Some sufficient conditions on hamilton graphs with toughness

Let G be a graph, and the number of components of G is denoted by c(G). Let t be a positive real number. A connected graph G is t-tough if tc(G − S) ≤ |S| for every vertex cut S of V(G). The toughness of G is the largest value of t for which G is t-tough, denoted by τ(G). We call a graph G Hamiltonian if it has a cycle that contains all vertices of G. Chvátal and other scholars investigate the relationship between toughness conditions and the existence of cyclic structures. In this paper, we establish some sufficient conditions that a graph with toughness is Hamiltonian based on the number of edges, spectral radius, and signless Laplacian spectral radius of the graph.