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arXiv · 2010.12649

Optimization of eigenvalue bounds for the independence and chromatic number of graph powers

Aida Abiad·Gabriel Coutinho·Miquel Angel Fiol·Bruno Nogueira·Sjanne Zeijlemaker
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Abstract

The $k^{\text{th}}$ power of a graph $G=(V,E)$, $G^k$, is the graph whose vertex set is $V$ and in which two distinct vertices are adjacent if and only if their distance in $G$ is at most $k$. This article proves various eigenvalue bounds for the independence number and chromatic number of $G^k$ which purely depend on the spectrum of $G$, together with a method to optimize them. Our bounds for the $k$-independence number also work for its quantum counterpart, which is not known to be a computable parameter in general, thus justifying the use of integer programming to optimize them. Some of the bounds previously known in the literature follow as a corollary of our main results. Infinite families of graphs where the bounds are sharp are presented as well.

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Aida Abiad, Gabriel Coutinho, Miquel Angel Fiol, Bruno Nogueira, Sjanne Zeijlemaker. 2020-10-23. Optimization of eigenvalue bounds for the independence and chromatic number of graph powers. https://arxiv.org/abs/2010.12649

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