Characteristic polynomials and zeta functions of equitably partitioned graphs

TL;DR

This paper investigates the characteristic polynomials and zeta functions of graphs with equitable partitions, introducing a deletion graph concept and providing decomposition formulas that unify several existing results in graph theory.

Contribution

It introduces the deletion graph over equitable partitions and establishes a similarity transformation that simplifies the adjacency matrix, leading to new decomposition formulas for zeta functions.

Findings

Decomposition formula for Ihara-Bartholdi zeta function over equitable partitions.

Unified framework encompassing previous results on generalized join graphs.

Block triangular matrix form for adjacency matrices compatible with equitable partitions.

Abstract

Let $\pi=\{V_1,\dots,V_r\}$ be an equitable partition of the vertex set of a directed graph (digraph) $X$. It is well known that the characteristic polynomial $\phi(X/\pi,x)$ of a quotient graph $X/\pi$ divides that of $X$, but the remainder part is not well investigated. In this paper, we define a deletion graph $X\backslash\pi$ over an equitable partition $\pi$, which is a signed directed graph defined for a fixed set of deleting vertices $\{\bar{v}_i\in V_i, i=1,\cdots,r\}$, and give a similarity transformation exchanging the adjacency matrix $A(X)$ which is compatible with the equitable partition for a block triangular matrix whose diagonal blocks are the adjacency matrix of the quotient graph and the deletion graph. In fact, we show the result for more general matrices including adjacency matrix of graphs, and as corollaries, we show the followings: (i) a decomposition formula of the reciprocal of the Ihara-Bartholdi zeta function over an equitably partitioned undirected graph into the quotient graph part and the deletion graph part, and (ii) Chen and Chen's result ([CC17, Theorem 3.1]) on the Ihara-Bartholdi zeta functions on generalized join graphs, and (iii) Teranishi's result [Ter03, Theorem 3.3].

Equations

$$A(X)$$

$$\varphi (A(X),x)$$

$$A(X)$$

$$\varphi (A(X),x)$$

$$A_1$$

$$\varphi (A(X),x)=(x-3)(x-1{)}^5(x+2{)}^4,\varphi (A(X/{\pi }_1),x)=(x-3)(x-1).$$

$$A(X)$$

$$\displaystyle A(X)_{\mbox{\tiny$v^{i}_{k}v^{j}_{l}$}}=\begin{cases}A(X_{i})_{v^{i}_{k}v^{i}_{l}}&\text{ if }i=j,\\ A(H)_{ij}&\text{ if }i\neq j.\end{cases}$$

$$A(X/\pi {)}_{ij}=\{\begin{array}{ll}{k}_i& \text{if }i=j,\\ A(H{)}_{ij}{n}_j& \text{if }i\ne j,\end{array}$$

$$M|V_i\times V_j\cdot 1=b_{ij}\cdot 1,i,j=1,\cdots ,r,$$

$$D|V_i\times V_i=d_i{I}_i,I_i\in \mathrm{Mat}(V_i\times V_i;\{0,1\})\text{ is the unit matrix.}$$

$$M/\pi =\alpha A(X/\pi )+({\delta }_{ij}{d}_i),$$

$$\text{(a) (i)}L(X)/\pi$$

$$\text{(ii)}Q(X)/\pi$$

$$\text{(b)}{Z}_{X/\pi }(u,t{)}^{-1}$$

$$D^{+}(X)$$

$$D^{-}(X)$$

$$V(X_1+X_2)$$

$$A\left(X_1+X_2\right)$$

$$\tilde{A}(X_i{)}_{uv}=\{\begin{array}{ll}A(X_i{)}_{uv}& \text{ if }u,v\in V(X_i),\\ 0& \text{ otherwise}.\end{array}$$

$$A{\left(B(\overline{v})▹C\right)}_{uv}=\{\begin{array}{ll}A(X{)}_{\overline{v}v}& \text{ if }u\in B\text{ and }v\in C,\\ 0& \text{ otherwise}.\end{array}$$

$$V^{\prime }:=V(X)\setminus \{{\overline{v}}_i{\}}_{i=1}^r,V_i^{\prime }:=V_i\setminus \{{\overline{v}}_i\}.$$

$$X\backslash \pi$$

$$P_{\mbox{\tiny$v^{i}_{k}j$}}=\delta_{ij}\text{ for }v^{i}_{k}\in V_{i}.$$

$$M\backslash \pi$$

$${|V^{\prime }\times \pi }_{v_k^i{v}_l^j}=\sum _{h=1}^r{P}_{v_k^{i}h}{M}_{{\overline{v}}_h{v}_l^j}=M_{{\overline{v}}_i{v}_l^j}.$$

$$A(X)\backslash \pi =A(X\backslash \pi ).$$

$$A{\left(\sum _{i,j=1}^r\left(V_i^{\prime }({\overline{v}}_i)▹V_j^{\prime }\right)\right)}_{v_k^i{v}_l^j}=A{\left(V_i^{\prime }({\overline{v}}_i)▹V_j^{\prime }\right)}_{v_k^i{v}_l^j}=A(X{)}_{{\overline{v}}_i{v}_l^j}.$$

$$P|V^{\prime }\times \pi \cdot A(X)|\{{\overline{v}}_i{\}}_{i=1}^r\times V^{\prime }$$

$$\displaystyle=\left(\begin{array}[]{@{\,}cccc|cccc@{\,}}\mathbf{0}&\mathbb{1}_{4}&\mathbb{1}_{4}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \hline\cr\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbb{1}_{4}&\mathbf{0}&\mathbf{0}&\mathbb{1}_{4}\end{array}\right)$$

$$=\left(\begin{array}{cc}{J}_{23}& 0\\ 0& J_{14}\end{array}\right),$$

$$A\left(\sum _{i,j=1}^2\left(V_i^{\prime }({\overline{v}}_i)▹V_j^{\prime }\right)\right)$$

$$=\left(\begin{array}{cc}{J}_{23}& 0\\ 0& J_{14}\end{array}\right).$$

$$P|V^{\prime }\times \pi \cdot A(X)|\{{\overline{v}}_i{\}}_{i=1}^r\times V^{\prime }$$

$$\displaystyle=\left(\begin{array}[]{@{\,}cc|ccccc@{\,}}\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbb{1}_{2}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbb{1}_{5}&\mathbf{0}&\mathbb{1}_{5}&\mathbf{0}\end{array}\right)=\begin{pmatrix}\mathbf{0}&J_{5}\\ \mathbf{0}&J_{24}\end{pmatrix}.$$

$$A\left(\sum _{i,j=1}^3\left(V_i^{\prime }({\overline{v}}_i)▹V_j^{\prime }\right)\right)$$

