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 \(W(G) = \sum\limits_{\{u, v\} \subseteq V(G)} d_G(u, v)\), 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.