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

Abstract

Let $α'$ and $μ_i$ denote the matching number of a non-empty simple graph $G$ with $n$ vertices and the $i$-th smallest eigenvalue of its Laplacian matrix, respectively. In this paper, we prove a tight lower bound $$α' \ge \min\left\{\Big\lceil\frac{μ_2}{μ_n} (n -1)\Big\rceil,\ \ \Big\lceil\frac{1}{2}(n-1)\Big\rceil \right\}.$$ This bound strengthens the result of Brouwer and Haemers who proved that if $n$ is even and $2μ_2 \ge μ_n$, then $G$ has a perfect matching. A graph $G$ is factor-critical if for every vertex $v\in V(G)$, $G-v$ has a perfect matching. We also prove an analogue to the result of Brouwer and Haemers mentioned above by showing that if $n$ is odd and $2μ_2 \ge μ_n$, then $G$ is factor-critical. We use the separation inequality of Haemers to get a useful lemma, which is the key idea in the proofs. This lemma is of its own interest and has other applications. In particular, we prove similar results for the number of balloons, spanning even subgraphs, as well as spanning trees with bounded degree.