An inequality using perfect matchings and Laplacian spread of a graph

Let G=(V,E) be a simple connected graph of order n. Let 0=μ1(G)≤μ2(G)≤⋯≤μn(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 |X|=|Y| and the set of all edges between X and Y decomposed into r disjoint perfect matchings, then, 2r-LS(G)2μ2(G)≤|X|n≤2r+LS(G)2μn(G), where LS(G)=μn(G)-μ2(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, |W|≥9|U| and μn(G)≤5μ2(G), then G has a matching that saturates U.