Wiener index of sum of shadowgraphs

Abstract

The Wiener index of a graph [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text] where the sum is taken through all unordered pairs of vertices of [Formula: see text] and [Formula: see text] is distance between two vertices [Formula: see text] and [Formula: see text] of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be two graphs. For a graph [Formula: see text], let [Formula: see text] be a copy of [Formula: see text] and [Formula: see text]. The [Formula: see text]-sum [Formula: see text] is a graph with the set of vertices [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] of [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text], where [Formula: see text] be one of the shadowgraph [Formula: see text] or closed shadowgraph [Formula: see text]. In this paper, we reported the Wiener index of these graphs.