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

TL;DR

This paper studies the expected values of graph parameters related to minimum rank in Erdős-Rényi random graphs, providing bounds that inform about average behaviors of these parameters across all labeled graphs.

Contribution

It introduces bounds for the expected minimum rank, maximum nullity, and related parameters in Erdős-Rényi graphs, advancing understanding of their average properties.

Findings

Bounds for expected minimum rank and nullity in G(v,p)

Analysis of average values over all labeled graphs

Insights into parameters related to graph rank

Abstract

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 ${\rm mr}(G(v,p))$, ${\rm M}(G(v,p))$, $\nu(G(v,p))$ and $\xi(G(v,p))$, which yield bounds on the average values of these parameters over all labeled graphs of order $v$.