Toughness, hamiltonicity and spectral radius in graphs

Abstract

The study of the existence of Hamiltonian cycles in a graph is a classical problem in graph theory. By incorporating toughness and spectral conditions, we can consider Chvátal’s conjecture from another perspective: What is the spectral condition to guarantee the existence of a Hamiltonian cycle among t-tough graphs? We first give the answer to 1-tough graphs, i.e. if ρ(G)≥ρ(Mn), then G contains a Hamiltonian cycle, unless G≅Mn, where Mn=K1∇Kn−4+3 and Kn−4+3 is the graph obtained from 3K1∪Kn−4 by adding three independent edges between 3K1 and Kn−4. The Brouwer’s toughness theorem states that every d-regular connected graph always has t(G)>dλ−1 where λ is the second largest absolute eigenvalue of the adjacency matrix. In this paper, we extend the result in terms of its spectral radius, i.e. we provide a spectral condition for a graph to be 1-tough with minimum degree δ and to be t-tough, respectively.