Characteristic polynomials and zeta functions of equitably partitioned graphs

Let pi = {V-1, ...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/pi over an equitable partition pi, which is a signed directed graph defined for a fixed set of deleting vertices {(v) over bar (i) is an element of V-i, i = 1= 1, ..., 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 ([4, Theorem 3.1]) on the IharaBartholdi zeta functions on generalized join graphs, and (iii) Teranishi's result [32, Theorem 3.3]. (C) 2019 Elsevier Inc. All rights reserved.