A Study of Hexagon Star Network with Vertex-Edge-Based Topological Descriptors

. There are many network topology designs that have emerged to fulﬁll the growing need for networks to provide a robust platform for a wide range of applications like running businesses and managing emergencies. Amongst the most famous network topology designs are star network, mesh network, hexagonal network, honeycomb network, etc. In a star network, a central computer is linked with various terminals and other computers over point-to-point lines. The other computers and terminals are directly connected to the central computer but not to one another. As such, any failure in the central computer will result in a failure of the entire network and computers in star network will not be able to communicate. The star topology design can be represented by a graph where vertices represent the computer nodes and edges represent the links between the computer nodes. In this paper, we study the vertex-edge-based topological descriptor for a newly designed hexagon star network.


Introduction
e world of information technology has been witnessing the evolvement of computer networks to serve many different real-life applications. As such, the evolvement of robust networks has become vital to provide a platform on which modern communications occur. Direct interconnection networks can be modeled by graphs, with nodes and edges corresponding to computer nodes and communication links between them, respectively. In [1], the authors present a survey of summarizing the pros and cons of these types of networks. A network topology like star network suffers from its vulnerability of having all computers nodes linked to a central computer node which as a result can cause the entire network to fail to communicate once a failure occurs in the central computer node.
In a fixed interconnection parallel architecture, vertices represent computer nodes while edges represent communication links between nodes. Researchers in parallel processing are motivated to utilize a new improved interconnection network wherein performance is evaluated under different variety of circumstances. Networks like hexagonal, honeycomb, and grid networks have interesting topological properties, such as bearing resemblance to atomic or molecular lattice structures [2]. Such a property allows flexibility in using the aforementioned networks in variety of different architecture designs for assessing performance in different contexts.
Cheminformatics is a recently evolving multidisciplinary subject, representing the intersection point of chemistry, mathematics, and information science. Graph theory has been heavily utilized in chemistry. In particular, chemical graph theory is the topology branch of mathematical chemistry that implements graph theory to mathematical modeling of chemical occurrence [3][4][5]. is theory contributes a major role in the domain of chemical sciences.

Preliminaries
Let E and V be the edge set and vertex set of a simple connected graph G � (V, E). e degree of a vertex θ, denoted by Ψ(θ), is the number of edges that are incident to θ. e open neighborhood of a vertex θ, denoted by N(θ), is a number of all vertices adjacent to θ. e closed neighborhood of θ, denoted by N[θ], is the union of θ and N(θ). e ev-degree, denoted by Ψ ev (e), of any edge e � θϑ ∈ E is the total number of vertices of the closed neighborhood union of θ and ϑ. e ve-degree, denoted by Ψ ve (θ), of any vertex θ ∈ V is the number of different edges that are incident to any vertex from the closed neighborhood of θ. For details, see [24,25,27,28]. e ve-degree-and ev-degreebased topological indices are defined in the following equations.

The Hexagon Star Network and Its Applications
Hexagonal networks belong to the family of networks modeled by planar graphs. ere are three regular plane tessellations composed of the same kind of regular (equilateral) polygons: triangular, square, and hexagon. ey formed the basis for designing the direct interconnection networks with highly competitive overall performance. Hexagonal networks are based on triangular plane tessellation or the partition of a plane into equilateral triangles provided that each node has up to six neighbors. e closest networks, in terms of design and structure, are those based on regular hexagons, called honeycomb networks. ose based on a regular square partition are called mesh networks [31,32].
A hexagonal network has a shape that is most suitable for cellular communication design of mobile networks for many reasons. In a hexagon grid, the frequencies overlapping from one cell to the other can be clearly seen. In this context, the hexagonal network is proposed as a viable alternative interconnection network to mesh connected computer, with nodes serving as processors. It is also to model cellular networks where nodes represent base stations. In this paper, we define a new interconnection network hexagon star network. e proposed network is a composition of triangles around a hexagon, as shown in Figure 1

Main Results
In this section, we determine the ev-degree Zagreb index, first ve-degree Zagreb α(β) index, second ve-degree Zagreb β index, ve-degree atom-bond connectivity (ABC ve ) index, ve-degree geometric-arithmetic (GA ve ) index, ve-degree Randic index, ev-degree Randic index, ve-degree sumconnectivity (χ ve ) index, and ve-degree harmonic (H ve ) for hexagon star network G l,w .
4.1. ev-Degree Zagreb Index. Using Table 1, we compute the ev-degree Zagreb index:

First ve-Degree
Zagreb α Index. Using Table 2, we compute the first ve-degree Zagreb α index: Table 3, we compute the first ve-degree Zagreb β index:  Table 3, we compute the second ve-degree Zagreb index:

Second ve-Degree Zagreb Index. Using
4.5. ve-Degree Randic Index. Using Table 3, we compute the ve-degree Randic index:

ve-Degree
Atom-Bond Connectivity Index. Using Table 3, we compute the ve-degree atom-bond connectivity index: 4.8. ve-Degree Geometric-Arithmetic Index. Using Table 3, we compute the ve-degree geometric-arithmetic index: 4.9. ve-Degree Harmonic Index. Using Table 3, we compute the ve-degree harmonic index: , 4.10. ve-Degree Sum-Connectivity Index. Using Table 3, we compute the ve-degree sum-connectivity index:

Numerical Results and Discussion of Hexagon Star Network
In this section, we calculate the numerical values for the hexagon star network (G l,w ) by using the ev-degree and ve-degree topological descriptors. We construct the numerical tables of ev-degree and ve-degree descriptors. For different values of l and w, we determined the numerical tables for the ev-degree and ve-degree indices.
In this section, we present numerical results related to the ev-degree and ve-degree topological descriptors for the hexagon star network (G l,w ). We have used different values of l and w to compute numerical tables for the ev-degree and ve-degree indices such as the ev-degree Zagreb index, first ve-degree Zagreb α index, first ve-degree Zagreb β index, the second ve-degree Zagreb index, ve-degree Randic index, ev-degree Randic index, ve-degree atom-bond connectivity (ABC ve ) index, ve-degree geometric-arithmetic (GA ve ) index, ve-degree harmonic (H ve ) index, and ve-degree sum-connectivity (χ ve ) for the hexagon star network (see Tables 4 and 5). e graphical representation is shown in the Figures 2 and 3. Complexity 5

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.