An improved spectral lower bound of treewidth

TL;DR

This paper improves spectral lower bounds on treewidth by relating the treewidth to Laplacian eigenvalues. It proves two main bounds: $\mathsf{tw}(G) \ge \frac{n \lambda_{2}}{\Delta + \lambda_{2}} - 1$ and $\mathsf{tw}(G) \ge \frac{2n \lambda_{2}}{3\lambda_{n} - \lambda_{2}} - 1$, with the first near-tight for complete bipartite graphs and the second tight for complete graphs. The proofs combine a Robertson--Seymour style partition with a three-partition construction and a Gu-Liu type inequality, linking the Rayleigh quotient to cut sizes. These results refine the spectral toolkit for decomposability and could inform algorithms exploiting treewidth bounds.

Abstract

We show that for every $n$-vertex graph with at least one edge, its treewidth is greater than or equal to $n λ_{2} / (Δ+ λ_{2}) - 1$, where $Δ$ and $λ_{2}$ 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.

Figures (1)

• Figure 1: A visual proof of \ref{['obs:triangle']}.

Theorems & Definitions (12)

• Lemma 2.1: RobertsonS86
• Lemma 2.2: See e.g., GruberH08
• Lemma 3.1
• Lemma 3.2
• proof
• Theorem 3.4
• proof
• Remark 3.5
• Lemma 4.1: GuL22
• Theorem 4.2