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

Authors: Eshrag A. Refaee; Ali Ahmad
Journal: Complexity
 (2021)

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## 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.


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## Body

## 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, “ve-degree and ev-degree.” The connection between “traditional degree-based” and “ve-degree and ev-degree” can be seen in [25–30].


## 2. Preliminaries

LetE and V be the edge set and vertex set of a simple connected graph G=V,E. The degree of a vertex θ, denoted by Ψθ, is the number of edges that are incident to θ. The open neighborhood of a vertex θ, denoted by Nθ, is a number of all vertices adjacent to θ. The closed neighborhood of θ, denoted by Nθ, is the union of θ and Nθ. The ev-degree, denoted by Ψeve, of any edge e=θϑ∈E is the total number of vertices of the closed neighborhood union of θ and ϑ. The ve-degree, denoted by Ψveθ, 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]. The ve-degree- and ev-degree-based topological indices are defined in the following equations.(1)MevG=∑ϑ∈EΨevθ2,(2)Mαve1G=∑ϑ∈VΨveϑ2,(3)Mβve1G=∑θϑ∈EΨveθ+Ψveϑ,(4)Mve2G=∑θϑ∈EΨveθ×Ψveϑ,(5)ℝveG=∑θϑ∈EΨveθ×Ψveϑ−1/2,(6)ℝevG=∑ϑEΨveϑ−1/2,(7)ABℂveG=∑θϑ∈EΨveθ+Ψveϑ−2Ψveθ×Ψveϑ1/2,(8)GAveG=∑θϑ∈E2Ψveθ×Ψveϑ1/2Ψveθ+Ψveϑ,(9)ℍveG=∑θϑ∈E2Ψveθ+Ψveϑ,(10)χveG=∑θϑ∈EΨveθ+Ψveϑ−1/2.Equations (1)–(10) are known as ev-degree Zagreb α index, first ve-degree Zagreb α index, first ve-degree Zagreb β index, second ve-degree Zagreb index, ve-degree Randic index, ev-degree Randic index, ve-degree atom-bond connectivity index, ve-degree geometric-arithmetic index, ve-degree harmonic index, and ve-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 Figure1.Figure 1
The hexagon star network sheet forl=4 and w=4.


## 4. Main Results

In this section, we determine theev-degree Zagreb index, first ve-degree Zagreb αβ index, second ve-degree Zagreb β index, ve-degree atom-bond connectivity (ABℂve) index, ve-degree geometric-arithmetic (GAve) index, ve-degree Randic index, ev-degree Randic index, ve-degree sum-connectivity (χve) index, and ve-degree harmonic (ℍve) for hexagon star network Gl,w.


### 4.1.ev-Degree Zagreb Index

Using Table1, we compute the ev-degree Zagreb index:(11)MevGl,w=∑e1∈EΨeve12,MevGl,w=8l+4w62+12lw−2l−4w82=768lw+160l−112w.Table 1
ev-Degree of Gl,w.
Ψθ,Ψϑev-DegreeFrequency2,468l+4w4,4812lw−2l−4w


### 4.2. Firstve-Degree Zagreb α Index

Using Table2, we compute the first ve-degree Zagreb α index:(12)Mαve1Gl,w=∑θ∈VΨveθ2,Mαve1Gl,w=4l+2w82+2l+4122+4l+4w−8142+6lw−5l−5w+4162=1536lw+48l−368w+32.Table 2
ve-Degree of Gl,w.
Ψθve-DegreeFrequency284l+2w4122l+44144l+4w−84166lw−5l−5w+4


### 4.3. Firstve-Degree Zagreb β Index

Using Table3, we compute the first ve-degree Zagreb β index:(13)Mβve1Gl,w=∑θϑ∈EΨveθ+Ψveϑ,Mβve1Gl,w=4l+820+4l+4w−822+424+4l26+4w−628+8l+4w−1230+12lw−14l−12w+1432=384lw+64l−64w.Table 3
End vertices’ve-degrees of each edge of Gl,w.
Ψθ,Ψϑve-DegreeFrequency2,48,124l+82,48,144l+4w−84,412,1244,412,144l4,414,144w−64,414,168l+4w−124,416,1612lw−14l−12w+14


### 4.4. Secondve-Degree Zagreb Index

Using Table3, we compute the second ve-degree Zagreb index:(14)Mve2Gl,w=∑θϑ∈EΨveθ×Ψveϑ,Mve2Gl,w=4l+896+4l+4w−8112+4144+4l168+4w−6196+8l+4w−12224+12lw−14l−12w+14256=3072lw−288l−944w+168.


