Spectral radius and matchings in graphs

Abstract

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.