# On Analysis and Computation of Degree-Based Topological Invariants for Cyclic Mesh Network

Authors: Nida Zahra; Muhammad Ibrahim; Muhammad Kamran Siddiqui; Hajar Shooshtri
Journal: Journal of Chemistry
 (2021)

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## Abstract

Recently, there has been increasing attention on the system network due to its promising applications in parallel hanging architectures such as distributed computing (Day (2004), Day and Al-Ayyoub (2002)). Related networks differ in the circumstances of topology, and the descriptors were freshly examined by Hayat and Imran (2014) and Hayat et al. (2014). Distance-based descriptors, counting-related descriptors, and degree-based descriptors are all examples of topological descriptors. These topological characteristics are linked to chemical features of a substance, such as stability, strain energy, and boiling point. The specifications for the 1st Zagreb alpha, 1st Zagreb beta, 2nd Zagreb, sum-connectivity, geometric-arithmetic, Randic, harmonic, and atom-bond connectivity indices for mesh networksMNm×n based on VE and EV degree are discussed in this paper.


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## Body

## 1. Introduction

Cheminformatics is a relatively new field that combines chemistry, mathematics, and information science. Cheminformatics is primarily used to store, index, and retrieve information on chemicals. In index factors, graph theory is very essential. Biological activity is used as an introduction to numerous structural properties of molecules in the study of (QSAR) models. Topological indicators are a fascinating subset of these factors. Topological indices can be calculated using simply nodes (atoms) and edges in a graph representation (chemical bonds) [1, 2]. A series of numbers, a polynomial, or a numeric number can all be used to identify a graph. A complete graph is represented by numbers or an array, and for those graphs, these interpretations are supposed to be unique. The topological index is a mathematical term that belongs to a graph and is unaffected by graph automorphism. It identifies the structure of the graph. Degree-based topological indices, counting-related polynomials, and graph indices are some of the most common types of topological indices in [3, 4]. A topological index is a function “Top” from Σ to the set of real numbers. Topological indicators in a network are, of course, the number of nodes and links present in the network. Numerous networks having an atomic or molecular structure, such as honeycomb, grid networks, and hexagonal, are comparable. Topological properties of this network are very interesting, which are studied in various aspects in [5, 6]. Hexagonal and honeycomb structures have also been identified as important in biological evolution, where intersecting triangles are critical for transmission of aid in societal problems, particularly for the development of collaboration [7, 8]. Appropriate research on this topic and further benefit from the new research findings are given in [9–12]. While working on paraffin breaking, Wiener’s approach [13] gave the impression of being a topological description. This identifier is known as the route number. After this, Wiener index is used to remember the route number. In terms of theory and practice, this topological descriptor served as the foundation for the topological index, see [14, 15] for details. Therefore, the topological lesions in the chemical and quantitative literature are Weiner, Zagreb, and Randic [16, 17]. By using the previous degree concepts, all of the above works were completed. In grid history, Chellali et al. [18] developed two additional degree theories, namely, VE degrees and EV degrees, after some time. These concepts are a twist on the previous degree-based concept.


## 2. Preliminaries

In this section, we recall some fundamental definitions concerned to network which is usually represented byN=P,C, where P is the set of points and C is the set of connections of network. The degree of a point is the number of connections joined to that point. The open neighborhood, indicated as Nv, for a point v, is written as Nv=u∈P|u,v∈C in networks. If we add the point v to the set of Nv, we get the same as the closed neighborhood of v, represented by Nv. The number of connections that are connected to any points from the closed neighbourhood of v is equal to the vertex-edge degree, denoted by ϕvev, specified in [19], of the point v∈P. Furthermore, the number of points in the union of the closed neighbourhoods of u and v is equal to the edge-vertex degree of the connection e=uv∈C, indicated by ϕeve specified in [19]. Let N be a simple network and e=uv∈CN. The EV and VE degree topological descriptor-related details can be seen in [18, 19].ConsiderN to be a basic network in all of the illustrations and v∈VN.The Zagreb indices for edge-vertex degree are specified as(1)ℳevN=∑e∈CNϕeve2.The first Zagreb alpha indices for vertex-edge degree are specified as(2)ℳ1αveN=∑u∈PNϕveu2.The first Zagreb beta indices for vertex-edge degree are specified as(3)ℳ1βveN=∑uv∈CNϕveu+ϕvev.The second Zagreb indices for vertex-edge degree are specified as(4)ℳ2veN=∑uv∈CNϕveu×ϕvev.The Randic indices for vertex-edge degree are specified as(5)ℛveN=∑uv∈CNϕveu×ϕvev−1/2.The Randic indices for edge-vertex degree are specified as(6)ℛevN=∑e∈CNϕvee−1/2.The atom-bond connectivity indices for vertex-edge degree are specified as(7)ℬCveN=∑uv∈CNϕveu+ϕvev−2ϕveu×ϕvev.The geometric-arithmeticve−GA indices for vertex-edge degree are specified as(8)GAveN=∑uv∈CN2ϕveu×ϕvevϕveu+ϕvev.The harmonicve−H indices for vertex-edge degree are specified as(9)ℋveN=∑uv∈CN2ϕveu+ϕvev.The sum-connectivity (ve-χ) indices for vertex-edge degree are specified as(10)χveN=∑uv∈CNϕveu+ϕvev−1/2.


