  Let $G$ be a distance-regular graph of order $v$ and size $e$. In this paper,
we show that the max-cut in $G$ is at most $e(1-1/g)$, where $g$ is the odd
girth of $G$. This result implies that the independence number of $G$ is at
most $\frac{v}{2}(1-1/g)$. We use this fact to also study the extendability of
matchings in distance-regular graphs. A graph $G$ of even order $v$ is called
$t$-extendable if it contains a perfect matching, $t<v/2$ and any matching of
$t$ edges is contained in some perfect matching. The extendability of $G$ is
the maximum $t$ such that $G$ is $t$-extendable. We generalize previous results
on strongly regular graphs and show that all distance-regular graphs with
diameter $D\geq 3$ are $2$-extendable. We also obtain various lower bounds for
the extendability of distance-regular graphs of valency $k$ that depend on $k$,
$\lambda$ and $\mu$, where $\lambda$ is the number of common neighbors of any
two adjacent vertices and $\mu$ is the number of common neighbors of any two
vertices in distance two.
