Spectral radius and fractional matchings in graphs

Abstract

A fractional matching of a graph G is a function f giving each edge a number in [0,1] so that ∑e∈Γ(v)f(e)≤1 for each v∈V(G), where Γ(v) is the set of edges incident to v. The fractional matching number of G, written α∗′(G), is the maximum of ∑e∈E(G)f(e) over all fractional matchings f. Let G be an n-vertex connected graph with minimum degree d, let λ1(G) be the largest eigenvalue of G, and let k be a positive integer less than n. In this paper, we prove that if λ1(G)<d1+2kn−k, then α∗′(G)>n−k2. As a result, we prove α∗′(G)≥nd2λ1(G)2+d2; we characterize when equality holds in the bound.