Research Article Vertex-Edge-Degree-Based Topological Properties for Hex-Derived Networks

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Introduction
e applications of topological descriptors of chemical structure are nowadays a normal process in the education of structure-property relations, specifically in QSPR/QSAR studies. Topological indices play a dynamic part in the QSPR/ QSAR study. ey associate certain physicochemical assets of chemical compounds. Graph theory has provided pharmacists with an assortment of suitable apparatuses, such as topological indices. Chemicals and chemical compounds are frequently displayed by chemical graphs. A chemical graph is an illustration of the structural formula of a chemical compound in terms of graph theory, in which atoms are denoted with vertices and edges show the chemical bonding between them. Lately, a latest topic that has piqued the interest of researchers is cheminformatics, which is a composite of chemistry, information science, and mathematics, in which the QSAR/QSPR relationship, bioactivity, and classification of chemical compounds are investigated. e topological descriptor is a real number associated with chemical compositions that maintains the correlation of chemical structures with a variety of physicochemical properties, chemical reactivity, or biological activity.
Topological indices are classified into three types: distancebased topological indices, degree-based topological indices, and counting-related topological indices. Numerous researchers have recently discovered topological indices for the study of fundamental properties of molecular graphs or networks. ese networks have very appealing topological properties, which have been considered in various characteristics such as [1][2][3][4][5][6][7][8].
Chen et al. [9] explained the construction of hexagonal mesh networks that consist of triangles, as shown in Figure 1. Furthermore, we gather the p th hexagonal mesh by putting p triangles around the boundary of each hexagon. Imran et al. defined the new hex-derived networks, namely, first type HDN 1 (p) (see Figure 2) and second type HDN 2 (p) (see Figure 3); for detailed construction, see [10]. Simonraj and George [11] created the new network which is named as third type of hex-derived networks. Koam et al. [12] computed the vertex-edge-based topological indices of some hex-derived networks. ere are some works related to hexderived networks which can be seen in [13][14][15]. Related research papers that contain the theoretical as well as application aspects for new research directions can be found in [16][17][18][19][20][21][22].

Preliminaries
Let G � (V, E) be a simple connected graph with E being the edge set and V being the vertex set. e symbol Ψ(θ) denoted the concept of degree of a vertex θ, and it is defined by the number of attached edges with θ. e symbol N(θ) denoted the number of all vertices adjacent to θ and is called as the open neighborhood of a vertex θ. On the contrary, the symbol N[θ] is the union of θ and N(θ) and called as the closed neighborhood of θ. e concept of ve-degree denoted by Ψ ve (θ), and can be defined as follows: for any vertex θ from the vertex set of a graph, is the number of different edges that are attached to any vertex from the closed neighborhood of θ. In this research work, we elaborated different ve-degree-associated topological descriptors. In [23], vertex/edge-degree-based topological indices are defined in which they computed the degree of an edge uv as d u + d v − 2. In this article, we consider ve-degree which is the degree of a vertex and is calculated by adding the degrees of its all-neighboring vertices.
is research work contains the computational exact results of given above descriptors.

Hex-Derived Network HDN 1 (p)
Let HDN 1 (p) be the notation for the hex-derived network of the first type, and it is shown in Figure 2. e original hexderived network HDN 1 (p) contains 9p 2 − 15p + 7 total number of vertices in which there are 6p 2 − 12p + 6 vertices of degree 3, 6 vertices of degree 5, 6p − 12 vertices of degree 7, and 3p 2 − 9p + 7 vertices of degree 12. ere are 27p 2 − 51p + 24 count of edges for the graph HDN 1 (p); all these edges are partitioned into eight subsets according to their degrees and corresponding ve-degrees of end vertices elaborated in equations (9)- (16).
Given below are some ve-degree-based indices, for example, M 1 βve index, M 2 ve index, R ve index, ABC ve index, GA ve index, H ve index, and χ ve index for HDN 1 (p). Theorem 1. Let HDN 1 (p) be the first type of hex-derived network; then, Proof. Let HDN 1 (p) be the notation for the hex-derived network of the first type, and it is shown in Figure 2. e original hex-derived network HDN 1 (p) contains 9p 2 − 15p + 7 total number of vertices in which there are 6p 2 − 12p + 6 vertices of degree 3, 6 vertices of degree 5, 6p − 12 vertices of degree 7, and 3p 2 − 9p + 7 vertices of degree 12. ere are 27p 2 − 51p + 24 count of edges for the graph HDN 1 (p); all these edges are partitioned into eight subsets according to their degrees and relative ve-degrees of both end vertices elaborated in equations (9)- (16). Evaluating equation (2), we can determine the first ve-degree Zagreb β index as Evaluating equations (9)-(16) and after simplifications, we will have the required results as Evaluating equation (3), we can compute the second ve-degree Zagreb index as Evaluating equations (9)-(16) and after simplifications, we will have the required result as Evaluating equation (4), we can determine the ve-degree harmonic index as Evaluating equations (9)- (16) and after simplifications, we get Proof. e ve-degree Randic index can be determined by evaluating the edge partitions in equation (5): e methodology of edge partitions can be determined from equations (9)-(16); after mathematical calculations, we get the following result: e ve-degree sum-connectivity index can be determined by using the values from equations (9)-(16) in equation (6); we get the following: Complexity 5 After simplification, we obtain  Proof. e numerical descriptor of the ve-degree atombond connectivity index can be calculated by evaluating the values of edge partitions in equation (7): Evaluating equations (9)

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. (22) e ve-degree geometric-arithmetic index can be calculated by evaluating the methodology of edge partitions in (1): Evaluating equations (9)