Some sufficient conditions on hamilton graphs with toughness - Cai, Gaixiang; Yu, Tao; Xu, Huan; Yu, Guidong (Frontiers in Computational Neuroscience, 2022)
Let $G$ be a graph, $ω(G)$ denote the number of components of a graph $G$, and $t$ is real number. If $ω(G-S)\ge2 \Rightarrow |S|\ge tω(G-S)$ holds for each set $S$ of vertex set $(G)$ of $G$, $G$ be said to be $t$-tough. The $toughness$ of $G$ is the maximum value of $t$ for which $G$ is $t$-tough, denoted with $τ(G)$. The graph $G$ is called Hamilton graph if it has a cycle which contains all vertices of $G$. Chv$\acute{a}$tal and other scholars have studied the relation between toughness conditions to the existence of cycle structures. In this paper, we first establish some sufficient conditions for a graph with toughness to be Hamiltonian in terms of the edge number, the spectral radius and the signless Laplacian spectral radius of the graph.