Wiener-type Invariants on Graph Properties

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 mainly give some sufficient
conditions for a connected graph to be $k$-connected, $\beta$-deficient, $k$-hamiltonian, $k$-edge-hamiltonian and
$k$-path-coverable.