## 3. Result for Mesh NetworkMNm×n

In this part, we calculate several mesh networkMNm×n topological indices based on EV and VE degree, seen in Figure 1. Let MNm×n be a mesh network. MNm×n has mn points and 2mn−m+n connections. Table 1 shows how we partition the set of connections of MNm×n into four components on the behalf of degrees of end points.Similarly, partition the collection of points into three components based on the degrees of points, as shown in Table 2.Figure 1
Mesh networkMN5×7.Table 1
Connection division ofMNm×n.
No. of connectionsdegu,degv82,32m+2n−123,32m+2n−83,42mn−5m−5n+124,4Table 2
Points partitioning ofMNm×n.
No. of pointsdegu422m+n−43mn−2m+n+44Depending on the EV degree of the links ofMNm×n for n≥5, we separate them in Table 3 and partition the points and connections in Table 4 and Table 5 depending on the VE degree of MNm×n for n≥5.Table 3
Link partitioning ofMNm×n.
No. of connectionsDegree of end pointsEV degrees82,352m+2n−123,362m+2n−83,472mn−5m+n+124,48Table 4
Points partitioning ofMNm×n.
No. of pointsDegree (u)VE degrees4268392m+n−1631044142m+n−16415mn−4m+n+16416Table 5
The VE degree of the end points of connections ofMNm×n.
No. of connectionsDegree of end pointsVE degree of end points82,36,983,39,102m+n−203,310,1083,49,142m+n−163,410,1584,414,152m+n−204,415,152m+n−164,415,162mn−9m+n+404,416,16


### 3.1. Edge-Vertex Degree-Based Indices

Now, we will calculate Zagreb and Randic indices for the mesh networkMNm×n depending on the EV degree.


#### 3.1.1. Zagreb Index

Employing Table3, the Zagreb index is calculated as follows:(11)ℳevMNm×n=∑e∈EMNm×nϕeve2,ℳevMNm×n=8×52+2m+n−12×62+2m+n−8×72+2mn−5m+n+12×82=200+72m+n−432+98m+n−392+128mn−320m+n+768=128mn−150m+n+144.


#### 3.1.2. The Randic Index

Employing Table3, the Randic index is calculated as follows:(12)ℛevMNm×n=∑e∈EMNm×nϕeve−1/2,ℛevMNm×n=8×5−1/2+2m+n−12×6−1/2+2m+n−8×7−1/2+2mn−5m+n+12×8−1/2=12mn+26+27−522m+n+85−126−87+62=0.707mn−0.195m+n−0.102.


### 3.2. Vertex-Edge Degree-Based Indices

Now, we will calculate the 1st Zagreb alpha, 1st Zagreb beta, 2nd Zagreb, geometric-arithmetic, sum-connectivity, Randic, harmonic, and atom-bond connectivity indices for the mesh networkMNm×n depending on VE.


#### 3.2.1. The 1st Zagreb Alpha Index

Employing Table4, the 1st Zagreb alpha index is calculated as follows:(13)ℳ1αveMNm×n=∑u∈VMNm×nϕveu2,ℳ1αveMNm×n=4×62+8×92+2m+n−16×102+4×142+2m+n−16×152+mn−4m+n+16×162=144+648+200m+n−1600+784+450m+n−3600+256mn−1024m+n+4096=256mn−374m+n+472.


