  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.
