Comparative Study of Valency-Based Topological Descriptor for Hexagon Star Network

A class of graph invariants referred to today as topological indices are inefficient progressively acknowledged by scientific experts and others to be integral assets in the depiction of structural phenomena. The structure of an interconnection network can be represented by a graph. In the network, vertices represent the processor nodes and edges represent the links between the processor nodes. Graph invariants play a vital feature in graph theory and distinguish the structural properties of graphs and networks. A topological descriptor is a numerical total related to a structure that portray the topology of structure and is invariant under structure automorphism. There are various uses of graph theory in the field of basic science. The main notable utilization of a topological descriptor in science was by Wiener in the investigation of paraffin breaking points. In this paper we study the topological descriptor of a newly design hexagon star network. More preciously, we have computed variation of the Randic0 R0, fourth Zagreb M4, fifth Zagreb M5, geometric-arithmetic GA; atom-bond connectivity ABC; harmonic H; symmetric division degree SDD; first redefined Zagreb, second redefined Zagreb, third redefined Zagreb, augmented Zagreb AZI, Albertson A; Irregularity measures, Reformulated Zagreb, and forgotten topological descriptors for hexagon star network. In the analysis of the quantitative structure property relationships (QSPRs) and the quantitative structure activity relationships (QSARs), graph invariants are important tools to approximate and predicate the properties of the biological and chemical compounds. We also gave the numerical and graphical representations comparisons of our different results.


Introduction
Cheminformatics is another field of modern sciences that connects chemistry, math, and other fields of science. Quantitative structure-activity relationship (QSAR) and Quantitative structure-activity relationship (QSPR) are the principle parts of cheminformatics which are useful to contemplate the physico-chemical properties of networks. A topological descriptor (TD) is a numerical total related to a structure that portray the topology of the structure and is invariant under structure automorphism. There are various uses of graph theory in the field of basic science. The main notable utilization of a TD in science was by Wiener in the investigation of paraffin breaking points [1]. From that point forward, to clarify physico-chemical properties, different TDs have been presented.
Topological descriptors (TD) are commonly partitioned into three sorts: degree, distance and spectrum based. The structures of networks can be scientifically demonstrated by a figure. The vertex represents the processor hub and an edge describes the links among processors. The topology of the figure of a network chooses the way by which any two vertices are linked by an edge. The topology of a network system can be used to obtained specific properties without a lot of stretches. The width is resolved as the most extreme separation between any two hubs in the system. The quantity of connections associated with a hub decides the level of that hub. If this number is the equivalent for all hubs in the system, the system is called regular.
TD can be effortlessly processed by utilizing the ideas of atomic topology (AT), an order dependent on the graph theory. Actually, AT has demonstrated to be a fantastic apparatus for quick and exact estimation of numerous physicochemical as well as biological properties [2,3]. So as to compute topological indices, basics of AT are utilized where chemical compound is changed over into a graph, considering the atoms and bonds are represented by vertices and edges of a graph. The basic definitions and notations are taken from the book [4]. The number of vertices adjacent to the vertex e is the degree of e, denoted as d e .

Degree-Based Indices
In this section, we define some degree based topological indices T H ð Þ: represents T H ð Þ as the general, second, and second modified Randic 0 indices if a 6 ¼ 0 2 R; a ¼ 1; and a ¼ À1 respectively.
represents T H ð Þ as the general sum connectivity, sum connectivity, Zagreb and hyper Zagreb indices, if a 6 ¼ 0 2 R; a ¼ À1 2 ; a ¼ 1 and a ¼ 2 respectively.

Hexagon Star Network Sheet
Interconnection systems are significant in PC systems administration and used to change information between the PC and processer. In the most recent couple of years, numerous specialists structured the new interconnection systems. In an equal PC framework, interconnection organize is accustomed to expanding the exhibition. In diagram hypothesis, organize is spoken to as a chart. In this articulation, the processer spoke to by vertex and association between the units spoke to by edges. From the topology of a system, we can decide certain properties. The level of a hub is characterized as the all outnumber of connections associated with that hub. The system is supposed to be regular if each hub in the system has the same degree. In this paper, we define a new interconnection network hexagon star network. This network is a composition of triangles around a hexagon, as shown in Fig. 1.

Main Results
In this section, we give results, which are used to obtained any degree-based topological descriptors. We obtained exact results of degree-based TD for hexagon star network sheet H. Vetrík [22] introduced a new method to calculate the topological indices and also in [23], we follow the same technique in this paper. Now, we presents a formula, which can be used to obtain any degree based TD. Proof. The graph H contains 6pq þ 5p þ q vertices and 12pq þ 6p edges. Each vertex of H has degree 2 or 4, vertices of H can be partitioned according to their degrees. Let This means that the set V i contains the vertices of degree i. The set of vertices with respect to their degrees are as follows: We partite the edges of H into sets based on degrees of its end vertices. Let Figure 1: The hexagon star network sheet for p ¼ 2, q ¼ 2 The number of edges incident to one vertex of degree 2 and other vertex of degree 4 is 8p þ 4q, so Ä 2;4 ¼ 8p þ 4q: Now, the remaining number of edges are those edges which are incident to two vertices of degree 4, i.e., Ä 4; Hence, After simplification, we get Now we obtained the well-known degree based TD of hexagon star network in the following theorem. the Randic 0 index of H is For a ¼ À1 2 , the Randic′ index After simplification, we get For a ¼ À1, the second modified Zagreb index is We gave graphical comparison of Theorem 4.2 in Fig. 2 and numerical values Tab. 1.
In the next theorem, we determined general sum-connectivity index, first Zagreb index and hyper-Zagreb index of the hexagon star network H.
We gave graphical comparison of Theorem 4.5 in Fig. 5 the second redefined Zagreb index of H,   We gave graphical comparison of Theorem 4.6 in Fig. 6 and numerical values Tab. 5.  with several challenging schemes. In the analysis of the quantitative structure property relationships (QSPRs) and the quantitative structureactivity relationships (QSARs), graph invariants are important tools to approximate and predicate the properties of the biological and chemical compounds. In this paper, we study the valency-based topological descriptor for hexagon star network.
Funding Statement: The authors received no specific funding for this study.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.