Full text

Characteristic polynomials and zeta functions of equitably partitioned graphs

Osamu Kada

Part-time Lecturer, Faculty of Science and Engineering,

Hosei University, Koganei, Tokyo 184-8584, Japan

e-mail: osamu.kada.77@hosei.ac.jp

November, 2019

Abstract Let $\pi=\{V_{1},\dots,V_{r}\}$ be an equitable partition of the vertex set of a directed graph (digraph) $X$. It is well known that the characteristic polynomial $\phi(X/\pi,x)$ of a quotient graph $X/\pi$ divides that of $X$, but the remainder part is not well investigated. In this paper, we define a deletion graph $X\backslash\pi$ over an equitable partition $\pi$, which is a signed directed graph defined for a fixed set of deleting vertices $\{\overline{v}_{i}\in V_{i},i=1,\cdots,r\}$, and give a similarity transformation exchanging the adjacency matrix $A(X)$ which is compatible with the equitable partition for a block triangular matrix whose diagonal blocks are the adjacency matrix of the quotient graph and the deletion graph. In fact, we show the result for more general matrices including adjacency matrix of graphs, and as corollaries, we show the followings: (i) a decomposition formula of the reciprocal of the Ihara-Bartholdi zeta function over an equitably partitioned undirected graph into the quotient graph part and the deletion graph part, and (ii) Chen and Chen’s result ([CC17, Theorem 3.1]) on the Ihara-Bartholdi zeta functions on generalized join graphs, and (iii) Teranishi’s result [Ter03, Theorem 3.3].

AMS classification: 05C50

Keywords: Equitable partition; characteristic polynomial; zeta function; graph; generalized join (composition) graph

1. Introduction

A partition $\pi=\{V_{1},\dots,V_{r}\}$ of the vertex set $V(X)$ of a directed graph (digraph) $X$ (we allow multiple loops and edges) is called equitable if for each $i,j=1,\dots,r$, there is a integer $b_{ij}$ such that each vertex in the cell $V_{i}$ has exactly $b_{ij}$ (oriented) edges to vertices in the cell $V_{j}$. The adjacency matrix of the (weighted) quotient graph over $\pi$ is defined by the matrix $(b_{ij})$ and is denoted by $A(X/\pi)$. It is well known that the characteristic polynomial of the (weighted) quotient graph over $\pi$ (front divisor of $X$) divides the characteristic polynomial of the adjacency matrix $A(X)$ of the graph ([GR01, 9.3]), but the remainder part is not well investigated.

We define the deletion graph $X\backslash\pi$ over $\pi$ which is a signed directed graph defined for a fixed set of deleting vertices $\{\overline{v}_{i}\in V_{i},i=1,\cdots,r\}$, and give a similarity transformation exchanging the adjacency matrix $A(X)$ which is compatible with the equitable partition for a block triangular matrix whose diagonal blocks are the adjacency matrices of the quotient graph and the deletion graph. Hence, we have that the remainder part is the characteristic polynomial of the adjacency matrix $A(X\backslash\pi)$ of the deletion graph. This answers the question posed by Deng and Wu [DW05, Section 5] of whether we can associate any geometrical meaning on the remainder part.

Moreover, we get the decomposition formula for the characteristic polynomials for the Laplacian matrix, and the reciprocal of the Ihara-Bartholdi zeta function. Since a generalized join (composition) (Schwenk [S74]) of regular graphs is a special case of an equitably partitioned graph, we get the decomposition formula of the reciprocal of the Ihara-Bartholdi zeta function of generalized joined graph by Chen and Chen [CC17] as a corollary of our Theorem.

Having an equitably partitioned graph is equivalent to having a covering projection ([DSW07, Lemma 3.1]), and when the graph is a covering of a voltage assignment, this is equivalent to having a free action (i.e. regular covering) ([GT77, Theorems 3,4], see also [DW05]). There is much in the literature on the decomposition of (a) the characteristic polynomial of the adjacency matrix of the graph and (b) the reciprocal of the Ihara-Bartholdi zeta functions. We list them. On the topic (a), there are results for graph covering with voltages in a finite group by Mizuno and Sato [MS95, Theorem 1], [MS97, Theorem 1] (see also [KL92], [Sat99 Theorem 24], [KL01], [FKL04]); for branched cover with branch index 1 by Deng and Wu [DW05, Theorem 4.2] assuming a semi-free action on digraph; for branched cover with branch index 1 by Deng, Sato and Wu [DSW07, Theorem 6.4]. On the topic (b), there are results for the reciprocal of the (weighted) Ihara-zeta (Bartholdi-zeta) function of a regular ($g$-cyclic $\Gamma$-, or irregular) cover by Mizuno and Sato [MS01, Theorem 5], [MS02, Theorem 7], [MS04, Theorem 4], and by Sato [Sat06, Theorem 3], [Sat07, Theorem 4].

When there is a symmetry (automorphism), so when the equitable partition is the orbit partition, [BFW16], [FSSW17],[FSW18] give a decomposition of any automorphism compatible matrix, which include the adjacency matrix, the Laplacian matrix, etc.

The remainder of the paper is organized as follows. In section 2, we give basic facts on equitably partitioned directed graphs. In section 3, we define the deletion graph. In section 4, we give our main theorem (Theorem 4.4) of a similarity transformation exchanging the adjacency matrix $A(X)$ for a block triangular matrix whose diagonal blocks are the adjacency matrix of the quotient graph and the deletion graph, giving the decomposition formula. In Section 5 and 6 we give applications of the decomposition formula to the reciprocal of the Ihara-Bartholdi zeta functions of equitably partitioned graphs, especially on generalized join graphs.

