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, Lova´sz’ parameter ϑ, Schrijver’s parameter ϑ′, 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 Cvetkovi´c.