Characteristic polynomials and zeta functions of equitably partitioned graphs

In this paper, we define the {\Italyc deletion graph} $X\setminus \pi$ over an equitable partition $\pi=\{V_1,\cdots, V_r\}$ of vertex set of a directed graph (digraph) $X$, which is an signed directed graph defined for a fixed set of deleting vertices $\{\overline{v}_i\in V_i, i=1,\cdots,r\}$. We show that the characteristic polynomial of adjacency matrix $A(X)$ is decomposed into the quotient graph part and the deletion graph part over the equitable partition. This answers the question posed by Deng and Wu [DW05, section 5] in a more general form. Furthermore, we have a decomposition formula of the reciprocal of Ihara-Bartholdi zeta function over an equitably partitioned undirected graph into the quotient graph part and the deletion graph part. As corollaries, we have the Chen and Chen's result ([CC17, Theorem 3.1]) on Ihara-Bartholdi zeta functions on generalized join graphs, and the Teranishi's result [Ter03, Theorem 3.3].