Research On Spectral Sufficient Conditions Of Traceable And Hamiltonian Graphs

Graph spectral theory is an important research topic in the graph theory and combi-natorial matrix theory. The problem of determining whether a given graph is traceable or hamiltonian is NP-complete, which is very meaningful to give a brief and useful sufficient condition. We will give spectral sufficient conditions of traceable and hamiltonian graphs.·Firstly, we introduce background of spectral graph theory and the present situation of the research problem in this paper. Secondly we introduce some basic concepts and symbols used in this paper. Finally, we also introduce some main results of this paper briefly.·We give sufficient conditions for a connected graph being traceable and Hamiltonian in terms of the Wiener index of a graph and the complement of a graph, which correct,and extend the result of Yang [45]. Furthermore, we also present sufficient conditions for a connected bipartite graph being traceable and Hamiltonian in terms of the Wiener index of a bipartite graph and its quasi-complement. Finally, we give some sufficient distance spectral conditions for a connected graph or a connected bipartite graph being traceable and Hamiltonian in terms of its distance spectral radius.·We give sufficient conditions for a connected graph being traceable and Hamiltonian in terms of the Harary index of a graph and the complement of a graph, which correct and extend the result of Hua and Wang [18]. Furthermore, we also present sufficient conditions for a bipartite graph being traceable and Hamiltonian in terms of the Harary index of a bipartite graph and its quasi-complement. Finally, we give some sufficient Harary spectral conditions for a graph or a bipartite graph being traceable and Hamiltonian in terms of the Harary spectral radius of its complement or quasi-complement.