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

Let $\alpha'$ and $\mu_i$ 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 $$\alpha' \ge \min\left\{\Big\lceil\frac{\mu_2}{\mu_n} (n -1)\Big\rceil,\ \ \Big\lceil\frac{1}{2}(n-1)\Big\rceil \right\}.$$ This bound strengthens the result of Brouwer and Haemers who proved that if $n$ is even and $2\mu_2 \ge \mu_n$, 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 $2\mu_2 \ge \mu_n$, 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.