Spectral Conditions For Hamiltonicity Of Graphs

In graph theory,it is important to find sufficient conditions for Hamiltonicity of graphs.Many existing results are in terms of the degree of vertex and the number of edges.In recent years,many sufficient conditions use the spectral radius of graphs.In this paper,inspired by Nikiforov's result,we first prove a sufficient condition for Hamiltonicity of a general graph in terms of the signless Laplacian spectral radius.Similarly,improving a result by Li and Ning,we establish a sufficient condition for Hamiltonicity of balanced bipartite graphs.In addition,we construct a family of graphs to show that results proved in this paper give new strength for one to determine the Hamiltonicity of graphs.It is surprising that theorems in this paper contain larger families of extremal graphs than previous results using the spectral radius of the adjacency matrix.