Optimization of eigenvalue bounds for the independence and chromatic number of graph powers

Abstract

The k th power of a graph G = (V, E), G k , is the graph whose vertex set is V and in which two distinct vertices are adjacent if and only if their distance in G is at most k. This article proves various eigenvalue bounds for the independence number and chromatic number of G k which purely depend on the spectrum of G, together with a method to optimize them. Our bounds for the k-independence number also work for its quantum counterpart, which is not known to be a computable parameter in general, thus justifying the use of integer programming to optimize them. Some of the bounds previously known in the literature follow as a corollary of our main results. Infinite families of graphs where the bounds are sharp are presented as well.

Characterizing and computing weight-equitable partitions of graphs

“…They were first used by Haemers in 1979 [5,Theorem 6] to provide a proof of Hoffman's lower bound for the chromatic number of a general graph, weakening the regularity assumption required in the well-known Hoffman result on the independence number [19]. Such a bound for general graphs has recently been extended to the distance k-chromatic number also using weight-equitable partitions, see [3,Theorem 4.3]. Fiol and Garriga [12,10] used them to obtain several sharp spectral bounds for parameters of non-regular graphs.…”