Spectral radius and matchings in graphs

A perfect matching in a graph G is a set of disjoint edges covering all vertices of G. Let ρ(G) be the spectral radius of a graph G, and let θ(n) be the largest root of x3−(n−4)x2−(n−1)x+2(n−4)=0. In this paper, we prove that for a positive even integer n≥8 or n=4, if G is an n-vertex graph with ρ(G...

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Published inLinear algebra and its applications Vol. 614; pp. 316 - 324
Main Author O, Suil
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Inc 01.04.2021
American Elsevier Company, Inc
Subjects
Apexes
Graph matching
Graph theory
Graphs
Integers
Linear algebra
Perfect matching
Spectral radius
05C70
05C50
Perfect matching
Spectral radius
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Summary:A perfect matching in a graph G is a set of disjoint edges covering all vertices of G. Let ρ(G) be the spectral radius of a graph G, and let θ(n) be the largest root of x3−(n−4)x2−(n−1)x+2(n−4)=0. In this paper, we prove that for a positive even integer n≥8 or n=4, if G is an n-vertex graph with ρ(G)>θ(n), then G has a perfect matching; for n=6, if ρ(G)>1+332, then G has a perfect matching. It is sharp for every positive even integer n≥4 in the sense that there are graphs H with ρ(H)=θ′(n) and no perfect matching, where θ′(n)=θ(n) if n=4 or n≥8 and θ′(6)=1+332.
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ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2020.06.004