  In this paper, we discuss the Hamiltonicity of graphs in terms of Wiener
index, hyper-Wiener index and Harary index of their quasi-complement or
complement. Firstly, we give some sufficient conditions for an balanced
bipartite graph with given the minimum degree to be traceable and Hamiltonian,
respectively. Secondly, we present some sufficient conditions for a nearly
balanced bipartite graph with given the minimum degree to be traceable.
Thirdly, we establish some conditions for a graph with given the minimum degree
to be traceable and Hamiltonian, respectively. Finally, we provide some
conditions for a $k$-connected graph to be Hamilton-connected and traceable for
every vertex, respectively.
