AUTHOR=Cai Gaixiang , Yu Tao , Xu Huan , Yu Guidong 

TITLE=Some sufficient conditions on hamilton graphs with toughness

JOURNAL=Frontiers in Computational Neuroscience

VOLUME=Volume 16 - 2022

YEAR=2022

URL=https://www.frontiersin.org/journals/computational-neuroscience/articles/10.3389/fncom.2022.1019039

DOI=10.3389/fncom.2022.1019039

ISSN=1662-5188

ABSTRACT=Let $G$ be a graph, $\omega(G)$ denote the number of components of a graph $G$, and $t$ is real number. If $\omega(G-S)\ge2 \Rightarrow |S|\ge t\omega(G-S)$ 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 $\tau(G)$. 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.