A note on the Ratio and Inertia Bounds for the $k$-Independence Number

Abstract

The $k$-th power $G^k$ of a graph $G$ is the graph on the same vertex set where the edge set consists of those pairs of distinct vertices of $G$ that are at distance at most $k$ from each other. A. Abiad, G. Coutinho, and M. A. Fiol [On the $k$-independence number of graphs, Discrete Mathematics 342 (2019), 2875--2885] proposed extensions of the classical ratio (for regular graphs) and inertia bounds to the independence number of $G^k$ for $k\ge 2$. Continuing a line of work comparing these two parameters with other known bounds, we show that the $\vartheta$-function of L. Lovász and the weighted inertia bound of A. R. Calderbank and P. Frankl, when applied directly to $G^k$, perform at least as well as the ratio and inertia bounds of Abiad-Coutinho-Fiol, respectively. In particular, $\vartheta (G^k)$ provides a polynomial-time computable upper bound on the independence number of $G^k$ that is at least as strong as the ratio bound when the latter applies (i.e.,\ when the graph $G$ is regular).