A note on topological indices and the twin classes of graphs

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.