Wiener index for Common Neighborhood Graph of Special tress

Abstract

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 $con(X-tree)$.