  In this paper we study relationships between the \emph{matching number},
written $\mu(G)$, and the \emph{independence number}, written $\alpha(G)$. Our
first main result is to show \[ \alpha(G) \le \mu(G) + |X| - \mu(G[N_G[X]]), \]
where $X$ is \emph{any} intersection of maximum independent sets in $G$. Our
second main result is to show \[ \delta(G)\alpha(G) \le \Delta(G)\mu(G), \]
where $\delta(G)$ and $\Delta(G)$ denote the minimum and maximum vertex degrees
of $G$, respectively. These results improve on and generalize known relations
between $\mu(G)$ and $\alpha(G)$. Further, we also give examples showing these
improvements.
