Spectral radius and Hamiltonian properties of graphs, II

In this paper, we first present spectral conditions for the existence of Cn-1 in graphs (2-connected graphs) of order n, which are motivated by a conjecture of Erdos. We also prove spectral conditions for the existence of Hamilton cycles in balanced bipartite graphs. This result presents a spectral analog of Moon-Moser's theorem on Hamilton cycles in balanced bipartite graphs, and extends a previous theorem due to Li and the second author for n sufficiently large. We conclude this paper with two problems on tight spectral conditions for the existence of long cycles of given lengths.