Unified bounds for the independence number of graphs

TL;DR

This paper introduces unified spectral bounds for the independence number of graphs, generalizing classical bounds like the Lovász theta and Hoffman bounds, with conditions for when these bounds are tight.

Contribution

It provides a unified framework using generalized inverses and eigenvalues to bound the independence number, connecting various classical bounds and characterizing extremal graphs.

Findings

Unified bounds encompass Lovász theta, Schrijver theta, and Hoffman bounds.

Necessary and sufficient conditions for graphs attaining these bounds.

Structural and spectral criteria for maximum independent sets and graph capacities.

Abstract

The Hoffman ratio bound, Lov\'{a}sz theta function and Schrijver theta function are classical upper bounds for the independence number of graphs, which are useful in graph theory, extremal combinatorics and information theory. By using generalized inverses and eigenvalues of graph matrices, we give bounds for independence sets and the independence number of graphs. Our bounds unify the Lov\'{a}sz theta function, Schrijver theta function and Hoffman-type bounds, and we obtain the necessary and sufficient conditions of graphs attaining these bounds. Our work leads to some simple structural and spectral conditions for determining a maximum independent set, the independence number, the Shannon capacity and the Lov\'{a}sz theta function of a graph.