An inequality using perfect matchings and Laplacian spread of a graph

Let G be a connected graph of order n. Let 0 = mu(1)(G) <= mu(2)(G) <= ... <= = mu(n)(G) be the Laplacian eigenvalues of G. In this paper, we show that if X and Y are two subsets of vertices of G such that vertical bar X vertical bar = vertical bar Y vertical bar and the set of all edges between X and Y decomposed into r disjoint perfect matchings, then, 2r - LS(G)/2 mu(2)(G) <= vertical bar X vertical bar/n <= 2r + LS(G)/2 mu(n)(G), where LS(G) = mu(n)(G) - mu(2)(G). Also, we determine a relation between the Laplacian eigenvalues and matchings in a bipartite graph by showing that if G = (U, W) is a bipartite graph, vertical bar W vertical bar >= 9 vertical bar U vertical bar and mu(n)(G) <= 5 mu(2)(G), then G has a matching that saturates U.