Caro-Wei bound ($\alpha (G)\ge {\sum }_v\frac{1}{1+\mathrm{deg}v}$)

For a finite simple graph, $\alpha (G)\ge {\sum }_{v\in V(G)}1/(1+\mathrm{deg}v)\ge |V(G){|}^2/(|V(G)|+2|E(G)|)$. Randomly order its vertices and retain each vertex preceding all its graph neighbours. This gives an independent set with the first expression as its expected value. The second inequality is the Cauchy-Schwarz inequality.