  The matching number of a graph $G$, denoted by $\alpha'(G)$, is the size of a
maximum matching in $G$. The matching number of regular graphs has been well
studied, however, not as much is known for general graphs. Haxell and Scott
gave a complete description for the lower bound on the matching number of a
graph with maximum degree at most 3 in terms of the number of vertices with
fixed degree. Similar idea was used by Henning and Yeo, who established a tight
lower bound on the matching number of a graph with given maximum degree in
terms of the numbers of vertices and edges. We discover a tight lower bound on
the matching number of general graphs via eigenvalues of combinatorial
Laplacian. Let $\mu_i$ denotes the $i$th smallest eigenvalue of the Laplacian
matrix of a graph. In this paper, we show that $\alpha'(G)\ge
\min\{\frac{\mu_2}{\mu_n}(n -1), \frac{1}{2}(n-1)\}$ for a nonempty graph $G$
on $n$ vertices, strengthening a theorem of Brouwer and Haemers who showed that
if $2\mu_2 \ge \mu_n$, then $G$ has a perfect matching. Moreover, the lower
bound is tight. We contribute a useful lemma in the proof, which is of its own
interest, and as applications, we prove similar results for factor-critical
graphs, the number of balloons, spanning even subgraphs, as well as spanning
trees with bounded degree.