2. Equitable Partitions of Directed Graphs

For totally ordered sets $U,V$ and a set $W$, we denote by $\mathrm{Mat\,}(U\times V;\,W)$ the set of matrices indexed by $U\times V$ whose components are in $W$, that is, the set $\{M:U\times V\to W\}$ of mappings from $U\times V$ to $W.$ For a square matrix $M$, $\phi(M,x)=\det(xI-M)$ is the characteristic polynomial of $M$. Let $X=(V(X),E(X))$ be a finite directed (multi)graph with a set $V(X)$ of vertices and a set $E(X)$ of directed edges. If $e\in E(X)$ implies $\overline{e}\in E(X)$ (here $\overline{e}$ is an inverse edge), then $X$ can be considered as an undirected graph. We allow $X$ to have multiple edges and multiple loops. For $u,v\in V(X)$, we denote by $u\to v$ if there is an edge that goes from $u$ to $v$, and $u\sim v$ if $u\to v$ and $v\to u$. The adjacency matrix of $X$ is denoted by $A(X)$, that is, $A(X)_{uv}$ is the number of edges from $u$ to $v$.

Definition 2.1**.**

Let $\pi=\{V_{1},\dots,V_{r}\}$ be a partition of the vertex set $V(X)$ of a directed graph $X$. For $i,j=1,\dots,r,$ for each vertex $u$ in the cell $V_{i}$, if the number $b_{ij}$ of edges that goes from $u$ to the vertices in $V_{j}$ does not depend on the choice of $u$, we say that $\pi$ is an equitable partition. In this case, the multi-directed graph $X/\pi$, called the (weighted) quotient (or front divisor) of $X$ over $\pi$, is such that the set of vertices is $\pi$, and there are $b_{ij}$ edges from $V_{i}$ to $V_{j}$ ([GR01, 9.3]). The adjacency matrix of $X/\pi$ is given by $A(X/\pi)=(b_{ij})\in\mathrm{Mat\,}(\pi\times\pi;\,\mathbb{Z}_{+}).$

Example 2.2**.**

1. (a)

Let $X$ be the following $C_{4}$: $\pi=\{V_{1},V_{2}\},V_{i}=\{v^{i}_{1},v^{i}_{2}\},i=1,2,$ $v^{1}_{j}=\raisebox{-5.0pt}{ \leavevmode\hbox to15.03pt{\vbox to15.29pt{\pgfpicture\makeatletter\hbox{\hskip 7.51318pt\lower-7.64452pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@curveto{7.11319pt}{3.92854pt}{3.92854pt}{7.11319pt}{0.0pt}{7.11319pt}\pgfsys@curveto{-3.92854pt}{7.11319pt}{-7.11319pt}{3.92854pt}{-7.11319pt}{0.0pt}\pgfsys@curveto{-7.11319pt}{-3.92854pt}{-3.92854pt}{-7.11319pt}{0.0pt}{-7.11319pt}\pgfsys@curveto{3.92854pt}{-7.11319pt}{7.11319pt}{-3.92854pt}{7.11319pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.52779pt}{-2.36708pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{j}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }\hskip 8.53581pt,v^{2}_{j}=\raisebox{-5.0pt}{ \leavevmode\hbox to13.89pt{\vbox to15.29pt{\pgfpicture\makeatletter\hbox{\hskip 6.944pt\lower-7.64452pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}{{}}{{}{}}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-6.544pt}{-6.544pt}\pgfsys@moveto{-6.544pt}{-6.544pt}\pgfsys@lineto{-6.544pt}{6.544pt}\pgfsys@lineto{6.544pt}{6.544pt}\pgfsys@lineto{6.544pt}{-6.544pt}\pgfsys@closepath\pgfsys@moveto{6.544pt}{6.544pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.52779pt}{-2.36708pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{j}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }\hskip 8.53581pt$.

1$$2$$1$$2$$X

V_{1}$$V_{2}$$X\setminus\pi

[TABLE] 2. (b)

Next, consider the following directed graph. $\pi=\{V_{1},V_{2}\},V_{1}=\{v^{1}_{j}|\ j=1,2,3\}$, $V_{2}=\{v^{2}_{k}|\ k=1,2\}$, $v^{1}_{j}=\raisebox{-5.0pt}{ \leavevmode\hbox to12.18pt{\vbox to15.29pt{\pgfpicture\makeatletter\hbox{\hskip 6.09045pt\lower-7.64452pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@curveto{5.69046pt}{3.14279pt}{3.14279pt}{5.69046pt}{0.0pt}{5.69046pt}\pgfsys@curveto{-3.14279pt}{5.69046pt}{-5.69046pt}{3.14279pt}{-5.69046pt}{0.0pt}\pgfsys@curveto{-5.69046pt}{-3.14279pt}{-3.14279pt}{-5.69046pt}{0.0pt}{-5.69046pt}\pgfsys@curveto{3.14279pt}{-5.69046pt}{5.69046pt}{-3.14279pt}{5.69046pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.52779pt}{-2.36708pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{j}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},v^{2}_{k}=\raisebox{-5.0pt}{ \leavevmode\hbox to12.18pt{\vbox to13.61pt{\pgfpicture\makeatletter\hbox{\hskip 6.09045pt\lower-6.80522pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{}}{{}{}}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-5.69046pt}{-5.69046pt}\pgfsys@moveto{-5.69046pt}{-5.69046pt}\pgfsys@lineto{-5.69046pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{-5.69046pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{5.69046pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.6389pt}{-3.47221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{k}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}.$

3$$1$$2$$1$$2$$X

V_{1}$$V_{2}$$X/\pi

[TABLE] 3. (c)

We recall examples from [GR01; Section 9.3]. Let $X=J(5,2,0)$ be the Petersen graph. (i) Let $\pi_{1}=\{V_{1},V_{2}\}$, $V_{i}=\{v^{i}_{k}|\ k=1,\dots,5\},\ i=1,2$, $v^{1}_{k}=$

k

, $v^{2}_{k}=$

k

.

1$$2$$3$$4$$5$$1$$2$$3$$4$$5

$\hskip 136.57323ptA(X)=\begin{pmatrix}A_{1}&I_{5}&\\ I_{5}&A_{2}\end{pmatrix},$

[TABLE]

(ii) Next consider the following distance partition $\pi_{2}=\{V_{1},V_{2},V_{3}\}$, $V_{1}=\{v^{1}_{1}\}$, $V_{2}=\{v^{2}_{k}|\ k=1,2,3\}$, $V_{3}=\{v^{3}_{l}|\ l=1,\dots,6\}$, $v^{1}_{1}=$

1

, $v^{2}_{k}=$

k

, $k=1,2,3,$ $v^{3}_{l}=$

l

, $l=1,\dots,6.$ $V_{i}$ is the set of vetrices such that their distances from $v^{1}_{1}$ are the same $i-1$.