### 4.5.ve-Degree Randic Index

Using Table3, we compute the ve-degree Randic index:(15)ℝveGl,w=∑θϑ∈EΨveθ×Ψveϑ−1/2,ℝveGl,w=4l+896−1/2+4l+4w−8112−1/2+4144−1/2+4l168−1/2+4w−6196−1/2+8l+4w−12224−1/2+12lw−14l−12w+14256−1/2=34lw+166+12142+177+1714−78l+177+11414−1328w+136−277−31414+131168.


### 4.6.ev-Degree Randic Index

Using Table1, we compute the ev-degree Randic index:(16)ℝevGl,w=∑e1EΨvee1−1/2,RevGl,w=8l+4w6−1/2+12lw−2l−4w8−1/2=32lw+436−122l+236−2w.


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

Using Table3, we compute the ve-degree atom-bond connectivity index:(17)ABℂveGl,w=∑θϑ∈EΨveθ+Ψveϑ−2Ψveθ×Ψveϑ1/2,ABℂveGl,w=4l+81896+4l+4w−820112+422144+4l24168+4w−626196+8l+4w−1228224+12lw−14l−12w+1430256=3430lw+3+2735+22+477−7830l+−3430+2+2735+2726w+23+7830−4735−3726−32+1322.


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

Using Table3, we compute the ve-degree geometric-arithmetic index:(18)GAveGl,w=∑θϑ∈E2Ψveθ×Ψveϑ1/2Ψveθ+Ψveϑ,GAveGl,w=4l+829620+4l+4w−8211222+4214424+4l216826+4w−6219628+8l+4w−12222430+12lw−14l−12w+14225632=12lw+856+81342+16117+321514−14l16117+161514−8w+1656−32117−16514+12.


### 4.9.ve-Degree Harmonic Index

Using Table3, we compute the ve-degree harmonic index:(19)ℍveGl,w=∑θϑ∈E2Ψveθ+Ψveϑ,ℍveGl,w=4l+8220+4l+4w−8222+4224+4l226+4w−6228+8l+4w−12230+12lw−14l−12w+14232=34lw+1252117160l+7674620w+971848.


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

Using Table3, we compute the ve-degree sum-connectivity index:(20)χveGl,w=∑θϑ∈EΨveθ+Ψveϑ−1/2,χveGl,w=4l+8120+4l+4w−8122+4124+4l126+4w−6128+8l+4w−12130+12lw−14l−12w+14132=322lw+255+21122+41530+21326−742l+−322+21530+21122+277w+455+742−41122−377−2530+136.


## 4.1.ev-Degree Zagreb Index

Using Table1, we compute the ev-degree Zagreb index:(11)MevGl,w=∑e1∈EΨeve12,MevGl,w=8l+4w62+12lw−2l−4w82=768lw+160l−112w.Table 1
ev-Degree of Gl,w.
Ψθ,Ψϑev-DegreeFrequency2,468l+4w4,4812lw−2l−4w


## 4.2. Firstve-Degree Zagreb α Index

Using Table2, we compute the first ve-degree Zagreb α index:(12)Mαve1Gl,w=∑θ∈VΨveθ2,Mαve1Gl,w=4l+2w82+2l+4122+4l+4w−8142+6lw−5l−5w+4162=1536lw+48l−368w+32.Table 2
ve-Degree of Gl,w.
Ψθve-DegreeFrequency284l+2w4122l+44144l+4w−84166lw−5l−5w+4


## 4.3. Firstve-Degree Zagreb β Index

Using Table3, we compute the first ve-degree Zagreb β index:(13)Mβve1Gl,w=∑θϑ∈EΨveθ+Ψveϑ,Mβve1Gl,w=4l+820+4l+4w−822+424+4l26+4w−628+8l+4w−1230+12lw−14l−12w+1432=384lw+64l−64w.Table 3
End vertices’ve-degrees of each edge of Gl,w.
Ψθ,Ψϑve-DegreeFrequency2,48,124l+82,48,144l+4w−84,412,1244,412,144l4,414,144w−64,414,168l+4w−124,416,1612lw−14l−12w+14


## 4.4. Secondve-Degree Zagreb Index

Using Table3, we compute the second ve-degree Zagreb index:(14)Mve2Gl,w=∑θϑ∈EΨveθ×Ψveϑ,Mve2Gl,w=4l+896+4l+4w−8112+4144+4l168+4w−6196+8l+4w−12224+12lw−14l−12w+14256=3072lw−288l−944w+168.


