Algebras, graphs and thetas

TL;DR

This paper generalizes the clique-coclique inequality and related bounds involving the Lovász theta number to broader classes of graphs, including homogeneous coherent configurations and 1-walk regular graphs, with characterizations of equality.

Contribution

It extends known inequalities from specific graph classes to more general structures and introduces stronger bounds involving the Lovász theta number.

Findings

Generalization of clique-coclique inequality to new graph classes

Introduction of stronger inequalities involving Lovász theta number

Characterizations of equality cases in these inequalities

Abstract

We extend the clique-coclique inequality, previously known to hold for graphs in association schemes and vertex-transitive graphs, to graphs in homogeneous coherent configurations and 1-walk regular graphs. We further generalize it to a stronger inequality involving the Lov\'asz theta number of such graphs, and some theta variants, including characterizations of the equality.