A solution to a conjecture on the signless Laplacian spectral radius for t-color-critical graphs

@misc{chen2026solution,
  title = {{A solution to a conjecture on the signless Laplacian spectral radius for $t$-color-critical graphs}},
  author = {Ming-Zhu Chen and Ya-Lei Jin and Peng-Li Zhang and Jian Zheng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.21367},
  eprint = {2609.21367},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21367},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - A solution to a conjecture on the signless Laplacian spectral radius for t-color-critical graphs
AU  - Ming-Zhu Chen
AU  - Ya-Lei Jin
AU  - Peng-Li Zhang
AU  - Jian Zheng
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.21367
UR  - https://arxiv.org/abs/2609.21367
AB  - An induced matching is a matching that forms an induced subgraph. A graph is t-color-critical if removing some induced matching of size t lowers its chromatic number, but removing any t−1 vertices does not. Let F be a t-color-critical graph with χ(F)=r+1. For sufficiently large n, Simonovits determined the unique edge-extremal F-free graph on n vertices. Recently, Zheng, Li and Li [Linear Algebra Appl.\ 730 (2026) 546--565] conjectured that, for t≥ 2 and r≥ 3, the join K_t−1∨ T_n−t+1,r uniquely maximizes the signless Laplacian spectral radius among all n-vertex F-free graphs when n is sufficiently large. In this paper, we prove this conjecture. In contrast to the usual spectral arguments, our proof of this conjecture relies on two techniques of a rather different flavour. Our first technique is an analogue of Zykov symmetrization for the signless Laplacian matrix. Our second technique is an induction on n, from which we obtain the lower bound on the smallest entry of the Perron vector of a signless Laplacian spectral extremal graph rather than a structural statement.
ER  -