_{1},\cdots,v_{n}$, $w:D(G)\longrightarrow{\bf C}$ a symmetric weight function and ${\bf W}$ the weighted matrix of $G$ corresponding to $w$. Then

[TABLE]

[TABLE]

[TABLE]

Proof. The argument is an analogue of Somodi’s method [22].

At first, let $1\leq k\neq r\leq n$. Then, let ${\bf M}{}^{tt}_{kk}$ denote the matrix ${\bf M}_{w}(u,t)$ with each entry of the $k$ th row replaced by its corresponding second partial derivative with respect to $t$. Furthermore, let ${\bf M}{}^{tt}_{kr}$ denote the matrix ${\bf M}_{w}(u,t)$ with each entry of the $k,r$ th rows replaced by their corresponding partial derivatives with respect to $t$. Then

[TABLE]

Since the non-diagonal entries of ${\bf M}{}^{tt}_{kk}$ are a linear expression of $t$, all non-diagonal entries in the $k$ th row of ${\bf M}{}^{tt}_{kk}$ are 0. Thus, the diagonal entry in the $k$ th row of ${\bf M}{}^{tt}_{kk}$ is

[TABLE]

and so,

[TABLE]

Therefore,

[TABLE]

Next, let $1\leq k\neq r\leq n$. Then the $k$ th row of ${\bf M}^{tt}_{kr}|_{(1-1/t,t)}$ is

[TABLE]

Furthermore, the $r$ th row of ${\bf M}^{tt}_{kr}|_{(1-1/t,t)}$ is

[TABLE]

Thus, we have

[TABLE]

where ${\bf L}^{[kr]}$ is the matrix obtained by $t$ times all rows of ${\bf L}$ except the $k,r$ th rows, and

[TABLE]

Moreover,

[TABLE]

[TABLE]

Now, we have

[TABLE]

Furthermore, we have

[TABLE]

Therefore, it follows that

[TABLE]

Hence,

[TABLE]

But,

[TABLE]

Furthermore,

[TABLE]

Therefore, it follows that

[TABLE]

Now, let $1\leq i\neq j\leq n$. Then, let ${\bf M}{}^{tu}_{ii}$ denote the matrix ${\bf M}_{w}(u,t)$ with each entry of the $i$ th row replaced by its corresponding second partial derivative with respect to $t$ and $u$. Furthermore, let ${\bf M}{}^{tu}_{ij}$ denote the matrix ${\bf M}_{w}(u,t)$ with each entry of the $i,j$ th rows replaced by their corresponding partial derivatives with respect to $t$ and $u$, respectively. Then

[TABLE]

Since the $i$ th row of ${\bf M}^{tu}_{ii}$ is

[TABLE]

[TABLE]

Next, let $1\leq i\neq j\leq n$. Then the $i$ th row of ${\bf M}^{tu}_{ij}|_{(1-1/t,t)}$ is

[TABLE]

Furthermore, the $j$ th row of ${\bf M}^{tu}_{ij}|_{(1-1/t,t)}$ is

[TABLE]

Thus, we have

[TABLE]

Similarly, we have

[TABLE]

Thus,

[TABLE]

Therefore, it follows that

[TABLE]

Now, we consider the second partial derivative for $f_{w}(u,t)$ by $u$. Then we have

[TABLE]

where ${\bf M}{}^{uu}_{ii}$ denote the matrix ${\bf M}_{w}(u,t)$ with each entry of the $i$ th row replaced by its corresponding second partial derivative with respect to $u$, and ${\bf M}{}^{uu}_{ij}$ denote the matrix ${\bf M}_{w}(u,t)$ with each entry of the $i,j$ th rows replaced by their corresponding partial derivatives with respect to $u$.

Since the $i$ th row of ${\bf M}^{uu}_{ii}$ is

[TABLE]

[TABLE]

Next, let $1\leq i\neq j\leq n$. Then the $i$ th row of ${\bf M}^{uu}_{ij}|_{(1-1/t,t)}$ is

[TABLE]

Furthermore, the $j$ th row of ${\bf M}^{uu}_{ij}|_{(1-1/t,t)}$ is

[TABLE]

Thus, we have

[TABLE]

Therefore,

[TABLE]

Hence,

[TABLE]

Q.E.D.

For $w={\bf 1}$, Theorem 12 implies the result of Li and Hou(Theorem 9). Furthermore, if $t=1$, then Theorem 12 implies Theorem 6. If $w={\bf 1}$ and $t=1$, the result of Somodi(Theorem 4) is obtained from Theorem 12.

Next, we state a weighted Kirchhoff index version of Hashimoto and Northshield Theorem.

Theorem 13

Let $G$ be a connected graph with $n$ vertices and $m$ edges, and ${\bf W}(G)$ a symmetric weighted matrix of $G$. Then

[TABLE]

[TABLE]

where $w(G)=\sum_{uv\in E(G)}w(u,v)$.

Proof. The argument is an analogue of Somodi’s method [22].

