**Haidar Ali 1, Muhammad Ahsan Binyamin 1, Muhammad Kashif Shafiq <sup>1</sup> and Wei Gao 2,\***


Received: 23 May 2019; Accepted: 22 June 2019; Published: 10 July 2019

**Abstract:** There are numeric numbers that define chemical descriptors that represent the entire structure of a graph, which contain a basic chemical structure. Of these, the main factors of topological indices are such that they are related to different physical chemical properties of primary chemical compounds. The biological activity of chemical compounds can be constructed by the help of topological indices. In theoretical chemistry, numerous chemical indices have been invented, such as the Zagreb index, the Randi´c index, the Wiener index, and many more. Hex-derived networks have an assortment of valuable applications in drug store, hardware, and systems administration. In this analysis, we compute the Forgotten index and Balaban index, and reclassified the Zagreb indices, *ABC*<sup>4</sup> index, and *GA*<sup>5</sup> index for the third type of hex-derived networks theoretically.

**Keywords:** forgotten index; balaban index; reclassified the zagreb indices; *ABC*<sup>4</sup> index; *GA*<sup>5</sup> index; *HDN*3(*m*); *THDN*3(*m*); *RHDN*3(*m*)

### **1. Introduction**

Topological indices are very useful tools for chemists which are provided by Graph Theory. In a molecular graph, vertices denotes the atoms and edges are represented as chemical bonds in the terms of graph theory. To predict bioactivity of the chemical compounds, the topological indices such as ABC index, Wiener index, Randi*c*´ index, Szeged index and Zagreb indices are very useful.

A graph *ξ* is a tuple, which consists of the n-connected vertex set |*V*(*ξ*)| and the edge set |*E*(*ξ*)|. *τ*(*m*) denotes the degree of a vertex '*m*' in a graph *ξ*. A graph can be represented by the polynomials, numeric numbers, a sequence of numbers, or a matrix. Throughout this article, all graphs examined are simple, finite, and connected.

As a chemical descriptor, the topological index has an integer attached to the graph which features the graph, and there is no change under graph automorphism. Previously, interest in the computing chemistry domain has grown in terms of topological descriptors and is mainly associated with the use of unusual quantities, the relationship between the structure property, and the relationship of the structure quantity. The topological indices that are based on distance, degree, and polynomials are some of the main classes of these indices. In a number of these segments, degree-based displayers are widely important and chemical graphs play an integral part in theory and theoretical chemistry.

In this article, we consider some important topological indices and some important derived graphs. We examine their chemical behavior by the help of topological indices. These topological indices are of use to chemists.

Chen et al. [1] gleaned a hexagonal mesh which consists of triangles. Triangle graphs are called oxide graphs in terms of chemistry. We can construct a *hexagonal mesh* by joining these triangles, as shown in Figure 1. There does not exist any hexagonal mesh whose dimension equals 1. By the joining of six triangles, we make a hexagonal mesh of dimension 2, *HX*(2) (see Figure 1 (1)). By putting

the triangles around the all sides of *HX*(2), we obtain hexagonal mesh of dimension 3, *HX*(3) (see Figure 1 (2)). Furthermore, we assemble the nth hexagonal mesh by putting *n* triangles around the boundary of each hexagon.
