On spectra of expansion graphs and matrix polynomials
From a directed graph \(D\), \textit{S. Friedland} and \textit{H. Schneider} [Electron. J. Linear Algebra 6, 2-10 (1999; Zbl 0930.05063)] formed an ``expansion graph'' \(E\) by replacing each arc by a chain of arcs. They studied the relationship between the spectrum of \(D\) and that of \(E\). The present authors generalise the results of Friedland and Schneider by allowing weights on the arcs and by allowing replacement of each arc by a number of chains of different lengths. They show that the adjacency matrix of an expansion graph is a partial linearization of a certain matrix polynomial. They then use Jordan chains to study the spectral properties of their expansion graphs. The last section is devoted to studying indices of imprimitivity (number of peripheral eigenvalues). It is shown that in general there is no relation between the index of imprimitivity of a graph and the indices of imprimitivity of its expansion graphs. However the indices of imprimitivity of some special expansion graphs are calculated.
scientific article; zbMATH DE number 1903560