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arXiv · 2404.08520

An improved spectral lower bound of treewidth

Tatsuya Gima·Tesshu Hanaka·Kohei Noro·Hirotaka Ono·Yota Otachi
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Abstract

We show that for every $n$-vertex graph with at least one edge, its treewidth is greater than or equal to $n \lambda_{2} / (\Delta + \lambda_{2}) - 1$, where $\Delta$ and $\lambda_{2}$ are the maximum degree and the second smallest Laplacian eigenvalue of the graph, respectively. This lower bound improves the one by Chandran and Subramanian [Inf. Process. Lett., 2003] and the subsequent one by the authors of the present paper [IEICE Trans. Inf. Syst., 2024]. The new lower bound is almost tight in the sense that there is an infinite family of graphs such that the lower bound is only $1$ less than the treewidth for each graph in the family. Additionally, using similar techniques, we also present a lower bound of treewidth in terms of the largest and the second smallest Laplacian eigenvalues.

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Tatsuya Gima, Tesshu Hanaka, Kohei Noro, Hirotaka Ono, Yota Otachi. 2024-04-12. An improved spectral lower bound of treewidth. https://arxiv.org/abs/2404.08520

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