Multiplicative Topological Properties on Degree Based for Fourth Type of Hex-Derived Networks

Chemical graph theory is a sub ﬁ eld of graph theory that uses a molecular graph to describe a chemical compound. When there is at least one connection between the vertices of a graph, it is said to be connected. Topology of graph has been expressed by numerical quantity which is known as topological index. Cheminformatics is a product ﬁ eld that combines chemistry, mathematics, and computer science. The graph plays a key role in modelling and coming up with any chemical arrangement. In this paper, we computed the multiplicative degree-based indices like Randi ć , Zagreb, Harmonic, augmented Zagreb, atom-bond connectivity, and geometric-arithmetic indices for newly developed fourth type of hex-derived networks and also present the graphical representations of results.


Introduction
Graph theory has provided chemists with a number of useful methods, such as topological indices. A molecular graph is often used to represent molecules and molecular compounds. A molecular graph is a graph theory definition of a compound's molecular formula, the vertices that correspond to the compound's atoms, and hence the edges that correspond to chemical bonds. Cheminformatics may be a trendy subject that would be a mix of chemistry, arithmetic, and data science. Topological indices, given by graph theory, square measure a vital tool. The topology index could be a quantity associated with a graph that unambiguously characterizes that graph. The chemical graph theory could be a combination of chemistry and graph theory. It is the mathematical chemistry branch that applies the graph hypothesis for modelling chemical structure. A graph will acknowledge a network, a meeting of numbers, a numeric variety, and a polynomial that speaks to the structure of that chart. The vertices and the edges of any chart moreover speak to the topological records. Cheminformatics is a modern scholarly field that brings together the fields of chemistry, mathematics, and information science. It examines the relationships between QSAR and QSPR, which are used to estimate biological activities and chemical compound properties. Wiener is the pioneer of TIs; he developed this theory in 1947, when he was working on the boiling points of Paraffins. Wiener called it a path number but afterwards; it is introduced by Wiener [1]. It is the first distance-based topological index. Topological indices are valuable in QSAR and QSPR studies, as they can alter the chemical structure into numerical values. More than 100 topological descriptors evaluated to get the connection between the atoms. There are a few strategies for evaluating atomic structures, the topological index of which is the most common since it can be derived specifically from atomic structures and measured effectively for a huge number of atoms.

Method for Drawing HDN4 Networks
Step 1. Let take a benzene network with dimension r.
Step 2. Place benzene graph in each K 3 subgraph of hexagonal network.
Step 3. Connect alternating vertices of each benzene graph to every corner of triangle.
Step 4. At the end, the derived result of the graph is called the fourth type of hex-derived network HDN4 (see Figure 1). In this way, we can also construct THDN4 (see Figure 2) and RHDN4 (see Figure 3). On the newly developed graphs, the degree-based TIs have been calculated in this paper. First of all, Randić index computes on the fourth type of hex-derived network.
Let Y be e simple graph. The general form of the Randić index R γ ðYÞ, where γ ∈ ℝ is the sum of ðκðĹÞκðḾÞÞ γ over all edges e =ćd ∈ EðYÞ, defined as follows: The Zagreb index, denoted by M 1 ðYÞ, introduced by Gutman and Das [2] was familiar, and mathematically, it can be written as follows: Zhong [3] calculated the harmonic index, and it can be written as follows: Furtula et al. [4] calculated the augmented Zagreb index, and this index is defined as follows: The mathematical form of ABC-index has been computed by Estrada et al. [5] and defined as follows: The mathematical form of GA-index has been developed by Vukičević and Furtula [6] and defined as follows:

Main Results
In this research article, we introduced the different kinds of fourth type of hex-derived networks. The degree-based topological indices have been calculated in this research on the above networks. Currently, there is exhaustive study on the topological indices and with different kinds, see [7][8][9].For the basic definitions about graph theory and notations, see [10,11].

Results for Fourth Type of the Hex-Derived Network
HDN4ðrÞ. For the first time, we compute the exact result on the above mentioned indices in Section 1 for newly developed graphs in this section.
Proof. The Y 1 ≅HDN4ðrÞ is shown in Figure 1, where 2 Journal of Function Spaces condition r ≥ 4. Thus, by using equation (1), it follows that For γ = 1, the general form of the Randić index R γ ðY 1 Þ has been computed as follows: Using Table 1, we get Forγ = 1/2, we apply the formula ofR γ ðY 1 Þ: Using Table 1, we get Forγ = −1, by using formula ofR γ ðY 1 Þ: Forγ = −1/2, we apply the formula ofR γ ðY 1 Þ: Theorem 2. For fourth type of hex-derived network Y 1 , the first form of the Zagreb index is equal to: Proof. Let Y 1 ≅HDN4ðrÞ be the hex-derived network. The below is the result of using the Table 1. Equation (2) can By doing some calculations, we get: The H-index, AZI-index, ABC-index, and GA-index have been computed for the fourth type of hex-derived network Y 1 .

Theorem 3.
Let Y 1 be the fourth form of the hex-derived network, then: Proof. Using Table 1 and equation (3) to calculate the Harmonic index.
By doing some calculations, we get By using equation (4) to calculate the augmented Zagreb index is equal to: By doing some calculations, we get    (5) to calculate the ABC-index: By doing some calculations, we get By using equation (6) to calculate the geometric arith- By doing some calculations, we get Proof. Let Y 2 ≅ ðTHDN4ðrÞÞ, using Table 2 and equation
By doing some calculations, we get By using equation (6) to calculate the atom bond connectivity index By using equation (7) to calculate the geometric arithmetic index: By doing some calculations, we get Proof. Let Y 3 ≅ RHDN4ðrÞ be seen in Figure 3, with condition r = s ≥ 4. The edge partition as seen in the table is as follows: For γ = 1, the general form of the Randić index R γ ðY 3 Þ can be calculated as follows: ð Þ κĹÞ · κḾÞÞ: ð ð ð ð47Þ Using Table 3, we get Forγ = 1/2, we apply the formula ofR γ ðY 3 Þ: Using Table 3, we get Forγ = −1, by using the formula ofR γ ðY 3 Þ: