On bounds for some graph invariants

Let $G$ be a graph without isolated vertices and let $\alpha (G)$ be its stability number and $\tau (G)$ its covering number. The {\it ${\alpha }_v$-cover} number of a graph, denoted by ${\alpha }_v(G)$, is the maximum natural number $m$ such that every vertex of $G$ belongs to a maximal independent set with at least $m$ vertices. In the first part of this paper we prove that $\alpha (G)\le \tau (G)[1+\alpha (G)-{\alpha }_v(G)]$. We also discuss some conjectures analogous to this theorem. In the second part we give a lower bound for the number of edges of a graph $G$ as a function of the stability number $\alpha (G)$, the covering number $\tau (G)$ and the number of connected components $c(G)$ of $G$. Namely, let $\alpha$ and $\tau$ be two natural numbers and let $$\mathrm{\Gamma }(\alpha ,\tau )=\min \sum _{i=1}^{\alpha }\text{bin}{z}_{i}2\mathrm{\mid }{z}_1+\mathrm{.}\mathrm{.}\mathrm{.}+z_{\alpha }=\alpha +\tau and{z}_i\ge 0\mathrm{\forall }i=1,\mathrm{.}\mathrm{.}\mathrm{.},\alpha \mathrm{.}$$ Then if $G$ is any graph, we have: $$\mathrm{\mid }E(G)\mathrm{\mid }\ge \alpha (G)-c(G)+\mathrm{\Gamma }(\alpha (G),\tau (G))\mathrm{.}$$