Unified bounds for the independence number of graphs
Jiang Zhou
TL;DR
This work develops a unified, matrix-analytic approach to bounding the independence number $\alpha(G)$ by leveraging generalized inverses of graph-associated PSD matrices. Through a general bound $F(M,x)$ and its variants, it encompasses and strengthens Hoffman-type bounds while recovering the Lovász theta and Schrijver theta bounds as special cases, together with exact equality characterizations. The framework yields simple structural and spectral conditions under which $\alpha(G)$, $\Theta(G)$, and $\vartheta(G)$ coincide, and extends to classical graph constructions and capacity considerations. By linking eigenvalue-based criteria with matrix inverses, the paper provides a versatile toolkit for determining maximum independent sets, the independence number, and Shannon capacity across a broad spectrum of graphs.
Abstract
The Hoffman ratio bound, Lovász 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 Lovász 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 Lovász theta function of a graph.