On spectra of expansion graphs and matrix polynomials
type: Item
instance of: scholarly article
title: On spectra of expansion graphs and matrix polynomials
zbMATH Open document ID: 1016.05051
DOI: 10.1016/S0024-3795(01)00595-X
author: Béla Nagy
author: K.-H. Förster
published in: Linear Algebra and its Applications
publication date: 2003-05-04T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 1903560
zbMATH Keywords: expansion graph
zbMATH Keywords: matrix polynomial
zbMATH Keywords: partial linearization
zbMATH Keywords: Jordan chain
zbMATH Keywords: peripheral eigenvalue
zbMATH Keywords: index of imprimitivity
reviewed by: Ian M. Wanless
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1903560