#### Abstract

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.

#### 1. Introduction

The 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 [3–5]. This theory contributes a major role in the domain of chemical sciences.

A topological index is a numeric amount related with a chemical constitution asserting a connection between chemical structures with numerous physicosynthetic properties and chemical reactivity. In this unique circumstance, topological indices are planned with the possibility of change of a chemical structure into a number that portrays the geography of that structure. Topological index has been utilized in science by Wiener in the investigation of paraffin breaking points [6], which has been followed by several attempts to clarify physicochemical properties, resulting in presenting many different topological descriptors. For detailed study of topological descriptors, see [7–23]. Chellali et al. [24] presented new degree ideas, namely, “-degree and -degree.” The connection between “traditional degree-based” and “-degree and -degree” can be seen in [25–30].

#### 2. Preliminaries

Let and be the edge set and vertex set of a simple connected graph . The degree of a vertex , denoted by , is the number of edges that are incident to . The open neighborhood of a vertex , denoted by , is a number of all vertices adjacent to . The closed neighborhood of , denoted by , is the union of and . The -degree, denoted by , of any edge is the total number of vertices of the closed neighborhood union of and . The -degree, denoted by , of any vertex is the number of different edges that are incident to any vertex from the closed neighborhood of . For details, see [24, 25, 27, 28]. The -degree- and -degree-based topological indices are defined in the following equations.

Equations (1)–(10) are known as -degree Zagreb index, first -degree Zagreb index, first -degree Zagreb index, second -degree Zagreb index, -degree Randic index, -degree Randic index, -degree atom-bond connectivity index, -degree geometric-arithmetic index, -degree harmonic index, and -degree sum-connectivity index, respectively.

#### 3. The Hexagon Star Network and Its Applications

Hexagonal networks belong to the family of networks modeled by planar graphs. There are three regular plane tessellations composed of the same kind of regular (equilateral) polygons: triangular, square, and hexagon. They 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. The closest networks, in terms of design and structure, are those based on regular hexagons, called honeycomb networks. Those 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. The proposed network is a composition of triangles around a hexagon, as shown in Figure 1.

#### 4. Main Results

In this section, we determine the -degree Zagreb index, first -degree Zagreb index, second -degree Zagreb index, -degree atom-bond connectivity () index, -degree geometric-arithmetic () index, -degree Randic index, -degree Randic index, -degree sum-connectivity () index, and -degree harmonic () for hexagon star network .

##### 4.1. -Degree Zagreb Index

Using Table 1, we compute the -degree Zagreb index:

##### 4.2. First -Degree Zagreb Index

Using Table 2, we compute the first -degree Zagreb index:

##### 4.3. First -Degree Zagreb Index

Using Table 3, we compute the first -degree Zagreb index:

##### 4.4. Second -Degree Zagreb Index

Using Table 3, we compute the second -degree Zagreb index:

##### 4.5. -Degree Randic Index

Using Table 3, we compute the -degree Randic index:

##### 4.6. -Degree Randic Index

Using Table 1, we compute the -degree Randic index:

##### 4.7. -Degree Atom-Bond Connectivity Index

Using Table 3, we compute the -degree atom-bond connectivity index:

##### 4.8. -Degree Geometric-Arithmetic Index

Using Table 3, we compute the *ve*-degree geometric-arithmetic index:

##### 4.9. -Degree Harmonic Index

Using Table 3, we compute the -degree harmonic index:

##### 4.10. -Degree Sum-Connectivity Index

Using Table 3, we compute the -degree sum-connectivity index:

#### 5. Numerical Results and Discussion of Hexagon Star Network

In this section, we calculate the numerical values for the hexagon star network () by using the -degree and -degree topological descriptors. We construct the numerical tables of -degree and -degree descriptors. For different values of and , we determined the numerical tables for the -degree and -degree indices.

In this section, we present numerical results related to the -degree and -degree topological descriptors for the hexagon star network (). We have used different values of and to compute numerical tables for the -degree and -degree indices such as the -degree Zagreb index, first -degree Zagreb index, first -degree Zagreb index, the second -degree Zagreb index, -degree Randic index, -degree Randic index, -degree atom-bond connectivity () index, -degree geometric-arithmetic () index, -degree harmonic () index, and -degree sum-connectivity () for the hexagon star network (see Tables 4 and 5). The graphical representation is shown in the Figures 2 and 3.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.