Wiener-type invariants and Hamiltonian properties of graphs

Abstract

The Wiener-type invariants of a simple connected graph $G=(V(G),E(G))$ can be expressed in terms of the
quantities $W_{f}=\sum_{\{u,v\}\subseteq V(G)}f(d_{G}(u,v))$ for various choices of the function $f(x)$, where
$d_{G}(u,v)$ is the distance between vertices $u$ and $v$ in $G$. In this paper, we give some sufficient
conditions for a bipartite graph to be Hamiltonian or a connected general graph to be Hamilton-connected and
traceable from every vertex in terms of the Wiener-type invariants of $G$ or the complement of $G$.