New results relating independence and matchings

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.