Spectral radius and fractional matchings in graphs

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