2$$1$$3$$4$$5$$1$$1$$2$$6$$3

[TABLE]

$A(X/\pi_{2})=\begin{pmatrix}0&3&0\\ 1&0&2\\ 0&1&2\end{pmatrix},\phi(A(X/\pi_{2}),x)=(x-3)(x-1)(x+2).$

For other examples of equitable partitions, see [GR01]. 4. (d)

Let $X=H[X_{1},\dots,X_{r}]$ be the generalized join (composition) graph of $X_{1},\dots,X_{r}$ determined by $H$ ([Sch74, 4.]), and assume $X_{i}$ are $k_{i}$-regular ([Sch74, section 4], [CC17]). That is, $\#V(H)=r,V(X)=\cup_{i=1}^{r}V(X_{i})$ and for $v^{i}_{k}\in V(X_{i})$, $v^{j}_{l}\in V(X_{j})$,

[TABLE]

Then $\pi=\{V(X_{1}),\dots,V(X_{r})\}$ is an equitable partition of $X$ by letting

[TABLE]

here $\ n_{j}=\#V_{j}$.

Definition 2.3**.**

Let $\pi=\{V_{1},\dots,V_{r}\}$ be a partition of $V(X)$ of a digraph $X$ and let $M\in\mathrm{Mat\,}(V(X)\times V(X);\,\mathbb{C})$. We say that the pair $(M,\pi)$ is *equitable * if there exists $B=(b_{ij})\in\mathrm{Mat\,}(\pi\times\pi;\,\mathbb{C})$ such that

[TABLE]

here $\mathbb{1}=\begin{pmatrix}1\\ \vdots\\ 1\end{pmatrix}\in\mathbb{Z}^{\#V_{j}}.$

We denote by $M/\pi:=B$, and call it the quotient matrix of $M$ over the partition $\pi.$ Note that $(A(X),\pi)$ is equitable means that $\pi$ is an equitable partition, and $A(X)/\pi=A(X/\pi)$.

Remark 2.4**.**

Let $\pi=\{V_{1},\dots,V_{r}\}$ be an equitable partition of $V(X)$, and let $M=\alpha A(X)+D$. Here, $\alpha\in\mathbb{C}$ and $D$ is a diagonal matrix such that

[TABLE]

Then $(M,\pi)$ is equitable with the quotient matrix

[TABLE]

here $\delta_{ij}=1$ if $i=j$ and $=0$ if $i\neq j$. Assume that $X$ is an undirected graph and let $D(X)$ be the degree matrix of $X$, that is, the diagonal matrix such that $D(X)_{uu}$ is the number of edges that go from $u$. Let $L(X)=-A(X)+D(X)$ (resp. $Q(X)=A(X)+D(X)$) be the Laplacian matrix (resp. the signless Laplacian matrix) of $X$, and let $Z_{X}(u,t)$ be the Ihara-Bartholdi zeta function (see section 5). Then by letting $(a)D=D(X)$, (i) $\alpha=-1,$ (ii) $\alpha=1,$ (b) $D=D^{Z}(X),\alpha=-t$, we have

[TABLE]

here $d_{i}(X)$ is the degree of vertices in $V_{i}$, and $d_{i}^{Z}(X)=s_{1}+s_{2}d_{i}(X),$ ${s_{1}=1-(1-u)^{2}t^{2}},$ $s_{2}=(1-u)t^{2}.$

3. The Deletion Graph over a partition of a graph

Definition 3.1**.**

We define a signed directed graph $X=(V(X),\mathrm{SD}(X))$ as follows: $\mathrm{SD}(X)$ is a set of a signed directed edges with its adjacency matrix $A(X)\in\mathrm{Mat\,}(V(X)\times V(X);\,\mathbb{Z})$ whose components are integers. That is, for $u,v\in V(X)$, if $A(X)_{uv}\geq 0$, there are $A(X)_{uv}$ positive edges from $u$ to $v$, if $A(X)_{uv}\leq 0$, there are $|A(X)_{uv}|$ negative edges from $u$ to $v$. $\mathrm{SD}(X)=(D^{+}(X),D^{-}(X))$ is such that

[TABLE]

For signed directed graphs $X_{i}=(V(X_{i}),\mathrm{SD}(X_{i})),i=1,2$, we define the signed directed graphs $X_{1}+X_{2}$ and $X_{1}-X_{2}$ by the following:

[TABLE]

here $\widetilde{A}(X_{i})$ is the extension of $A(X_{i})$ to $V(X_{1})\cup V(X_{2})$ by:

[TABLE]

Definition 3.2**.**

Let $X=(V(X),E(X))$ be a directed graph, and $B,C\subset V(X),\overline{v}\in V(X)$. We define the signed directed graph $B(\overline{v})\triangleright C$ by: $V(B(\overline{v})\triangleright C)=B\cup C$, and

[TABLE]

Let $\pi=\{V_{1},\dots,V_{r}\}$ be a partition of vertices $V(X)$ of a directed graph $X$, and fix a set of vertices $\{\overline{v}_{i}\in V_{i},i=1,\cdots,r\}$. Let

[TABLE]

We define a signed directed graph $X\backslash\pi$ and call it the deletion graph over the partition $\pi$, as follows:

[TABLE]

here, X\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|V^{\prime}}\kern-1.25pt is the restriction of $X$ to $V^{\prime}$.

Let $P\in\mathrm{Mat\,}(V(X)\times\pi;\,\{0,1\})$ be the characteristic matrix, that is,

[TABLE]

For $M\in\mathrm{Mat\,}(V(X)\times V(X);\,\mathbb{C})$, define the deletion matrix over the partition $\pi$, $M\backslash\pi$ $\in\mathrm{Mat\,}(V^{\prime}\times V^{\prime};\,\mathbb{C})$ by the following:

[TABLE]

The following holds.

