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 ₂ / (+ ₂) - 1, where and 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.