SOME NEW/OLD DEGREE-BASED TOPOLOGICAL INDICES OF NANOSTAR DENDRIMERS

Zeinab Foruzanfar1, Muhammad Kamran Jamil2, Mohammad Reza Farahani3 , Muhammad Imran4, Xiujun Zhang5 1Department of Engineering Sciences and Physics Buein Zahra Technical University Buein Zahra, Qazvin, IRAN 2Department of Mathematics Riphah Institute of Computing and Applied Sciences (RICAS) Riphah International University 14 Ali Road, Lahore, PAKISTAN 3Department of Applied Mathematics Iran University of Science and Technology (IUST) Narmak, Tehran 16844, IRAN 4Department of Mathematical Sciences United Arab Emirates University P.O. Box 15551, Al Ain, UNITED ARAB EMIRATES 4Department of Mathematics School of Natural Sciences (SNS) National University of Sciences and Technology (NUST) Sector H-12, Islamabad, PAKISTAN 5School of Information Science and Technology Chengdu University Chengdu, 610106, P.R. CHINA


Introduction
Graph theory applied in the study of molecular structure represents an interdisciplinary science, called chemical graph theory or molecular topology.By using tools taken from graph theory, set theory and statistics it attempt to identify structural features involved in structure-property activity relationship.Topological indices is a subsection of chemical graph theory, which correlates certain physico-chemical properties of the underlying chemical compound.Hundreds of papers have been published on topological indices so far.
Let is the set of all finite simple graphs.A topological index is a function T op : → R with the property that T op(G 1 ) = T op(G 2 ), if G 1 and G 2 are isomorphic.Due to their chemical significance a lot of research has been done on topological indices of different graph families.
Nano-biotechnology is a rapidly advancing area of scientific and technological opportunity that applies the tools and processes of nano-fabrication to build devices for studying bio-systems.One of the main object of this area is Dendrimers.Dendrimers are highly ordered branched macromolecules which have attracted much theoretical and experimental attention.
In this paper, G be a simple connected graph with vertex set V (G) and edge set E(G).The number of elements in V (G) and E(G) is represented as |V (G)| and |E(G)|, respectively.For a vertex u ∈ V (G), the number of vertices adjacent to the vertex u is called the degree of u, denoted as d(u).In 1975 Milan Randic introduced the very first vertex-degree based topological index [35], defined as The first and second Zagreb indices are among the oldest and most famous topological indices, introduced by Gutman and Trinajstic [34] defined as follows Recently, Todeschini et.al. [37,38] proposed the multiplicative variants of ordinary Zagreb indices, defined as Zhou et.al. [40] replaced the term d(u)d(v) by d(u) + d(v) and introduced the sum connectivity index as and the general sum connectivity index is defined as where α is any real number.For recent progress on these vertex-degree based topological indices see [8,9,10,21,23,30,33,36,39] I. Gutman et.al. [22,24] presented the neglected topological indices that earlier have been considered in the chemical and/or mathematical literature, but, that evaded the attention of most mathematical chemists.Recently, they succeed to demonstrate that these indices also have very promising applicative potential.The new/old topological indices studied by I. Gutman et.al. are the following: The reciprocal Randic index is defined as where α is any real number.The reduced reciprocal Randic index is defined as The reduced second Zagreb index is defined as In a study on the structure-dependency of the total π-electron energy, beside the first Zagreb index, it was indicated that another term on which this energy depends is of the form Recently, this sum was named forgotten index, or shortly the F − index.

Nanostar Dendrimers
Dendrimer is a synthetic 3-dimensional macromolecule that is prepared in a stepwise fashion from simple branched monomer units.The nanostar dendrimer is a part of a new group of macromolecules that appear to photon funnels just like artificial antennas.From polymer chemistry point of view, dendrimers are nearly perfect mono-disperse macromolecules with a regular and highly branched three dimensional architecture.Some topological indices of polyomino chains are discussed in [1]- [7], [29,32], [11]- [20].In this section, we will compute the reciprocal Randic index, reduced reciprocal Randiìndex, reduced second Zagreb index and forgotten index of nanostar dendrimers.

Nanostar Dendrimer
which is the required (1) result.
which is the required (2) result.
which is the required (3) result.
which is the required (4) result, and the proof is complete.

Nanostar Dendrimer
which is the required (5) result.
which is the required (6) result.

Concluding Remarks and Open Problems
Recently, Gutman, Furtula and Elphick introduced reciprocal Rindic index, reduced reciprocal Randic index, reduced second Zagreb index and forgotten index with great significant applications in chemical graph theory.In this article, we computed these topological indices for Nanostar Dendrimers.Some future work that can be done is to find these topological indices for Polyomino chains, benzenoid systems or some other chemical structures.

1 d 1 2
(u)d(v) In 1998, Bollobàs et.al. introduced the general Randic index by replacing − by any real number α as follows,

Theorem 1 .
the first class of these macromolecules, where n > 1 is the defining parameter, Fig 1.There are 140 • 2 n − 127 edges.The technique to find the certain topological indices we partitions the edge set of graph N S 1 [n] based on degrees of end vertices of each edge.Consider G be the graph of N S 1 [n], then