A conjecture by Erds and Hajnal [Ramsey-type theorems. Discrete Appl Math. 1989;25:37–52] pointed out that the structure of graphs with forbidden induced subgraphs is very different from general graphs. More precisely, they contain much larger cliques or independent sets. Inspired by this, we prove that non-Hamilton-connected graphs satisfying certain conditions contain a large clique. From the perspective of looking for a maximum clique, we present some sufficient conditions for a graph to be Hamilton-connected in terms of size, spectral radius and signless Laplacian spectral radius, respectively, which extend some corresponding results.

中文翻译：

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汉密尔顿连通图谱条件的改进

Erds 和 Hajnal 的猜想 [Ramsey 型定理。离散应用数学。1989;25:37–52]指出具有禁止诱导子图的图的结构与一般图有很大不同。更准确地说，它们包含更大的集团或独立集。受此启发，我们证明了满足一定条件的非汉密尔顿连通图包含一个大团。从寻找最大团的角度出发，我们分别在大小、谱半径和无符号拉普拉斯谱半径方面给出了图成为哈密顿连通的充分条件，并扩展了一些相应的结果。