## 4.5.ve-Degree Randic Index

Using Table3, we compute the ve-degree Randic index:(15)ℝveGl,w=∑θϑ∈EΨveθ×Ψveϑ−1/2,ℝveGl,w=4l+896−1/2+4l+4w−8112−1/2+4144−1/2+4l168−1/2+4w−6196−1/2+8l+4w−12224−1/2+12lw−14l−12w+14256−1/2=34lw+166+12142+177+1714−78l+177+11414−1328w+136−277−31414+131168.


## 4.6.ev-Degree Randic Index

Using Table1, we compute the ev-degree Randic index:(16)ℝevGl,w=∑e1EΨvee1−1/2,RevGl,w=8l+4w6−1/2+12lw−2l−4w8−1/2=32lw+436−122l+236−2w.


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

Using Table3, we compute the ve-degree atom-bond connectivity index:(17)ABℂveGl,w=∑θϑ∈EΨveθ+Ψveϑ−2Ψveθ×Ψveϑ1/2,ABℂveGl,w=4l+81896+4l+4w−820112+422144+4l24168+4w−626196+8l+4w−1228224+12lw−14l−12w+1430256=3430lw+3+2735+22+477−7830l+−3430+2+2735+2726w+23+7830−4735−3726−32+1322.


## 4.8.ve-Degree Geometric-Arithmetic Index

Using Table3, we compute the ve-degree geometric-arithmetic index:(18)GAveGl,w=∑θϑ∈E2Ψveθ×Ψveϑ1/2Ψveθ+Ψveϑ,GAveGl,w=4l+829620+4l+4w−8211222+4214424+4l216826+4w−6219628+8l+4w−12222430+12lw−14l−12w+14225632=12lw+856+81342+16117+321514−14l16117+161514−8w+1656−32117−16514+12.


## 4.9.ve-Degree Harmonic Index

Using Table3, we compute the ve-degree harmonic index:(19)ℍveGl,w=∑θϑ∈E2Ψveθ+Ψveϑ,ℍveGl,w=4l+8220+4l+4w−8222+4224+4l226+4w−6228+8l+4w−12230+12lw−14l−12w+14232=34lw+1252117160l+7674620w+971848.


## 4.10.ve-Degree Sum-Connectivity Index

Using Table3, we compute the ve-degree sum-connectivity index:(20)χveGl,w=∑θϑ∈EΨveθ+Ψveϑ−1/2,χveGl,w=4l+8120+4l+4w−8122+4124+4l126+4w−6128+8l+4w−12130+12lw−14l−12w+14132=322lw+255+21122+41530+21326−742l+−322+21530+21122+277w+455+742−41122−377−2530+136.


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

In this section, we calculate the numerical values for the hexagon star network (Gl,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 theev-degree and ve-degree topological descriptors for the hexagon star network (Gl,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 (ABℂve) index, ve-degree geometric-arithmetic (GAve) index, ve-degree harmonic (ℍve) index, and ve-degree sum-connectivity (χve) for the hexagon star network (see Tables 4 and 5). The graphical representation is shown in the Figures 2 and 3.Table 4
Numerical representation of results.
l,wMevMαve1Mβve1Mve2ℝve5,5194403683296007080823.4656,627936534081382410336832.6517,737968730561881614207243.3368,849536957762457618692055.5219,9626401215683110423791269.20510,10772801504323840029504884.39111,119345618236846464358328101.0812,1211116821737655296427752119.2713,1313041625545664896503320138.9414,1415120029660875264585032160.13Table 5
Numerical representation of results.
l,wℝevABℂveGAveℍveχve5,5119.95119.83328.0523.28161.7576,6169.40168.44465.6432.42786.8287,7227.33225.27627.2243.072116.148,8293.75290.30812.7955.218149.699,9368.65363.571022.468.864187.4910,10452.04445.041255.9484.009229.5311,11543.91534.731513.6100.65275.8212,12644.27632.631795.2118.80326.3413,13753.11738.752100.7138.45381.1114,14870.44853.062430.2159.59440.11Figure 2
The graphical representation ofMev, Mαve1, Mβve1, Mve2, and ℝve.Figure 3
The graphical representation ofℝev, ABℂve, GAve, ℍve, and χve.



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*Source: 9911308-2021-04-27.xml*