# 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.

## Key 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].