Wiener-Type Invariants of some Graph Operations

S. Hossein-Zadeh, A. Hamzeh, A. R. Ashrafi

Let $d(G,k)$ be the number of pairs of vertices of a graph $G$ that are at distance $k$, $\lambda$ a real number, and $W_{\lambda}(G)=\Sigma_{k\geq1}d(G,k)k^{lambda}$. $W_{\lambda}(G)$ is called the Wiener-type invariant of $G$ associated to real number $\lambda$. In this paper, the Wiener-type invariants of some graph operations are computed. As immediate consequences, the formulae for reciprocal Wiener index, Harary index, hyper-Wiener index and Tratch--Stankevich--Zefirov index are calculated. Some upper and lower bounds are also presented.