Lower bounds on the independence number of a graph in terms of degrees

Abstract: Given an integer $Δ\ge 3$, let ${\cal G}<em>{Δ}$ be the set of connected graphs $G\neq K</em>{Δ+1}$ with maximum degree $Δ$ and, for $i=1,\cdots, Δ$, let $V_i(G)$ be the set of vertices of $G$ of degree $i$. Using a result of T. Kelly and L. Postle, we prove that $\sum\limits_{i=1}<sup>Δc_i|V_i(G)|$ is a lower bound on the independence number $α(G)$ of $G\in {\cal G}<em>Δ$, where $c</em>Δ=\frac{1}Δ$ and $ic_{i}=1-c_{i+1}$ for $i=1,\cdots,Δ-1$. Moreover, if $\varepsilon &gt;0$ and $j\in {1,\cdots, Δ}$, then the inequality $$α(G)\ge \varepsilon|V_j(G)|+\sum\limits_{i=1}<sup>Δc_i|V_i(G)|$$ does not hold for infinitely many graphs $G\in {\cal G}_Δ$. Finally, further lower bounds on $α(G)$ in terms of degrees of $G$ are presented.