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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2606.01761 |
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Table of Contents:
- 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).