Computing Degree Based Topological Properties of Third Type of Hex-Derived Networks

In chemical graph theory, a topological index is a numerical representation of a chemical network, while a topological descriptor correlates certain physicochemical characteristics of underlying chemical compounds besides its chemical representation. The graph plays a vital role in modeling and designing any chemical network. Simonraj et al. derived a new type of graphs, which is named a third type of hex-derived networks. In our work, we discuss the third type of hex-derived networks H D N 3 ( r ) , T H D N 3 ( r ) , R H D N 3 ( r ) , C H D N 3 ( r ) , and compute exact results for topological indices which are based on degrees of end vertices.


Introduction and Preliminary Results
Graph theory has provided chemists with a variety of useful tools, such as topological indices.Molecules and molecular compounds are often modeled by molecular graph.A molecular graph is a representation of the structural formula of a chemical compound in terms of graph theory, whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds.Cheminformatics is new subject which is a combination of chemistry, mathematics, and information science.It studies quantitative structure-activity (QSAR) and structure-property (QSPR) relationships that are used to predict the biological activities and properties of chemical compounds.Biological indicators such as the Randi ć Index, Zagreb Index, Wiener Index, and Balaban index are used to predict and study the physical and chemical properties of chemical structures.The topological index is a numeric quantity associated with chemical constitutions purporting the correlation of chemical structures with many physicochemical properties, chemical reactivity or biological activity.Topological indices are made on the grounds of the transformation of a chemical network into a number that characterizes the topology of the chemical network.Some of the major types of topological indices of graphs are distance-based topological indices, degree-based topological indices, and counting-related topological indices.
Recently, many researchers have found topological indices vital for the study of structural properties of molecular graph or network or chemical tree.An acyclic connected graph is called a tree graph.The degree 3 or greater of every vertex of a tree is called the branching point of the tree.A chemical tree is a connected acyclic graph having maximum degree 4. The first and second Zagreb index of star-like trees and sun-like graphs and also caterpillar trees containing the hydrocarbons, especially ethane, propane, and butane, was studied and computed in Reference [1].Imran et al. [2] also computed the topological indices of fractal and cayley tree type dendrimers.
For any graph, G = (V, E) where V is be the vertex set and E to be the edge set of G.The degree κ(x) of vertex x is the amount of edges of G episode with x.A graph can be spoken by a polynomial, a numerical esteem or by network shape.
In the present paper, we consider the topological indices of hex-derived networks which are derived from a hexagonal graph that include molecular graphs of unbranched benzenoid hydrocarbons [3].Graphs of hexagonal systems consist of mutually fused hexagons.Since this class of chemical compounds is attracting the great attention of theoretical chemists, the theory of the topological index of the respective molecular graphs have been intensively developed in the last 4 decades.Benzenoid hydrocarbons are important raw materials of the chemical industry (used, for instance, for the production of dyes and plastics) but are also dangerous pollutants [3][4][5].A hexagonal mesh was derived by Chen et al. [6].A set of triangles made a hexagonal mesh, as shown in Figure 1.No hexagonal mesh with dimension 1 exists.A composition of six triangles made a 2-dimensional hexagonal mesh HX(2) (see Figure 1 (1)).By adding a new layer of triangles around the boundary of HX( 2), we have a 3-dimensional hexagonal mesh HX(3) (see Figure 1 (2)).Similarly, we formed HX(n) by adding n layers around the boundary of each proceeding hexagonal mesh.

Drawing algorithm of HDN3 networks
Step-1: First, we draw a hexagonal network of dimension r.
Step-2: Replace all K 3 subgraphs into a planar octahedron POH once.The resulting graph is called an HDN3 (see Figure 2) network.
Step-3: From the HDN3 network, we can easily form THDN3 (see Figure 3), RHDN3 (see Figure 4), and CHDN3 (see Figure 5).In this article, we consider G as a network, with V(G) as the set of vertices and edge set E(G); the degree of any vertex ṕ ∈ V(G) is denoted by κ( ṕ).
The Estrada index is a graph-spectrum-based topological index, which is defined as [7]: In full analogy with the Estrada index, Fath-Tabar et al. [8] proposed the Laplacian Estrada index, which is defined as: The Randić index [9] was denoted by R − 1

2
(G) and acquainted by Milan Randić and written as: The general Randić index R α (G) is the sum of (κ( ṕ)κ( q)) α over all edges e = ṕ q ∈ E(G), defined as: Gutman and Trinajstić were acquainted with a substantial topological index, which is the Zagreb index denoted by M 1 (G) and formalised as: ( The augmented Zagreb index was presented by Furtula et al. [10], and it is defined as: The harmonic index was presented by Zhong [11], and it is defined as: The Atom-bond connectivity (ABC) index is one of the famous degree-based topological indices denoted by Estrada et al. in Reference [12] and formalised as: The Geometric-arithmetic (GA) index is another famous connectivity topological descriptor, which was introduced by Vukičević et al. in Reference [13] and written as: By taking α = 1, the general Randić index is the second Zagreb index for any graph G.

