Wiener index of sum of shadowgraphs.

The Wiener index of a graph G , denoted by W (G) , is defined as W (G) = ∑ u ≠ v d (u , v) , where the sum is taken through all unordered pairs of vertices of G and d (u , v) is distance between two vertices u and v of G. Let G = (V 1 , E 1) and H = (V 2 , E 2) be two graphs. For a graph G , let v ′ be a copy of v and V ′ (G) = { v ′ : v ∈ V (G) }. The F -sum G + F H is a graph with the set of vertices V (G + F H) = (V (G) ∪ V ′ (G)) × V (H) and two vertices u = (u 1 , u 2) and v = (v 1 , v 2) of G + F H are adjacent if and only if [ u 1 = v 1 ∈ V 1  and  (u 2 , v 2) ∈ E (H) ] or [ u 2 = v 2  and  (u 1 , v 1) ∈ E (F (G)) ] , where F (G) be one of the shadowgraph D 2 (G) or closed shadowgraph D 2 [ G ]. In this paper, we reported the Wiener index of these graphs. [ABSTRACT FROM AUTHOR]