#### 3.2.2. The 1st Zagreb Beta Index

Employing Table5, the 1st Zagreb beta index is calculated as follows:(14)ℳ1βveMNm×n=∑uv∈EMNm×nϕveu+ϕvev,ℳ1βveMNm×n=8×15+8×19+2m+n−20×20+8×23+2m+n−16×25+8×29+2m+n−20×30+2m+n−16×31+2mn−9m+n+40×32=120+152+40m+n−400+184+50m+n−400+232+60m+n−600+62m+n−496+64mn−288m+n+1280=64mn−76m+n+72.


#### 3.2.3. The 2nd Zagreb Index

Employing Table5, the 2nd Zagreb index is calculated as follows:(15)ℳ2veMNm×n=∑uv∈EMNm×nϕveu×ϕvev,ℳ2veMNm×n=8×54+8×90+2m+n−20×100+8×126+2m+n−16×150+8×210+2m+n−20×225+2m+n−16×240+2mn−9m+n+40×256=432+720+200m+n−2000+1008+300m+n−2400+1680+450m+n−4500+480m+n−3840+512mn−2304m+n+10240=512mn−874m+n+1340.


#### 3.2.4. The Randic Index

Employing Table5, the Randic index is calculated as follows:(16)ℛveMNm×n=∑uv∈EMNm×nϕveu×ϕvev−1/2,ℛveMNm×n=8×54−1/2+8×90−1/2+2m+n−20×100−1/2+8×126−1/2+2m+n−16×150−1/2+8×210−1/2+2m+n−20×225−1/2+2m+n−16×240−1/2+2mn−9m+n+40×256−1/2=18mn+15+256+215+1215−916m+n+836+8310−2+8314−1656+8210−43−415+520.125mn+0.063m+n+0.0241.


#### 3.2.5. The Atom-Bond Connectivity Index

Employing Table5, we calculate the above said index as follows:(17)ℬCveMNm×n=∑uv∈EMNm×nϕveu+ϕvev−2ϕveu×ϕvev,ℬCveMNm×n=8×15−254+8×19−290+2m+n−20×20−2100+8×23−2126+2m+n−16×25−2150+8×29−2210+2m+n−20×30−2225+2m+n−16×31−2240+2mn−9m+n+40×32−2256=308mn+325+22356+4715+29215−93016m+n+81336+817310−62+8332−162356+2470−4283−42915+5302=0.68mn−0.048m+n−0.138.


#### 3.2.6. The Geometric-Arithmetic Index

Employing Table5, the above said index is calculated as follows:(18)GAveMNm×n=∑uv∈EMNm×n2ϕveu×ϕvevϕveu+ϕvev,GAveMNm×n=8×25415+8×29019+2m+n−20×210020+8×212623+2m+n−16×215025+8×221029+2m+n−20×222530+2m+n−16×224031+2mn−9m+n+40×225632=2mn+2+465+2+161531−9m+n+1665+481019−20+481423−3265+1621029−20−1281531+40=2mn−1.041m+n−0.037.


#### 3.2.7. The Harmonic Index

Employing Table5, the harmonic index is calculated as follows:(19)ℋveMNm×n=∑uv∈EMNm×n2ϕveu+ϕvev,ℋveMNm×n=8×215+8×219+2m+n−20×220+8×223+2m+n−16×225+8×229+2m+n−20×230+2m+n−16×231+2mn−9m+n+40×232=18mn+15+425+215+431−916m+n+1615+1619−2+1623−3225+1629−43−3231+52=0.125mn+0.063m+n+0.0241.


#### 3.2.8. The Sum-Connectivity Index

Employing Table5, the sum-connectivity index is calculated as follows:(20)χveMNm×n=∑uv∈EMNm×nϕveu+ϕvev−1/2,χveMNm×n=8×15−1/2+8×19−1/2+2m+n−20×20−1/2+8×23−1/2+2m+n−16×25−1/2+8×29−1/2+2m+n−20×30−1/2+2m+n−16×31−1/2+2mn−9m+n+40×32−1/2=122mn+15+25+230+231−942m+n+815+819−105+823−165+829−2030−1631+102=0.3533mn−0.0194m+n−0.071.


## 3.1. Edge-Vertex Degree-Based Indices

Now, we will calculate Zagreb and Randic indices for the mesh networkMNm×n depending on the EV degree.


### 3.1.1. Zagreb Index

Employing Table3, the Zagreb index is calculated as follows:(11)ℳevMNm×n=∑e∈EMNm×nϕeve2,ℳevMNm×n=8×52+2m+n−12×62+2m+n−8×72+2mn−5m+n+12×82=200+72m+n−432+98m+n−392+128mn−320m+n+768=128mn−150m+n+144.


