On spectra of expansion graphs and matrix polynomials

Abstract

An expansion graph of a directed weighted graph G 0 is obtained by replacing certain edges of G 0 by disjoint chains. The adjacency matrix of the expansion graph is a partial linearization of a monic matrix polynomial. We prove results on common properties of a monic operator polynomial and its partial linearization. The graph G 0 is connected if and only if each expansion graph of G 0 is connected; in this case we compute the index of imprimitivity of the adjacency matrix of some special expansion graphs of G 0.