Authors: Anđelić, Milica
Ashraf, Firouzeh
da Fonseca, Carlos
Simić, Slobodan 
Title: Vertex types in some lexicographic products of graphs
Journal: Linear and Multilinear Algebra
Volume: 67
Issue: 11
First page: 2282
Last page: 2296
Issue Date: 2-Nov-2019
Rank: M21
ISSN: 0308-1087
DOI: 10.1080/03081087.2018.1490689
Abstract: 
Let M = [mij] be a symmetric matrix, or equivalently, a weighted graph (Formula presented.) whose edge ij has the weight (Formula presented.). The eigenvalues of mij are the eigenvalues of M. We denote by (Formula presented.) the principal submatrix of M obtained by deleting from M both the ith row and the ith column. If μ is an eigenvalue of M, and thus of (Formula presented.), of multiplicity (Formula presented.), then vertex i of k ≥ 1 is a downer, or a neutral, or a Parter vertex, depending whether the multiplicity of μ in (Formula presented.) or, equivalently, in (Formula presented.), is k−1, k, or k+1, respectively. In this paper, for a fixed μ, we consider vertex types according to the above classification in graphs which are generalized lexicographic products of an arbitrary graph over cliques and co-cliques, or connected regular graphs. In addition, we add some comments on constructions of large families of cospectral and integral graphs.
Keywords: Adjacency matrix | cospectral graphs | downer vertex | generalized lexicographic product | integral graph | neutral vertex | Parter vertex
Publisher: Taylor & Francis

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