Results for Third Type of Hex-Derived Network HDN3(r)
In this section, we discuss the newly derived third type of hex-derived network and compute the exact results for Randi ć, Zagreb, Harmonic, augmented Zagreb, atom-bond connectivity and geometric-arithmetic indices for the very first time.
We apply the formula of R α (G 1 ): We apply the formula of R α (G 1 ): In the following theorem, we compute the first Zagreb index of hex-derived network G 1 .
Theorem 2. For hex-derived network G 1 , the first Zagreb index is equal to: Proof.Let G 1 be the hex-derived network HDN3(r).Using the edge partition from Table 1, the result follows.The Zagreb index can be calculated using Equation ( 5) as follows: By doing some calculations, we get: Now, we compute H, AZI, ABC, and GA indices of the third type of hex-derived network G 1 .
Theorem 3. Let G 1 be the third type of hex-derived network, then: • GA(G 1 ) = 30 + 96 Proof.Using the edge partition given in Table 1, The Harmonic index can be calculated using Equation (7) as follows: By doing some calculations, we get: The augmented Zagreb index can be calculated from Equation (6) as follows: By doing some calculations, we get: The atom-bond conectivity index can be calculated from Equation ( 8) as follows: By doing some calculations, we get: The geometric-arithmetic index can be calculated from Equation (9) as follows: 2 κ( ṕ)κ( q) (κ( ṕ) + κ( q)) .
By doing some calculations, we get: + 36
We apply the formula of R α (G 2 ): In the following theorem, we compute the first Zagreb index of the third type of triangular hex-derived network G 2 .
Theorem 5.For the third type of triangular hex-derived network G 2 , the first Zagreb index is equal to: Proof.Let G 2 be the triangular hex-derived network THDN3(r).Using the edge partition from Table 2, the result follows.The first Zagreb index can be calculated using Equation (5) as follows: By doing some calculations, we get: Now, we compute the H, AZI, ABC, and GA indices of the third type of triangular hex-derived network G 2 .Theorem 6.Let G 2 be the third type of a triangular hex-derived network, then: ; √ 10(−5 + 3r)).
The augmented Zagreb index can be calculated from Equation ( 6) as follows: By doing some calculations, we get: The atom-bond connectivity index can be calculated from Equation ( 8) as follows: By doing some calculations, we get: The geometric-arithmetic index can be calculated from Equation (9) as follows: 2 κ( ṕ)κ( q) (κ( ṕ) + κ( q)) .
We apply the formula of R α (G 3 ): We apply the formula of R α (G 3 ): In the following theorem, we compute the first Zagreb index of rectangular hex-derived network G 3 .
Theorem 8.For the third type of rectangular hex-derived network G 3 , the first Zagreb index is equal to: Proof.Let G 3 be the hex-derived network RHDN3(r).Using the edge partition from Table 3, the result follows.The Zagreb index can be calculated using Equation (5) as follows: By doing some calculations, we get: Now, we compute the H, AZI, ABC, and GA indices of the third type of rectangular hex-derived network G 3 .
Proof.Using the edge partition given in Table 3, The Harmonic index can be calculated using Equation (7) as follows: By doing some calculations, we get: The augmented Zagreb index can be calculated from Equation ( 6) as follows: By doing some calculations, we get: The atom-bond connectivity index can be calculated from Equation (8) as follows: By doing some calculations, we get: =⇒ ABC(G 3 ) = The geometric-arithmetic index can be calculated from Equation (9) as follows: 2 κ( ṕ)κ( q) (κ( ṕ) + κ( q)) .
By doing some calculations, we get: In this section, we compute certain degree-based topological indices of the third type of chain hex-derived network, CHDN3(r) of dimension r.We compute general Randić index R α (CHDN3(r)) with the α = {1, −1, 1 2 , − 1 2 }, M 1 , H, AZI ABC, and GA indices in the coming theorems of CHDN3(r).
Theorem 10.Consider the chain hex-derived network of type 3, CHDN3(r), the general Randić index is equal to: Proof.Let G 4 be the chain hex-derived network of type 3, CHDN3(r) shown in Figure 5, where r ≥ 2.
Table 4 shows such an edge partition of G 4 .Thus, from Equation (3), it follows that: For α = 1 The general Randi ć index R α (G 4 ) can be computed as follows: Using the edge partition given in Table 4, we get: We apply the formula of R α (G 4 ): Using the edge partition given in Table 4, we get: We apply the formula of R α (G 4 ): We apply the formula of R α (G 4 ): In the following theorem, we compute the first Zagreb index of chain hex-derived network G 4 .
Theorem 11.For the third type of chain hex-derived network G 4 , the first Zagreb index is equal to: Proof.Let G 4 be the hex-derived network CHDN3(r).Using the edge partition from Table 4, the result follows.The Zagreb index can be calculated using Equation ( 5) as follows: By doing some calculations, we get: Now, we compute the H, AZI, ABC and GA indices of the third type of chain hex-derived network G 4 .
By doing some calculations, we get: For the comparison of topological indices numerically for HDN3, THDN3, RHDN3, and CHDN3, we computed all indices for different values of r.From Tables 5-8, we can easily see that all indices are in increasing order as the values of r increases.
The Zagreb and augmented Zagreb indices were found to occur for the computation of the total π-electron energy of molecules [30].Thus, the total π-electron energy is in increasing order in the case of all networks.

Table 1 .
Edge partition of hex-derived network of type 3 HDN3(r) based on degrees of end vertices of each edge. (κ

Table 2 .
Edge partition of a hex-derived network of type 3 HDN3(r) based on degrees of end vertices of each edge.

Table 3 .
Edge partition of rectangular hex-derived network of type 3, RHDN3(r) based on degrees of end vertices of each edge. (κ

Table 4 .
Edge partition of chain hex-derived network of type 3, CHDN3(r) based on degrees of end vertices of each edge.

Table 6 .
Numerical computation of all indices for THDN3(r).

Table 7 .
Numerical computation of all indices for RHDN3(r).