At first, we have

[TABLE]

Note that $f_{w}(0,1)=0$. Then we have

[TABLE]

But, substituting $t=1$, Theorem 5 implies that the numerator is equal to

[TABLE]

Therefore, by Theorems 11 and 12, it follows that

[TABLE]

Q.E.D.

4 Weighted Kirchhoff indices of regular coverings

Let $G$ be a connected graph, and let $N(v)=\{w\in V(G)\mid(v,w)\in E(G)\}$ for any vertex $v$ in $G$. A graph $H$ is called a covering of $G$ with projection $\pi:H\longrightarrow G$ if there is a surjection $\pi:V(H)\longrightarrow V(G)$ such that $\pi{\mid}_{N(v^{\prime})}:N(v^{\prime})\longrightarrow N(v)$ is a bijection for all vertices $v\in V(G)$ and $v^{\prime}\in{\pi}^{-1}(v)$. When a finite group $\Pi$ acts on a graph $G$, the quotient graph $G/\Pi$ is a simple graph whose vertices are the $\Pi$-orbits on $V(G)$, with two vertices adjacent in $G/\Pi$ if and only if some two of their representatives are adjacent in $G$. A covering $\pi:H\longrightarrow G$ is said to be regular if there is a subgroup B of the automorphism group $Aut\ H$ of $H$ acting freely on $H$ such that the quotient graph $H/{\it B}$ is isomorphic to $G$.

Let $G$ be a graph and $\Gamma$ a finite group. Then a mapping $\alpha:D(G)\longrightarrow\Gamma$ is called an ordinary voltage assignment if $\alpha(v,u)=\alpha(u,v)^{-1}$ for each $(u,v)\in D(G)$. The pair $(G,\alpha)$ is called an ordinary voltage graph. The derived graph $G^{\alpha}$ of the ordinary voltage graph $(G,\alpha)$ is defined as follows:

$V(G^{\alpha})=V(G)\times\Gamma$ and $((u,h),(v,k))\in D(G^{\alpha})$ if and only if $(u,v)\in D(G)$ and $k=h\alpha(u,v)$.

The natural projection $\pi:G^{\alpha}\longrightarrow G$ is defined by $\pi(u,h)=u,(u,h)\in V(G^{\alpha})$. The graph $G^{\alpha}$ is called a derived graph covering of $G$ with voltages in $\Gamma$ or a $\Gamma$-covering of $G$. Then, a $\Gamma$-covering $G^{\alpha}$ is a $\mid\Gamma\mid$-fold regular covering of $G$ with covering transformation group $\Gamma$. Furthermore, every regular covering of a graph $G$ is a $\Gamma$-covering of $G$ for some group $\Gamma$(see [9]).

In the $\Gamma$-covering $G^{\alpha}$, set $v_{g}=(v,g)$ and $e_{g}=(e,g)$, where $v\in V(G),e\in D(G),g\in\Gamma$. For $e=(u,v)\in D(G)$, the arc $e_{g}$ emanates from $u_{g}$ and terminates at $v_{g\alpha(e)}$. Note that $e{}^{-1}_{g}=(e^{-1})_{g\alpha(e)}$.

Let ${\bf W}={\bf W}(G)$ be a weighted matrix of $G$. Then we define the weighted matrix $\tilde{{\bf W}}={\bf W}(G^{\alpha})=(\tilde{w}(u_{g},v_{h}))$ of $G^{\alpha}$ derived from ${\bf W}$ as follows:

[TABLE]

For $g\in\Gamma$, let the matrix ${\bf W}{}_{g}=(w^{(g)}_{uv})$ be defined by

[TABLE]

If ${\bf M}_{1},{\bf M}_{2},\cdots,{\bf M}_{s}$ are square matrices, then let ${\bf M}_{1}\oplus\cdots\oplus{\bf M}_{s}$ be the block diagonal sum of ${\bf M}_{1},\cdots,{\bf M}_{s}$ and 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 Kronecker product ${\bf A}\bigotimes{\bf B}$ of matrices A and B is considered as the matrix A having the element $a_{ij}$ replaced by the matrix $a_{ij}{\bf B}$.

Mizuno and Sato [20] presented a relation between $f_{G{}^{\alpha}}(\tilde{w},u)$ and $f_{G}(w,u)$.

Theorem 14** (Mizuno and Sato)**

Let $G$ be a connected graph with $n$ vertices, $\Gamma$ a finite group and $\alpha:D(G)\longrightarrow\Gamma$ an ordinary voltage assignment. Moreover, let ${\bf W}(G)$ be a symmetric weighted matrix of $G$. Furthermore, let ${\rho}_{1}=1,{\rho}_{2},\cdots,{\rho}_{k}$ be all inequivalent irreducible representations of $\Gamma$, and $f_{i}$ the degree of ${\rho}_{i}$ for each $i$, where $f_{1}=1$. Then

[TABLE]

By Theorem 14, $f_{w}(G,u,t)$ divides $f_{\tilde{w}}(G^{\alpha},u,t)$.

