A contraction theorem for the largest eigenvalue of a multigraph

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Abstract

Let G be a multigraph with loops, and let e be an edge in G. Let H be the multigraph obtained by contracting along the edge e. Let l_G and l_H be the largest eigenvalues of G and H respectively. A characterisation theorem is given of precisely when l_H<l_G, l_H = l_G, or l_H>l_G. In the case where H happens to be a simple graph, then so is G, and the theorem subsumes those of Hoffman-Smith and Gumbrell for subdivision of edges or splitting of vertices of a graph.
Original languageEnglish
Pages (from-to)339-349
Number of pages11
JournalLinear Algebra and Its Applications
Volume557
Early online date30 Jul 2018
DOIs
Publication statusPublished - 15 Nov 2018

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