Iranian Journal of Mathematical Chemistry
Volume:1 Issue: 1, Spring-Summer 2010

In this survey article a brief account on the development of Padmakar-Ivan (PI) index in that applications of Padmakar-Ivan (PI) index in the fascinating field of nano-technology are discussed.

The Padmakar-Ivan (PI) index is a first-generation topological index (TI) based on sums over all edges between numbers of edges closer to one endpoint and numbers of edges closer to the other endpoint. Edges at equal distances from the two endpoints are ignored. An analogous definition is valid for the Wiener index W, with the difference that sums are replaced by products. A few other TIs are discussed, and comparisons are made between them. The best correlation is observed between indices G and PI; satisfactory correlations exist between W/n3 and PI/n2, where n denotes the number of vertices in the hydrogen-depleted graph.

Omega polynomial is defined on opposite edge strips ops in a graph G = G(V,E). Thefirst and second derivatives, in X = 1, of Omega polynomial provide the Cluj-Ilmenau CIindex. Close formulas for calculating these topological descriptors in an infinite latticeconsisting of all R[8] faces, related to the famous Dyck graph, is given.

The Sadhana index (Sd) is a newly introduced cyclic index. Efficient formulae for calculating the Sd (Sadhana) index of linear phenylenes are given and a simple relation is established between the Sd index of phenylenes and of the corresponding hexagonal sequences.

Structural codes vis-a-vis structural counts, like polynomials of a molecular graph, areimportant in computing graph-theoretical descriptors which are commonly known astopological indices. These indices are most important for characterizing carbon nanotubes(CNTs). In this paper we have computed Sadhana index (Sd) for phenylenes and theirhexagonal squeezes using structural codes (counts). Sadhana index is a very simple W-Sz-PItype topological index obtained by summing the number of edges on both sides of theelementary cuts of benzenoid graphs. It has the similar discriminating power as that of theWeiner (W)-, Szeged (Sz)-, and PI-indices.

The Padmakar-Ivan (PI) index is a Wiener-Szeged-like topological index which reflectscertain structural features of organic molecules. The PI index of a graph G is the sum of alledges uv of G of the number of edges which are not equidistant from the vertices u and v. In this paper we obtain the second and third extremals of catacondensed hexagonal systems with respect to the PI index.

The topological index of a graph G is a numeric quantity related to G which is invariant under automorphisms of G. The vertex PI polynomial is defined as PIv (G) = Σe=uv nu (e) + nv (e). Then Omega polynomial Ω(G,x) for counting qoc strips in G is defined as Ω(G,x) = Σcm(G,c)xc with m(G,c) being the number of strips of length c. In this paper, a new infinite class of fullerenes is constructed. The vertex PI, omega and Sadhana polynomials of this class of fullerenes are computed for the first time.

In this paper some upper and lower bounds for the greatest eigenvalues of the PI and vertex PI matrices of a graph G are obtained. Those graphs for which these bounds are best possible are characterized.

In this paper, at first we introduce a new index with the name Co-PI index and obtain some properties related this new index. Then we compute this new index for TUC4C8(R) nanotubes.

General formulas are obtained for the vertex Padmakar-Ivan index (PIv) of tetrathiafulvalene (TTF) dendrimer, whereby TTF units we are employed as branching centers. The PIv index is a Wiener-Szeged-like index developed very recently. This topological index is defined as the summation of all sums of nu(e) and nv(e), over all edges of connected graph G.

Let G and H be two graphs. The corona product G o H is obtained by taking one copy of G and |V(G)| copies of H; and by joining each vertex of the i-th copy of H to the i-th vertex of G, i = 1, 2, …, |V(G)|. In this paper, we compute PI and hyper-Wiener indices of the corona product of graphs.