Sufficient conditions for certain structural properties of graphs based on Wiener-type indices

Abstract

Let $G=(V,E)$ be a simple connected graph with the vertex set $V$
and the edge set $E$. The Wiener-type invariants of $G=(V,E)$ can be
expressed in terms of the quantities $W_{f}=\sum_{\{u,v\}\subseteq V}f(d_{G}(u,v))$ for various choices of the function $f$, where
$d_{G}(u,v)$ is the distance between vertices $u$ and $v$ in $G$. In
this paper, we establish sufficient conditions based on Wiener-type
indices under which every path of length $r$ is contained in a
Hamiltonian cycle, and under which a bipartite graph on $n+m$
($m>n$) vertices contains a cycle of size $2n$