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

Abstract

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 $t\ge 2$ and $r\ge 3$, the join $K_{t-1}\vee T_{n-t+1,r}$ 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.