## Suil O, Spectral radius and fractional matchings in graphs

A fractional matching of a graph G is a function f giving each edge a number between 0 and 1 so that $$\sum_{e \in \Gamma(v)} f(e) \le 1$$ for each $$v \in V(G)$$, where $$\Gamma(v)$$ is the set of edges incident to v. The fractional matching number of G, written $$\alpha’_*(G)$$, is the maximum of $$\sum_{e \in E(G)} f(e)$$ over all fractional matchings f. Let G be an n-vertex graph with minimum degree d, and let $$\lambda_1(G)$$ be the largest eigenvalue of G. In this talk, we prove that if k is a positive integer and $$\lambda_1(G) < d\sqrt{1+\frac{2k}{n-k}}[/latex], then [latex]\alpha'_*(G) > \frac{n-k}{2}$$.