Wiener Index, Hyper-wiener Index, Harary Index and Hamiltonicity of graphs

TL;DR

This paper explores how various graph indices like Wiener, hyper-Wiener, and Harary relate to Hamiltonian properties of graphs, providing new sufficient conditions for traceability and Hamiltonicity based on these indices.

Contribution

It introduces novel sufficient conditions linking graph indices to Hamiltonian properties in bipartite and general graphs, extending existing criteria.

Findings

Conditions for bipartite graphs to be traceable and Hamiltonian

Criteria for nearly balanced bipartite graphs' traceability

Conditions for k-connected graphs to be Hamilton-connected

Abstract

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.