in this survey article a brief account on the development of padmakar-ivan (pi) index in thatapplications of padmakar-ivan (pi) index in the fascinating field of nano-technology arediscussed.

Given an ordered partition Π = {P1, P2, ..., Pt} of the vertex set V of a connected graph G = (V,E), the partition representation of a vertex v ∈ V with respect to the partition Π is the vector r(v|Π) = (d(v, P1), d(v, P2), ..., d(v, Pt)), where d(v, Pi) represents the distance between the vertex v and the set Pi. A partition Π of V is a resolving partition of G if different vertices of G have ...

A dendrimer is an artificially manufactured or synthesized molecule built up from branched units called monomers. In this paper, we shall give the exact formulas for the first-connectivity, second-connectivity indices and polynomials, PI and vertex-PI indices and polynomials of an infinite class of dendrimer nanostars.

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and let $d_u$ denote the degree of vertex $u$ in $G$. The Randi'c index of $G$ is defined as${R}(G) =sum_{uvin E(G)} 1/sqrt{d_ud_v}.$In this paper, we investigate the relationships between Randi'cindex and several topological indices.

In this paper, the concepts of Wiener index of a vertex weighted and edge weighted graphs are discussed. Vertex weight and edge weight of a clique are introduced. Wiener index of a vertex weighted partial cube is also discussed. Also a new concept known as Connectivity index is introduced. A relation between Connectivity index and Wiener index for different graphs are discussed.

An arborescence graph is a directed in which, for vertex u called the root, and any other v, there exactly one path from to v. The pathos of an Ar defined as collection minimum number arc disjoint open paths whose union Ar. In [6], Ar, total digraph Q = DP T(Ar)has set V (Q) (Ar) ∪ A(Ar) P(Ar), where set, P(Ar) A(Q) consists following arcs: ab such that a, b ∈ head coincides with tail b; uv u, ...