Proposition 3.3**.**

[TABLE]

(Proof)  Let C^{\prime}(X)=P\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|V^{\prime}\times\pi}\kern-1.25pt\cdot A(X)\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|{\overline{v}{i}}{i=1}^{r}\times V^{\prime}}\kern-1.25pt. Then for $v^{i}_{k}\in V_{i}^{\prime}:=V_{i}\setminus\{\overline{v}_{i}\}$ and $v^{j}_{l}\in V_{j}^{\prime}$, we have $C^{\prime}(X)_{v^{i}_{k}v^{j}_{l}}=A(X)_{\overline{v}_{i}v^{j}_{l}},$ and

[TABLE]

Since A\left(X\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|V^{\prime}}\kern-1.25pt\right)=A(X)\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|V^{\prime}\times V^{\prime}}\kern-1.25pt, we have the assertion.  ∎

Example 3.4**.**

Consider Example 2.2 (c) (i). Let $\overline{v}_{1}=v^{1}_{5}=$

5

, $\overline{v}_{2}=v^{2}_{5}=\,$

5

. Then

[TABLE]

here $J_{k_{1}\cdots k_{s}}$ is the matrix such that $k_{1},\cdots,k_{s}$-column vectors are $\mathbb{1}$, and others are $\mathbf{0}.$ On the other hand,

[TABLE]

Next, consider Example 2.2 (c) (ii). Let \overline{v}_{1}=v^{1}_{1}=\raisebox{-5.0pt}{ \leavevmode\hbox to16.5pt{\vbox to16.5pt{\pgfpicture\makeatletter\hbox{\hskip 8.24751pt\lower-8.24751pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{tikz@radial@inner}{rgb}{1,1,1}\definecolor[named]{tikz@radial@outer}{rgb}{1,1,0}{}{{}}{}{{}}{{}}{{}{}}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{-8.04752pt}\pgfsys@moveto{0.0pt}{-8.04752pt}\pgfsys@lineto{-8.04752pt}{0.0pt}\pgfsys@lineto{0.0pt}{8.04752pt}\pgfsys@lineto{8.04752pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{8.04752pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@transformcm{0.32074}{0.0}{0.0}{0.32074}{0.0pt}{0.0pt}\pgfsys@invoke{ }\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{-8.04752pt}\pgfsys@moveto{0.0pt}{-8.04752pt}\pgfsys@lineto{-8.04752pt}{0.0pt}\pgfsys@lineto{0.0pt}{8.04752pt}\pgfsys@lineto{8.04752pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{8.04752pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }\ ,\ \overline{v}_{2}=v^{2}_{3}=\raisebox{-5.0pt}{ \leavevmode\hbox to12.18pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip 6.09045pt\lower-6.55522pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\color[rgb]{1,0.04,0.04}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.04,0.04}\pgfsys@color@rgb@stroke{1}{0.04}{0.04}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{1}{0.04}{0.04}\pgfsys@invoke{ }\definecolor{pgffillcolor}{rgb}{1,0.04,0.04}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@curveto{5.69046pt}{3.14279pt}{3.14279pt}{5.69046pt}{0.0pt}{5.69046pt}\pgfsys@curveto{-3.14279pt}{5.69046pt}{-5.69046pt}{3.14279pt}{-5.69046pt}{0.0pt}\pgfsys@curveto{-5.69046pt}{-3.14279pt}{-3.14279pt}{-5.69046pt}{0.0pt}{-5.69046pt}\pgfsys@curveto{3.14279pt}{-5.69046pt}{5.69046pt}{-3.14279pt}{5.69046pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{.}{rgb}{1,0.04,0.04}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.04,0.04}\pgfsys@color@rgb@stroke{1}{0.04}{0.04}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{1}{0.04}{0.04}\pgfsys@invoke{ }\hbox{{\definecolor{.}{rgb}{1,0.04,0.04}\color[rgb]{1,0.04,0.04}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.04,0.04}3}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\overline{v}_{3}=v^{3}_{6}=\raisebox{-5.0pt}{ \leavevmode\hbox to12.18pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip 6.09045pt\lower-6.55522pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{}}{{}{}}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-5.69046pt}{-5.69046pt}\pgfsys@moveto{-5.69046pt}{-5.69046pt}\pgfsys@lineto{-5.69046pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{-5.69046pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{5.69046pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}6}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}.

Then

[TABLE]

On the other hand,

[TABLE]

Hence, $A(X)\backslash\pi=A(X\backslash\pi).$ ∎

4. The Similarity Transformation

Lemma 4.1**.**

Let $(M,\pi)$ be an equitable pair, here $M\in\mathrm{Mat\,}(V(X)\times V(X);\,\mathbb{C})$ and $\pi=\{V_{1},\dots,V_{r}\}$ is a partition of vertex set $V(X)$ of a graph $X$. Then

[TABLE]

[TABLE]

Definition 4.2**.**

Let $\pi=\{V_{1},\dots,V_{r}\}$ be a partition of vertices $V(X)$ of a graph $X$. Fix $\{\overline{v}_{i}\in V_{i},i=1,\cdots,r\}$. Define

[TABLE]

here $V^{\prime}=V(X)\setminus\{\overline{v}_{i}\}_{i=1}^{r}$, by the following:

[TABLE]

For instance, in Example 2.2 (a), letting $\overline{v}_{i}=v^{i}_{2},i=1,2,$

[TABLE]

Lemma 4.3**.**

The column vectors of $(P,Q)\in\mathrm{Mat\,}(V(X)\times(\pi\times V^{\prime});\,\{0,1\})$ form a basis of $\mathbb{C}^{V(X)}$, that is, $(P,Q)$ is invertible.

(Proof) Assume

[TABLE]

here $P_{\cdot i}$ (resp. $Q_{\cdot v^{j}_{l}}$) is the $i$-th (resp. $v^{j}_{l}$-th) column vector of $P$ (resp. $Q$). For $s\in\{1,\dots,r\}$, $P_{\overline{v}_{s}i}=\delta_{si}$, and since $v^{j}_{l}\in V_{j}^{\prime}\subset V^{\prime}=V(X)\setminus\{\overline{v}_{i}\}_{i=1}^{r}$, we have $v^{j}_{l}\neq\overline{v}_{s}$, implying $Q_{\overline{v}_{s}v^{j}_{l}}=0.$ So, considering the entry corresponding to $\overline{v}_{s}$ we have $\lambda_{s}=0,s=1,\dots,r.$ And we have $\mu_{v^{j}_{l}}=0,v^{j}_{l}\in V^{\prime},j=1,\dots,r.$ ∎

The following is our main theorem.

