A note on topological indices and the twin classes of graphs

Abstract: Topological indices are parameters associated with graphs that have many applications in different areas such as mathematical chemistry. Among various topological indices, the Wiener index is classical \cite{w}. In this paper, we prove a formula for the Wiener index and more general $m$-Steiner Wiener index of an arbitrary graph $G$ in terms of the cardinalities of its twin classes. In particular, we will show that calculating these parameters for the graph $G$ can be reduced to calculating the same for a much smaller graph (in general) called the reduced graph of $G$. As applications of our main result, the $m$-Steiner Wiener index is explicitly calculated for various important classes of graphs from the literature including \begin{enumerate} \item[(a)] Power graphs associated with finite groups, \item[(b)] Zero divisor graphs and the ideal-based zero divisor graphs associated with commutative rings with unity, and \item[(c)] Comaximal ideal graphs associated with commutative rings with unity. \end{enumerate} We have also found an upper bound on the $m$-Steiner Wiener index of an infinite class of graphs called the completely joined graphs. As a corollary of this result, we explicitly calculate the $m$-Steiner Wiener index of the complete multipartite graphs.