**Journal of Prime Research in Mathematics**

Vol. 17 (2021), Issue 2, pp. 138 – 148

ISSN: 1817-3462E (Online) 1818-5495 (Print)

ISSN: 1817-3462E (Online) 1818-5495 (Print)

# Computation of Topological Indices for Inner Dual

Graph of Honeycomb and Graphene Network.

**FM Bhatti \(^{a,*}\), Iqra Zaman \(^{a}\), Sawaira Sikander \(^{b,}\)**

\(^a\) Department of Mathematics, Riphah Institute of Computing and Applied Sciences(RICAS), Riphah International University, Raiwind Campus, Pakistan.

Correspondence should be addressed to: fmbhatti@riphah.edu.pk

Copyright © 2022 FM Bhatti, Iqra Zaman, Sawaira Sikander. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Published:**Received: 25 October 2021; Accepted:3 December 2021; Published Online: June 2022.

### Abstract

In QSPR/QSAR study, the molecular structure indices are now standard methods for studying structureproperty relations. Due to the chemical significance of these indices, the number of proposed molecular descriptors is quickly rising in the last few years. A topological index is a transformation of a chemical structure into a real number. In mathematics, honeycomb networks are widely used because of their extreme importance in computer graphics, image processing, cellular phone base stations, and in chemistry to represent benzenoid hydrocarbons. They are formed by recursively using hexagonal tiling in a particular pattern. HC(n) represents the honeycomb network of dimension n, where n is the number of hexagons between boundary and central hexagon. An atomic-scale honeycomb structure composed of carbon atoms is known as graphene. Professor Andre Geim and Professor Kostya Novoselov separated it from graphite in 2004. It is the first 2D material that is one million times thinner than human hair, two hundred times stronger than steel, and the world’s most conductive material. The graph 2D graphene is expressed as G(r, s) where “r” means the number of rows, and “s” is the number of hexagons in a row. This paper uses the inner dual graph of honeycomb networks and 2D graphene network, which are named as HcID(n) and GID(r, s) respectively. We derive some results related to topological indices for these graphs. We compute degree-based indices, first general Zagreb index, general Randi´c connectivity index, general sum-connectivity index, first Zagreb index, Second Zagreb index, Randi´c index, Atom-bond Connectivity (ABC) index, and Geometric-Arithmetic (GA) index of inner dual graphs of honeycomb networks and graphene network.

#### Keywords:

Honeycomb structure, Inner dual graph, Graphene network, Topological indices.