Theorem 4.4**.**

Let $(M,\pi)$ be an equitable pair, here $M\in\mathrm{Mat\,}(V(X)\times V(X);\,\mathbb{C})$ and $\pi=\{V_{1},\dots,V_{r}\}$ is a partition of vertex set $V(X)$ of a graph $X$. Then letting $\overline{P}=(P,Q)$ we have

[TABLE]

hence,

[TABLE]

here $M\backslash\pi$ is defined for a fixed $\{\overline{v}_{i}\}_{i=1}^{r},\overline{v}_{i}\in V_{i}$, and $\phi(M\backslash\pi,x)$ does not depend on $\{\overline{v}_{i}\}_{i=1}^{r}$. In particular, if $\pi$ is equitable, then

[TABLE]

(Proof) We show that

[TABLE]

For v^{i}_{k}\in V_{i},\mbox{\footnotesizev^{j}_{l}}\in V_{j}^{\prime}=V_{j}\setminus\{\overline{v}_{j}\},

[TABLE]

[TABLE]

So that we have (PC+Q\cdot M\backslash\pi)_{\mbox{\tinyv^{i}{k}v^{j}{l}}}=M_{\mbox{\tinyv^{i}{k}v^{j}{l}}}=(MQ)_{\mbox{\tinyv^{i}{k}v^{j}{l}}} (by (4.11)).  ∎

Example 4.5**.**

Let $X,\pi$ be as in Example 2.2.

(a) Delete $\{\overline{v}_{1},\overline{v}_{2}\}=\{\raisebox{-5.0pt}{ \leavevmode\hbox to11.67pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip 5.83301pt\lower-6.55522pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{5.12128pt}{0.0pt}\pgfsys@curveto{5.12128pt}{2.82843pt}{2.82843pt}{5.12128pt}{0.0pt}{5.12128pt}\pgfsys@curveto{-2.82843pt}{5.12128pt}{-5.12128pt}{2.82843pt}{-5.12128pt}{0.0pt}\pgfsys@curveto{-5.12128pt}{-2.82843pt}{-2.82843pt}{-5.12128pt}{0.0pt}{-5.12128pt}\pgfsys@curveto{2.82843pt}{-5.12128pt}{5.12128pt}{-2.82843pt}{5.12128pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} },\raisebox{-5.0pt}{ \leavevmode\hbox to11.67pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip 5.83301pt\lower-6.55522pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}{{}}{{}{}}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-4.55211pt}{-4.55211pt}\pgfsys@moveto{-4.55211pt}{-4.55211pt}\pgfsys@lineto{-4.55211pt}{4.55211pt}\pgfsys@lineto{4.55211pt}{4.55211pt}\pgfsys@lineto{4.55211pt}{-4.55211pt}\pgfsys@closepath\pgfsys@moveto{4.55211pt}{4.55211pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }\}$.

X\setminus\pi=X\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|,\raisebox{-5.0pt}{ \leavevmode\hbox to9.34pt{\vbox to10.49pt{\pgfpicture\makeatletter\hbox{\hskip 4.66772pt\lower-5.24419pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ 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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }\right)\Bigr{)}

$=$

1$$1

   $-$

$1$

$+$

1$$1

$+$

1$$1

$+$

$1$

$=$

1$$1

,

here we denoted the negative edeges by the red lines.

Hence, $A(X\setminus\pi)=\begin{pmatrix}-1&1\\ 1&-1\end{pmatrix}.$ By our Theorem, we have $\phi\left(A(X),\,x\right)\\ =\begin{vmatrix}x-1&-1\\ -1&x-1\end{vmatrix}\cdot\begin{vmatrix}x+1&-1\\ -1&x+1\end{vmatrix}=((x-1)^{2}-1)\cdot((x+1)^{2}-1)=x^{2}(x+2)(x-2).$

(b) Delete $\{\overline{v}_{1},\overline{v}_{2}\}=\{\raisebox{-5.0pt}{ \leavevmode\hbox to11.67pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip 5.83301pt\lower-6.55522pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{5.12128pt}{0.0pt}\pgfsys@curveto{5.12128pt}{2.82843pt}{2.82843pt}{5.12128pt}{0.0pt}{5.12128pt}\pgfsys@curveto{-2.82843pt}{5.12128pt}{-5.12128pt}{2.82843pt}{-5.12128pt}{0.0pt}\pgfsys@curveto{-5.12128pt}{-2.82843pt}{-2.82843pt}{-5.12128pt}{0.0pt}{-5.12128pt}\pgfsys@curveto{2.82843pt}{-5.12128pt}{5.12128pt}{-2.82843pt}{5.12128pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} },\raisebox{-5.0pt}{ \leavevmode\hbox to11.67pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip 5.83301pt\lower-6.55522pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}{{}}{{}{}}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-4.55211pt}{-4.55211pt}\pgfsys@moveto{-4.55211pt}{-4.55211pt}\pgfsys@lineto{-4.55211pt}{4.55211pt}\pgfsys@lineto{4.55211pt}{4.55211pt}\pgfsys@lineto{4.55211pt}{-4.55211pt}\pgfsys@closepath\pgfsys@moveto{4.55211pt}{4.55211pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }\}$.

$X\setminus\pi$

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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }\}\right)\Bigr{)}

$=$

1$$2$$1

           $-$

1$$2

             $+$

1$$2$$1

          $+$

1$$2$$1

          $+$

$1$

$=$

1$$2$$1

.

[TABLE]

[TABLE]

(c) (i) By (3.6) we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since $A_{1}+A_{2}=A_{1}A_{2}=-I_{4}$, we have $\phi\left(A(X\backslash\pi),\,x\right)=|(x^{2}+x-2)I_{4}|=(x^{2}+x-2)^{4}.$ By $\phi\left(A(X/\pi_{1}),\,x\right)=(x-3)(x-1)$ we have

[TABLE]

To get the characteristic polynomial of the adjacency matrix of the Petersen graph by the another method, see [GR01, section 9.1 and exercise 8.9].

5. Applications to the Ihara-Bartholdi zeta functions

In this section, we assume $X=(V(X),E(X))$ is an undirected graph, that is, $E(X)$ is a set of symmetric directed edges. As corollaries of our Theorem, we have the decomposition of the characteristic polynomial of Laplacian matrix, and the reciprocal of the Ihara-Bartholdi zeta functions of equitably partitioned graphs.

