Expected values of parameters associated with the minimum rank of a graph

Tracy Hall, Leslie Hogben, Ryan Martin, Bryan Shader

Published 2016-05-18Version 1

We investigate the expected value of various graph parameters associated with the minimum rank of a graph, including minimum rank/maximum nullity and related Colin de Verdi\`ere-type parameters. Let $G(v,p)$ denote the usual Erd\H{o}s-R\'enyi random graph on $v$ vertices with edge probability $p$. We obtain bounds for the expected value of the random variables $\mr(G(v,p))$, $\M(G(v,p))$, $\nu(G(v,p))$ and $\xi(G(v,p))$, which yield bounds on the average values of these parameters over all labeled graphs of order $v$.