An improved spectral lower bound of treewidth

We show that for every $n$-vertex graph with at least one edge, its treewidth is greater than or equal to $n{\lambda }_2\mathrm{/}(\mathrm{\Delta }+{\lambda }_2)-1$, where $\mathrm{\Delta }$ 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.