Zeta functions of a graph are defined as follows. For an edge $e\in E(X)$, we denote by $o(e)$ (resp. $t(e)$) the origin (resp. terminus) of $e$. A closed path in $X$ is an sequence of edges $C=(e_{1},\dots,e_{k})$ with $t(e_{i})=o(e_{i+1})$ for $i\in\mathbb{Z}/k\mathbb{Z}$. We denote by $k=|C|$, the length of $C$, and by $cbc(C)=\#\{i\in\{1,\dots,k\}|\ e_{i+1}=\overline{e_{i}}\}$, the cyclic bump count of $C$. A cycle $[C]$ is the equivalence class of a closed path $C$ under cyclic permutation of its edges (that is, $(e_{1},\dots,e_{k})\sim(e_{2},\dots,e_{k},e_{1})$). A cycle is prime if none of its representatives can be written as $C^{k}$ for some $k\geq 2$. We denote by $\mathcal{C}$ the set of prime cycles. Bartholdi zeta function is defined by

[TABLE]

$Z_{X}(0,t)=Z_{X}(t)$ is the Ihara zeta function defined by Ihara [Iha66] in which he considered a zeta function of a regular graph and gave its reciprocal as a polynomial. It was generalized to general graphs by [Bas92] (see also [Ser80], [Has89], [Has90], [Sun96a], [Sun96b], [ST96], [FZ99], [KS00]). Bartholdi generalized Bass’s Theorem as the following.

Theorem 5.1**.**

[Bar99] Let $X$ be a connected graph with $n$ vertices and $m$ (non-oriented) edges. Then the reciprocal of the Bartholdi-Ihara zeta function of $X$ is given by

[TABLE]

where $A(X)$ is the adjacency matrix of $X$, and $D(X)=(d_{i}\delta_{ij})$ is the degree matrix which is diagonal with $d_{i}=\mathrm{deg}v_{i},V(X)=\{v_{1},\dots,v_{n}\}$.

For $u,t\in\mathbb{C},$ let

[TABLE]

be the diagonal matrix, here $s_{1}=1-(1-u)^{2}t^{2},\ s_{2}=(1-u)t^{2}.$ Then we have

[TABLE]

As a Corollary of Theorem 4.4, we have the following.

Corollary 5.2**.**

Let $\pi=\{V_{1},\dots,V_{r}\}$ be an equitable partition of the vertex set $V(X)$ of a graph $X$ and fix a set of deleting vertices $\{\overline{v}_{i}\in V_{i},i=1,\cdots,r\}$. Let $D$ be a diagonal matrix such that D\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|V_{i}\times V_{i}}\kern-1.25pt=d_{i}I_{i} (for instance, $D={\large O},D(X)$ or $D^{Z}(X)$). Then we have

[TABLE]

In particular,

[TABLE]

$\text{here }s_{1}=1-(1-u)^{2}t^{2}.$**

(Proof)  

Let $M=\alpha A(X)+D$. Then $M/\pi=\alpha A(X/\pi)+(\delta_{ij}d_{i})$. For $v^{j}_{l}\in V_{j}^{\prime}:=V_{j}\setminus\{\overline{v}_{j}\},$ \left(D\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|{\overline{v}{i}}{i=1}^{r}\times V^{\prime}}\kern-1.25pt\right)_{\mbox{\tiny\overline{v}{i}v^{j}{l}}}=D_{\mbox{\tiny\overline{v}{i}v^{j}{l}}}=0. So, we have M\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|{\overline{v}{i}}{i=1}^{r}\times V^{\prime}}\kern-1.25pt=\alpha A(X)\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|{\overline{v}{i}}{i=1}^{r}\times V^{\prime}}\kern-1.25pt. Hence,

[TABLE]

So, we have the first assertion. By letting (a) $D=D(X)$, (i) $\alpha=-1,$ (ii) $\alpha=1,$ (b) $D=D^{Z}(X),\alpha=-t$, we have the other assertions.  ∎

Remark 5.3**.**

Note that D(X)\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|V^{\prime}\times V^{\prime}}\kern-1.25pt\neq D(X\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|V^{\prime}}\kern-1.25pt) in general. Let $X$ be a cycle $C_{4}$ in Example 2.2 (a). Then $V^{\prime}=\{\raisebox{-5.0pt}{ \leavevmode\hbox to11.67pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip 5.83301pt\lower-6.55522pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{5.12128pt}{0.0pt}\pgfsys@curveto{5.12128pt}{2.82843pt}{2.82843pt}{5.12128pt}{0.0pt}{5.12128pt}\pgfsys@curveto{-2.82843pt}{5.12128pt}{-5.12128pt}{2.82843pt}{-5.12128pt}{0.0pt}\pgfsys@curveto{-5.12128pt}{-2.82843pt}{-2.82843pt}{-5.12128pt}{0.0pt}{-5.12128pt}\pgfsys@curveto{2.82843pt}{-5.12128pt}{5.12128pt}{-2.82843pt}{5.12128pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} },\raisebox{-5.0pt}{ \leavevmode\hbox to11.67pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip 5.83301pt\lower-6.55522pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}{{}}{{}{}}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-4.55211pt}{-4.55211pt}\pgfsys@moveto{-4.55211pt}{-4.55211pt}\pgfsys@lineto{-4.55211pt}{4.55211pt}\pgfsys@lineto{4.55211pt}{4.55211pt}\pgfsys@lineto{4.55211pt}{-4.55211pt}\pgfsys@closepath\pgfsys@moveto{4.55211pt}{4.55211pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }\},D(X)=2I_{4}$, and D(X)\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|V^{\prime}\times V^{\prime}}\kern-1.25pt=2I_{2}, but D(X\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|V^{\prime}}\kern-1.25pt)=D(C_{2})=I_{2}.

6. Generalized join graphs

In this section, graphs considered are assumed to be simple. We consider the case when $X$ is the generalized join (composition) of $X_{1},\dots,X_{r}$ (see Example 2.2 (d)). The following Corollary of Theorem 4.4 includes results of [Sch74, Theorem 7] and [CC17, Theorem 3.1].

Corollary 6.1**.**

Let $X=H[X_{1},\dots,X_{r}]$ be a generalized join (composition) with each $X_{i}$ being $k_{i}$-regular. Let $D\in\mathrm{Mat\,}(V(X)\times V(X);\,\mathbb{C})$ be a diagonal matrix such that D\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|V(X_{i})\times V(X_{i})}\kern-1.25pt=d_{i}I_{i}. Then, letting $\pi=\{V(X_{1}),\dots,V(X_{r})\}$, for any $\alpha\in\mathbb{C},$

[TABLE]

In particular,

[TABLE]

[TABLE]

here $\overline{D}\in\mathrm{Mat\,}(r\times r;\,\mathbb{C})$ is a diagonal matrix such that $\overline{D}_{ii}=\frac{1}{n_{i}}(-tk_{i}+d^{Z}_{i}(X))\in\mathbb{C},d^{Z}_{i}(X)=s_{1}+s_{2}d_{i}(X)=s_{1}+s_{2}(k_{i}+\sum_{j\neq i}A(H)_{ij}n_{j}),s_{1}={1-(1-u)^{2}t^{2},}$ $s_{2}=(1-u)t^{2},d_{i}(X)=\deg v,v\in V(X_{i}).$

(Proof) Let $M=\alpha A(X)+D$.

