Wiener Index of a Type of Composite Graph

Abstract

The Wiener index, one of the oldest molecular topological descriptors used in mathematical chemistry, was well-studied during the past decades. For a graph $G$, its Wiener index is defined as $W(G)=\sum _{\{u,v\}\subseteq V(G)}{d}_G(u,v)$, where $d_G(u,v)$ is the distance between two vertices $u$ and $v$ in $G$. In this paper, we study the Wiener index of a class of composite graph, namely, double graph. We reveal the relation between the Wiener index of a given graph and the one of its double graph as well as the relation between Wiener index of a given graph and the one of its $k$-iterated double graph. As a consequence, we determine the graphs with the maximum and minimum Wiener index among all double graphs and $k$-iterated double graphs of connected graphs of the same order, respectively.