Mizuno and Sato [19] explicitly expressed the weighted complexity of a connected regular covering of $G$ by using that of $G$.

Theorem 15** (Mizuno and Sato)**

Let $G$ be a connected graph with $n$ vertices, $\Gamma$ a finite group and $\alpha:D(G)\longrightarrow\Gamma$ an ordinary voltage assignment. Furthermore, let ${\bf W}(G)$ be a symmetric weighted matrix of $G$. Furthermore, let ${\rho}_{1}=1,{\rho}_{2},\cdots,{\rho}_{k}$ be the irreducible representations of $\Gamma$, and $f_{i}$ the degree of ${\rho}_{i}$ for each $i$, where $f_{1}=1$. Suppose that the $\Gamma$-covering $G^{\alpha}$ of $G$ is connected. Then the weighted complexity of $G^{\alpha}$ is

[TABLE]

By Theorems 14 and 15, we explicitly express the weighted Kirchhoff index of a connected regular covering of $G$ by using that of $G$.

Theorem 16

Let $G$ be a connected graph with $n$ vertices and $m$ edges, $\Gamma$ a finite group and $\alpha:D(G)\longrightarrow\Gamma$ an ordinary voltage assignment. Let ${\bf W}(G)$ be a symmetric weighted matrix of $G$. Set $|\Gamma|=r$. Let ${\rho}_{1}=1,{\rho}_{2},\cdots,{\rho}_{k}$ be the irreducible representations of $\Gamma$, and $f_{i}$ the degree of ${\rho}_{i}$ for each $i$, where $f_{1}=1$. Suppose that the $\Gamma$-covering $G^{\alpha}$ of $G$ is connected. Then

[TABLE] 2. 2.

[TABLE] 3. 3.

[TABLE]

Here,

[TABLE]

Proof. By Theorem 14, we have

[TABLE]

Now, let

[TABLE]

and

[TABLE]

Then we have

[TABLE]

Thus, we have

[TABLE]

Since

[TABLE]

we have

[TABLE]

But,

[TABLE]

and

[TABLE]

Note that $|\Gamma|=r=1+f^{2}_{2}+\cdots+f^{2}_{k}$. By Theorems 11 and 12, we have

[TABLE]

By Theorem 15, we have

[TABLE]

But,

[TABLE]

Thus,

[TABLE]

Therefore, it follows that

[TABLE]

Hence

[TABLE]

Next, we have

[TABLE]

Then we have

[TABLE]

By Theorems 11 and 12, we have

[TABLE]

By Theorem 15, we have

[TABLE]

But,

[TABLE]

Thus,

[TABLE]

Therefore, it follows that

[TABLE]

Hence

[TABLE]

Finally, we have

[TABLE]

Then we have

[TABLE]

By Theorems 11 and 12, we have

[TABLE]

By Theorem 15, we have

[TABLE]

Thus,

[TABLE]

Therefore, it follows that

[TABLE]

Q.E.D.

In the case of $t=1$, the following result holds(see [18]).

Corollary 2** (Mitsuhashi, Morita and Sato)**

Let $G$ be a connected graph with $n$ vertices and $m$ edges, $\Gamma$ a finite group and $\alpha:D(G)\longrightarrow\Gamma$ an ordinary voltage assignment. Furthermore, let ${\bf W}(G)$ be a symmetric weighted matrix of $G$. Set ${\bf Q}_{w}={\bf D}_{w}-{\bf I}_{n}$. Let ${\rho}_{1}=1,{\rho}_{2},\cdots,{\rho}_{k}$ be the irreducible representations of $\Gamma$, and $f_{i}$ the degree of ${\rho}_{i}$ for each $i$, where $f_{1}=1$. Suppose that the $\Gamma$-covering $G^{\alpha}$ of $G$ is connected. Then

[TABLE]

Proof. By the first formula of Theorem 16, we have

[TABLE]

But,

[TABLE]

Therefore, the result follows. Q.E.D.

Next, in the case of $w={\bf 1}$, we have

[TABLE]

and

[TABLE]

For $g\in\Gamma$, let

[TABLE]

Thus,

Corollary 3

Let $G$ be a connected graph with $n$ vertices and $m$ edges, $\Gamma$ a finite group and $\alpha:D(G)\longrightarrow\Gamma$ an ordinary voltage assignment. Let ${\rho}_{1}=1,{\rho}_{2},\cdots,{\rho}_{k}$ be the irreducible representations of $\Gamma$, and $f_{i}$ the degree of ${\rho}_{i}$ for each $i$, where $f_{1}=1$. Suppose that the $\Gamma$-covering $G^{\alpha}$ of $G$ is connected. Then the Kirchhoff index of $G^{\alpha}$ is

[TABLE]

[TABLE]

Acknowledgments

We would like to thank the referees for many useful suggestions and comments. The first author is partially supported by the Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of Science (Grant No. 16K05187). 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 forth author is partially supported by the Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of Science (Grant No. 15K04985).