(i) First we prove that

[TABLE]

which implies (6.13) by Theorem 4.4 and (2.3). By Example 2.2 (d) and Remark 2.4, $(M,\pi)$ is an equitable pair. Let $V_{i}^{\prime}:=V(X_{i})\setminus\{\overline{v}_{i}\},i=i,\dots,r.$ As the notation in Theorem 4.4, for $i\neq j$, $v^{i}_{k}\in V_{i}^{\prime},v^{j}_{l}\in V_{j}^{\prime},$ since \left(P\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|V^{\prime}\times\pi}\kern-1.25pt\cdot A(X)\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|{\overline{v}{i}}{i=1}^{r}\times V^{\prime}}\kern-1.25pt\right)_{v^{i}_{k}v^{j}_{l}}=A(X)_{\overline{v}_{i}v^{j}_{l}}=A(H)_{ij}, we have $A(X\backslash\pi)_{v^{i}_{k}v^{j}_{l}}=A(X)_{v^{i}_{k}v^{j}_{l}}-A(H)_{ij}=0,$ which implies $(M\backslash\pi)_{v^{i}_{k}v^{j}_{l}}=\alpha A(X\backslash\pi)_{v^{i}_{k}v^{j}_{l}}+D_{v^{i}_{k}v^{j}_{l}}=0.$

Hence, M\backslash\pi=\begin{pmatrix}C_{1}&&\raisebox{-8.61108pt}{ \huge O}\\ &\ddots&\\ \raisebox{8.61108pt}{\hskip 5.69054pt\huge O}&&C_{r}\end{pmatrix}, here C_{i}=\left(M\backslash\pi\right)\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|V_{i}^{\prime}\times V_{i}^{\prime}}\kern-1.25pt. So,

[TABLE]

We prove that

[TABLE]

which implies (6.16). For a subgraph $X_{i}$, let $\overline{\pi}_{i}=\{V(X_{i})\}$ be the trivial partition. Since $M_{i}\mathbb{1}=(\alpha k_{i}+d_{i})\mathbb{1}$, we have $M_{i}/\overline{\pi}_{i}=\alpha k_{i}+d_{i}\in\mathbb{C}$. So, by Theorem 4.4 we have

[TABLE]

We show M_{i}\backslash\overline{\pi}_{i}=(M\backslash\pi)\kern-1.66702pt\raisebox{-2.15277pt}{\footnotesize,|V_{i}^{\prime}\times V_{i}^{\prime}}\kern-1.25pt=C_{i}, which implies (6.18). Since

[TABLE]

and

[TABLE]

So, we have the assertion.

(ii) Next we prove that

[TABLE]

which implies (6.14). By (2.1) and (2.2),

[TABLE]

Hence, letting

[TABLE]

we have  $xI-M/\pi=(\rho_{ij}n_{j}).$  Since

[TABLE]

we have (6.19). ∎

Remark 6.2**.**

We can generalize Corollary 6.1 to the following Teranishi’s result [Ter03, Theorem 3.3].

Let $X=H[X_{1},\dots,X_{r}]$ be a generalized join (composition) and for each $i=1,\dots,r$, let $\pi_{i}=(V_{i}^{1},\dots,V_{i}^{\alpha_{i}})$ be an equitable partition of $V(X_{i}).$ Then letting $\pi=\left(\pi_{i}\right)_{i=1}^{r}=\left(\left(V_{i}^{k}\right)_{k=1}^{\alpha_{i}}\right)_{i=1}^{r}$, $\pi$ is an equitable partition of $V(X)$ and

[TABLE]

When $X_{i}$ is $k_{i}$-regular, letting $\pi_{i}=\overline{\pi}_{i}=\{V(X_{i})\}$ the trivial parition, $\pi_{i}$ is an equitable partition of $V(X_{i})$ with $\phi\left(A(X_{i}/\pi_{i}),\,x\right)=x-k_{i}$, we get (6.13) for $\alpha=1,D=0$, and similarly for all $\alpha$ and $D=(d_{i}I_{i}).$ The same proof also applies in the proof of Corollary 6.1, considering the parition $\pi_{i}$ instead of the trivial partition $\overline{\pi}_{i}$.

Remark 6.3**.**

As the notation in [CC17, Theorem 3.1],

[TABLE]

here $N_{i}=\sum_{j\neq i}A(H)_{ij}n_{j}.$ Let $M=-tA(X)+D^{Z}(X)$. Letting $\alpha=-t,D=D^{Z}(X)$ in (6) we have $(M/\pi)_{ij}=\begin{cases}\gamma_{i}&\text{if }i=j,\\ -t\rho_{ij}n_{j}&\text{if }i\neq j,\end{cases}$

here $\rho_{ij}=A(H)_{ij}$. Let $N_{ij}=\begin{cases}\gamma_{i}&\text{if }i=j,\\ -t\sqrt{n_{i}n_{j}}\rho_{ij}&\text{if }i\neq j.\end{cases}$

Then, by letting $\rho^{\prime}_{ij}=\begin{cases}\frac{\gamma_{i}}{n_{i}}&\text{if }i=j,\\ -t\rho_{ij}&\text{if }i\neq j,\end{cases}$

we have $M/\pi=(\rho^{\prime}_{ij}n_{j})$ and $N=(\rho^{\prime}_{ij}\sqrt{n_{i}n_{j}}).$ Then

[TABLE]

Let

[TABLE]

Since $D(X_{i})+N_{i}I_{i}=(k_{i}+N_{i})I_{i}=d_{i}(X)I_{i}$, we have

[TABLE]

So, by (6.1),(6.3),(6.23),(6.24), we have

[TABLE]

which is [CC17, Theorem 3.1].

Acknowledgement I would like to thank Yusuke Ide, Norio Konno, Masashi Kosuda, Hideo Mitsuhashi and Iwao Sato for their help writing this paper. I am indebted to the anonymous referee for valuable comments and helpful suggestions, which led to an improvement of the original manuscript.

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