Weighted matrix eigenvalue bounds on the independence number of a graph

Abstract

Weighted generalizations of Hoffman's ratio bound on the independence number of a regular graph are surveyed. Several known bounds are reviewed as special cases of modest extensions. Comparisons are made with the Shannon capacity Θ, Lovász' parameter θ{symbol}, Schrijver's parameter θ{symbol}′, and the ultimate independence ratio for categorical products. The survey concludes with some observations on graphs that attain a weighted version of a bound of Cvetković.