### 3.1.2. The Randic Index

Employing Table3, the Randic index is calculated as follows:(12)ℛevMNm×n=∑e∈EMNm×nϕeve−1/2,ℛevMNm×n=8×5−1/2+2m+n−12×6−1/2+2m+n−8×7−1/2+2mn−5m+n+12×8−1/2=12mn+26+27−522m+n+85−126−87+62=0.707mn−0.195m+n−0.102.


## 3.1.1. Zagreb Index

Employing Table3, the Zagreb index is calculated as follows:(11)ℳevMNm×n=∑e∈EMNm×nϕeve2,ℳevMNm×n=8×52+2m+n−12×62+2m+n−8×72+2mn−5m+n+12×82=200+72m+n−432+98m+n−392+128mn−320m+n+768=128mn−150m+n+144.


## 3.1.2. The Randic Index

Employing Table3, the Randic index is calculated as follows:(12)ℛevMNm×n=∑e∈EMNm×nϕeve−1/2,ℛevMNm×n=8×5−1/2+2m+n−12×6−1/2+2m+n−8×7−1/2+2mn−5m+n+12×8−1/2=12mn+26+27−522m+n+85−126−87+62=0.707mn−0.195m+n−0.102.


## 3.2. Vertex-Edge Degree-Based Indices

Now, we will calculate the 1st Zagreb alpha, 1st Zagreb beta, 2nd Zagreb, geometric-arithmetic, sum-connectivity, Randic, harmonic, and atom-bond connectivity indices for the mesh networkMNm×n depending on VE.


### 3.2.1. The 1st Zagreb Alpha Index

Employing Table4, the 1st Zagreb alpha index is calculated as follows:(13)ℳ1αveMNm×n=∑u∈VMNm×nϕveu2,ℳ1αveMNm×n=4×62+8×92+2m+n−16×102+4×142+2m+n−16×152+mn−4m+n+16×162=144+648+200m+n−1600+784+450m+n−3600+256mn−1024m+n+4096=256mn−374m+n+472.


### 3.2.2. The 1st Zagreb Beta Index

Employing Table5, the 1st Zagreb beta index is calculated as follows:(14)ℳ1βveMNm×n=∑uv∈EMNm×nϕveu+ϕvev,ℳ1βveMNm×n=8×15+8×19+2m+n−20×20+8×23+2m+n−16×25+8×29+2m+n−20×30+2m+n−16×31+2mn−9m+n+40×32=120+152+40m+n−400+184+50m+n−400+232+60m+n−600+62m+n−496+64mn−288m+n+1280=64mn−76m+n+72.


### 3.2.3. The 2nd Zagreb Index

Employing Table5, the 2nd Zagreb index is calculated as follows:(15)ℳ2veMNm×n=∑uv∈EMNm×nϕveu×ϕvev,ℳ2veMNm×n=8×54+8×90+2m+n−20×100+8×126+2m+n−16×150+8×210+2m+n−20×225+2m+n−16×240+2mn−9m+n+40×256=432+720+200m+n−2000+1008+300m+n−2400+1680+450m+n−4500+480m+n−3840+512mn−2304m+n+10240=512mn−874m+n+1340.


### 3.2.4. The Randic Index

Employing Table5, the Randic index is calculated as follows:(16)ℛveMNm×n=∑uv∈EMNm×nϕveu×ϕvev−1/2,ℛveMNm×n=8×54−1/2+8×90−1/2+2m+n−20×100−1/2+8×126−1/2+2m+n−16×150−1/2+8×210−1/2+2m+n−20×225−1/2+2m+n−16×240−1/2+2mn−9m+n+40×256−1/2=18mn+15+256+215+1215−916m+n+836+8310−2+8314−1656+8210−43−415+520.125mn+0.063m+n+0.0241.


### 3.2.5. The Atom-Bond Connectivity Index

Employing Table5, we calculate the above said index as follows:(17)ℬCveMNm×n=∑uv∈EMNm×nϕveu+ϕvev−2ϕveu×ϕvev,ℬCveMNm×n=8×15−254+8×19−290+2m+n−20×20−2100+8×23−2126+2m+n−16×25−2150+8×29−2210+2m+n−20×30−2225+2m+n−16×31−2240+2mn−9m+n+40×32−2256=308mn+325+22356+4715+29215−93016m+n+81336+817310−62+8332−162356+2470−4283−42915+5302=0.68mn−0.048m+n−0.138.


