A graph for which the inertia bound is not tight

TL;DR

This paper demonstrates that the inertia bound, which estimates the independence number of a graph using matrix inertia, is not always tight, providing counterexamples involving Paley graphs.

Contribution

It presents the first known instances where the inertia bound fails to be tight, specifically for the Paley graph on 17 vertices and related graphs.

Findings

Inertia bound is not tight for Paley graph on 17 vertices.

Counterexamples show the bound's limitations beyond small graphs.

The result answers an open question about the bound's universality.

Abstract

The inertia bound gives an upper bound on the independence number of a graph by considering the inertia of matrices corresponding to the graph. The bound is known to be tight for graphs on 10 or fewer vertices as well as for all perfect graphs. The question has been asked as to whether the bound is always tight. We show that the bound is not tight for the Paley graph on 17 vertices as well as for the graph obtained from Paley 17 by deleting a vertex.