On the $k$-independence number of graphs

TL;DR

This paper introduces new spectral bounds for the $k$-independence number of graphs, unifying and improving upon previous bounds, with applications to infinite graphs and graph diameter estimates.

Contribution

It generalizes existing spectral bounds on the $k$-independence number, providing tighter results and unification of prior bounds, including applications to infinite graphs.

Findings

New spectral bounds outperform previous bounds in most cases.

Bounds are tight for certain infinite graphs.

Derived lower spectral bounds for graph diameter.

Abstract

This paper generalizes and unifies the existing spectral bounds on the $k$-independence number of a graph, which is the maximum size of a set of vertices at pairwise distance greater than $k$. The previous bounds known in the literature follow as a corollary of the main results in this work. We show that for most cases our bounds outperform the previous known bounds. Some infinite graphs where the bounds are tight are also presented. Finally, as a byproduct, we derive some lower spectral bounds for the diameter of a graph.