Wiener index for Common Neighborhood Graph of Special tress

Molecular graphs are models of molecules in which atoms are represented by vertices and chemical bonds by edges of a graph. Graph invariant numbers reflecting certain structural features of a molecule that are derived from its molecular graph are known as topological indices. A topological index is a numerical descriptor of a molecule, based on a certain topological feature of the corresponding molecular graph. One of the most widely known topological descriptor is the Wiener index. The Weiner index \(w(G)\) of a graph G is defined as the half of the sum of the distances between every pair of vertices of \(G\). The construction and investigation of topological is one of the important directions in mathematical chemistry. The common neighborhood graph of G is denoted by con(\(G\)) has the same vertex set as G, and two vertices of con(\(G\)) are adjacent if they have a common neighbor in \(G\). In this paper we investigate the Wiener index of \(Y-tree,\, X-tree,\, con(Y-tree)\) and \(con(X-tree)\).