Spectral Condition for a Graph to be Hamiltonian with respect to   Normalized Laplacian
  Let G be a graph and let \Delta,\delta be the maximum and minimum degrees of G respectively, where \Delta/\delta<c<\sqrt{2} and c is a constant. In this paper we establish a sufficient spectral condition for the graph G to be Hamiltonian, that is, the nontrivial eigenvalues of the normalized Laplacian of G are sufficiently close to 1. 
Yi-Zheng Fan, Gui-Dong Yu
math.CO