On inertia and ratio type bounds for the $k$-independence number of a graph and their relationship

TL;DR

This paper explores the relationship between inertia and ratio bounds for the $k$-independence number in graphs, providing exact values and new bounds for special classes like walk-regular and partially walk-regular graphs.

Contribution

It establishes a connection between the polynomials used in inertia and ratio bounds for $k$-independence numbers and derives new bounds for specific graph classes.

Findings

Exact values for $eta_k$ in graphs with regularity

Relationship between inertia and ratio bounds in regular graphs

New sharp bounds for partially walk-regular graphs

Abstract

For $k\ge 1$, the $k$-independence number $\alpha_k$ of a graph is the maximum number of vertices that are mutually at distance greater than $k$. The well-known inertia and ratio bounds for the (1-)independence number $\alpha(=\alpha_1)$ of a graph, due to Cvetkovi\'c and Hoffman, respectively, were generalized recently for every value of $k$. We show that, for graphs with enough regularity, the polynomials involved in such generalizations are closely related and give exact values for $\alpha_k$, showing a new relationship between the inertia and ratio type bounds. Additionally, we investigate the existence and properties of the extremal case of sets of vertices that are mutually at maximum distance for walk-regular graphs. Finally, we obtain new sharp inertia and ratio type bounds for partially walk-regular graphs by using the predistance polynomials.