Abstract: Let G be an undirected simple graph and A(G) be the adjacency matrix of G. This paper gives some sufficient conditions for G to have a Hamiltonian path or cycle or to be Hamilton-connected in terms of eigenvalues of the complement of G, and gives a sufficient condition for a bipartite graph to have Hamiltonian cycles in terms of eigenvalues of its quasi-complement. These results improve some known results.

Key words: energy of a graph, Hamiltonian path, Hamiltonian cycle, Hamilton-connected graph