发表机构
Hainan University; Shanghai Normal University; Yangtze University（海南大学; 上海师范大学; 长江大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了关于$t$-色临界图$F$的$n$顶点$F$-自由图中符号拉普拉斯谱半径由$K_{t-1}\vee T_{n-t+1,r}$唯一最大化的猜想，采用Zykov对称化类比与对$n$归纳两种新技术。
AI 中文摘要
诱导匹配是一种形成诱导子图的匹配。如果移除某个大小为$t$的诱导匹配会降低其色数，但移除任意$t-1$个顶点不会，则称图$G$为$t$-色临界图。设$F$是一个满足$\chi(F)=r+1$的$t$-色临界图。对于足够大的$n$，Simonovits确定了$n$个顶点上唯一的边极值$F$-自由图。最近，Zheng、Li和Li [Linear Algebra Appl.\\ 730 (2026) 546--565]猜想：对于$t\ge 2$和$r\ge 3$，当$n$足够大时，在所有$n$个顶点的$F$-自由图中，联图$K_{t-1}\vee T_{n-t+1,r}$唯一地最大化符号拉普拉斯谱半径。在本文中，我们证明了这个猜想。与通常的谱论证相反，我们对该猜想的证明依赖于两种风格迥异的技术。第一种技术是符号拉普拉斯矩阵的Zykov对称化的类比。第二种技术是对$n$的归纳，由此我们获得符号拉普拉斯谱极值图的Perron向量最小分量的下界，而非结构性的陈述。
英文摘要
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.
