Study of HCP (Hexagonal Close-Packed) Crystal Structure Lattice through Topological Descriptors

School of Software, Pingdingshan University, Pingdingshan, Henan 467000, China Henan International Joint Laboratory for Multidimensional Topology and Carcinogenic Charateristics Analysis of Atmospheric Particulate Matter PM2.5, Pingdingshan, Henan 467000, China Department of Mathematics, Lahore College for Women University, Lahore, Pakistan University of Engineering and Technology, Lahore (RCET) 54000, Lahore, Pakistan Department of Mathematics, Government College University, Faisalabad, Pakistan


Introduction
Chemical graph theory plays a considerate role to investigate a wide range of inorganic and organic chemical structures by studying them through graphical representation. A topological descriptor is a mathematical entity based on certain topological features of chemical structure which correlates with corresponding chemical properties [1][2][3][4][5][6][7]. In this research work, we study the most symmetrical and complex close-packed structure de ned as HCP(n) using two-dimensional topological descriptors. In graph theory [8], degree of a vertex s in a corresponding graph is the number of edges incident with that vertex de ned as d s . Reverse degree of a vertex s is de ned as R s Δ(G) + 1 − d s , where Δ(G) is the maximum degree of a vertex in a graph G. Neighborhood degree of a vertex s in G is de ned as δ s t∈N s d t , where N S represents the neighborhood degree of a vertex s.
Hexagonal Close Packing (HCP) consists of alternating layers of spheres of atoms (vertices) arranged in a hexagon, with one additional atom (vertex) at the center as shown in Figure 1. Another layer of atoms is sandwiched between these two hexagonal layers which are triangular (three atoms form a triangle by three edges), and the atoms (vertices) of this layer ll the tetrahedral holes created by the top and bottom [9]. e edge set E(G) of HCP contains the edges that connect the atoms that are nearest to each other by an edge. In this context, we represent edges by both a lled and a dotted line to clarify the bonding between two atoms in a 3D HCP(n). e middle layer atoms also share a bond with both hexagonal layers by following a symmetrical pattern.
Elements that form an HCP crystal structure are zirconium, ruthenium, hafnium, and many more. It is the most strong and brittle structure that is highly found in metals that do not have a very smooth symmetry as in cubic structures, but it contains a vast and stronger metallic property than a usual hexagonal crystal structure.

Preliminaries
We give a brief view of some well-known TIs for which we compute closed formulas corresponding to each crystal structure.
Every index is defined as In Table 1, we give a brief view for each descriptor in terms of its function defining in above equation as f(H, K).HCP(n) is a very densely symmetric crystal structure with a complex bonding between its atoms. As it is not a tough thing to study any planar chem structure via descriptors but, here our goal is to perform evaluation for not a planar but a 3-dimensional structure with a complex symmetry in comparison to cubic structure. We regard the study for the work done to date for complex and 3D crystals, including diamond crystal structures [23], FCC [24], and BCC [25] crystal unit cells that have been investigated by utilizing the definitions from chemical graph theory [5], and allows to study organic chemistry, cordially here, metals and minerals attaining crystal unit cells.

Formulation
In this section, we consider that a graph G is HCP(n) defined as hexagonal close-packed crystal structure lattice consisting of n unit cells arranged in one dimension. V(HCP(n)) is defined as vertex set of hexagonal close-packed lattice, and edge set is defined as E(HCP(n)) where for n ordered lattice V(HCP(n)) � 10n + 7, E(HCP(n)) � 40n + 12.
3.1. Degree Based. We utilize Table 2 to compute closed forms for the defined descriptors dependent on degree.

Neighborhood Degree Based.
We utilize Table 4 to evaluate closed formulas for the neighborhood degree-based descriptors representing topological properties.
(1) Neighborhood version of first Zagreb index of HCP(n) is defined as follows: (2) e neighborhood second Zagreb index of HCP(n) is defined as follows: Computational Intelligence and Neuroscience (3) e neighborhood forgotten topological index of HCP(n) is defined as follows: (4) e neighborhood second modified Zagreb index of HCP(n) is defined as follows: (5) e third and fifth NDe index of HCP(n) is defined as follows:    Computational Intelligence and Neuroscience