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In this paper, with respect to the Wiener index, hyper-Wiener index, and Harary index, it gives some sufficient conditions for some graphs to be traceable, Hamiltonian, Hamilton-connected, or traceable for every vertex. Firstly, we discuss balanced bipartite graphs with , where is the minimum degree of G, and gain some sufficient conditions for the graphs to be traceable or Hamiltonian, respectively. Secondly, we discuss nearly balanced bipartite graphs with and present some sufficient conditions for the graphs to be traceable. Thirdly, we discuss graphs with and obtain some conditions for the graphs to be traceable or Hamiltonian, respectively. Finally, we discuss t-connected graphs and provide some conditions for the graphs to be Hamilton-connected or traceable for every vertex, respectively.

Sufficient Conditions for Hamiltonicity of Graphs with Respect to Wiener Index, Hyper-Wiener Index, and Harary Index