Computing Padmakar-Ivan index of four classes of dendrimers

A topological index or graph invariant for a molecular graph G is a number reflecting certain structural features of molecules that are obtained from the molecular graph. Here, a molecular graph is a labeled graph whose vertices correspond to the atoms and the edges correspond to chemical bonds. Suppose x and y are vertices of G. Then d(x,y) denotes the length of the shortest path connecting x and y. A topological index defined by the distance function d(,) is called a distancebased topological index. One of the oldest topological index is the Wiener index, introduced by Harold Wiener [1]. It is defined as the summation of distances between all pairs of vertices in G. Another molecular graph invariant, referred to as the PI index, and denoted by PI is put forward by Padmakar V. Khadikar based on distances on edges of the molecular graph under consideration. The PI index is the first topological index considering the distance between edges as those of vertices. It is defined as PI(G) = Σ[nu(e) + nv(e)], where nu(e) is the number of edges of G lying closer to u than v, nv(e) is the number of edges of G lying closer to v than u and summation goes over all edges of G[2]. Many topological indices have been defined and several of them have found applications as means to model physical, chemical, pharmaceutical and other properties of molecules. Khadikar and his team [39] investigated the behavior of some physicochemical quantities under PI index. Diudea and his coworkers [1017] considered the problem of computing distancebased topological indices of nanostructured materials. After proposing the PI index, one of us (ARA) [1828] computed the PI index of some classes of nanotubes, nanotori, nanocones and also of dendrimers. The motivation of this study is taken from the leading works of Diudea and his team on the problem of computing Wiener index of nanostructured materials. For the mathematical properties of the PI index we encourage the interested readers to consult papers [2939] and references therein for background material as well as basic computational techniques.