An improved spectral lower bound of treewidth
<|reference_start|>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 $n \lambda_{2} / (\Delta + \lambda_{2}) - 1$, 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.<|reference_end|>
@article{gima2024an,
    title={An improved spectral lower bound of treewidth},
    author={Tatsuya Gima, Tesshu Hanaka, Kohei Noro, Hirotaka Ono, Yota Otachi},
    journal={arXiv preprint arXiv:2404.08520},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.08520},
    primaryClass={math.CO cs.DS}
}