Spectral bounds for the $k$-independence number of a graph

TL;DR

This paper introduces two new spectral upper bounds for the $k$-independence number of a graph, improving upon previous bounds and demonstrating cases where the bounds are tight.

Contribution

The paper presents novel spectral bounds for the $k$-independence number and constructs graphs that attain the first bound, advancing understanding of graph spectral properties.

Findings

Constructed graphs attain equality for the first bound

Second bound compares favorably to previous bounds

New bounds improve estimation of $k$-independence number

Abstract

In this paper, we obtain two spectral upper bounds for the $k$-independence number of a graph which is is the maximum size of a set of vertices at pairwise distance greater than $k$. We construct graphs that attain equality for our first bound and show that our second bound compares favorably to previous bounds on the $k$-independence number.