A tight lower bound on the matching number of graphs via Laplacian eigenvalues

Let α′ and μi denote the matching number of a non-empty simple graph G with n vertices ith smallest eigenvalue its Laplacian matrix, respectively. In this paper, we prove tight lower bound α′≥min⌈μ2μn(n−1)⌉,⌈12(n−1)⌉.This strengthens result Brouwer Haemers who proved that if is even 2μ2≥μn, then has perfect matching. A factor-critical for every vertex v∈V(G), G−v We also an analogue to mentioned above by showing odd factor-critical. use separation inequality get useful lemma, which key idea in proofs. This lemma own interest other applications. particular, similar results balloons, spanning subgraphs, as well trees bounded degree.