On the Wiener Index of Rooted Product of Graphs

Abstract

Let \(H\) be a simple graph with \(n\) vertices and \(\mathcal{G} = \{G_1, G_2, \ldots, G_n\}\) be a sequence of \(n\) rooted graphs.
Following Godsil and McKay (Bull. Austral. Math. Soc. \(18 (1978) 21-28\)) defined the the rooted product \(H({G})\) of \(H\) by \({G}\) is defined by identifying the root of \(G_i\) with the \(i\)th vertex of \(H\).In this paper, we calculate the Wiener index of \(H({G})\), i.e., the sum of distances between all pairs of vertices, in terms of the Wiener indices of \(G_i\), \(i = 1, 2, \ldots, k\).As an application, we derive a recursive relation for computing the Wiener index of Generalized Bethe trees.