### 3.2.6. The Geometric-Arithmetic Index

Employing Table5, the above said index is calculated as follows:(18)GAveMNm×n=∑uv∈EMNm×n2ϕveu×ϕvevϕveu+ϕvev,GAveMNm×n=8×25415+8×29019+2m+n−20×210020+8×212623+2m+n−16×215025+8×221029+2m+n−20×222530+2m+n−16×224031+2mn−9m+n+40×225632=2mn+2+465+2+161531−9m+n+1665+481019−20+481423−3265+1621029−20−1281531+40=2mn−1.041m+n−0.037.


### 3.2.7. The Harmonic Index

Employing Table5, the harmonic index is calculated as follows:(19)ℋveMNm×n=∑uv∈EMNm×n2ϕveu+ϕvev,ℋveMNm×n=8×215+8×219+2m+n−20×220+8×223+2m+n−16×225+8×229+2m+n−20×230+2m+n−16×231+2mn−9m+n+40×232=18mn+15+425+215+431−916m+n+1615+1619−2+1623−3225+1629−43−3231+52=0.125mn+0.063m+n+0.0241.


### 3.2.8. The Sum-Connectivity Index

Employing Table5, the sum-connectivity index is calculated as follows:(20)χveMNm×n=∑uv∈EMNm×nϕveu+ϕvev−1/2,χveMNm×n=8×15−1/2+8×19−1/2+2m+n−20×20−1/2+8×23−1/2+2m+n−16×25−1/2+8×29−1/2+2m+n−20×30−1/2+2m+n−16×31−1/2+2mn−9m+n+40×32−1/2=122mn+15+25+230+231−942m+n+815+819−105+823−165+829−2030−1631+102=0.3533mn−0.0194m+n−0.071.


## 3.2.1. The 1st Zagreb Alpha Index

Employing Table4, the 1st Zagreb alpha index is calculated as follows:(13)ℳ1αveMNm×n=∑u∈VMNm×nϕveu2,ℳ1αveMNm×n=4×62+8×92+2m+n−16×102+4×142+2m+n−16×152+mn−4m+n+16×162=144+648+200m+n−1600+784+450m+n−3600+256mn−1024m+n+4096=256mn−374m+n+472.


## 3.2.2. The 1st Zagreb Beta Index

Employing Table5, the 1st Zagreb beta index is calculated as follows:(14)ℳ1βveMNm×n=∑uv∈EMNm×nϕveu+ϕvev,ℳ1βveMNm×n=8×15+8×19+2m+n−20×20+8×23+2m+n−16×25+8×29+2m+n−20×30+2m+n−16×31+2mn−9m+n+40×32=120+152+40m+n−400+184+50m+n−400+232+60m+n−600+62m+n−496+64mn−288m+n+1280=64mn−76m+n+72.


## 3.2.3. The 2nd Zagreb Index

Employing Table5, the 2nd Zagreb index is calculated as follows:(15)ℳ2veMNm×n=∑uv∈EMNm×nϕveu×ϕvev,ℳ2veMNm×n=8×54+8×90+2m+n−20×100+8×126+2m+n−16×150+8×210+2m+n−20×225+2m+n−16×240+2mn−9m+n+40×256=432+720+200m+n−2000+1008+300m+n−2400+1680+450m+n−4500+480m+n−3840+512mn−2304m+n+10240=512mn−874m+n+1340.


## 3.2.4. The Randic Index

Employing Table5, the Randic index is calculated as follows:(16)ℛveMNm×n=∑uv∈EMNm×nϕveu×ϕvev−1/2,ℛveMNm×n=8×54−1/2+8×90−1/2+2m+n−20×100−1/2+8×126−1/2+2m+n−16×150−1/2+8×210−1/2+2m+n−20×225−1/2+2m+n−16×240−1/2+2mn−9m+n+40×256−1/2=18mn+15+256+215+1215−916m+n+836+8310−2+8314−1656+8210−43−415+520.125mn+0.063m+n+0.0241.


## 3.2.5. The Atom-Bond Connectivity Index

Employing Table5, we calculate the above said index as follows:(17)ℬCveMNm×n=∑uv∈EMNm×nϕveu+ϕvev−2ϕveu×ϕvev,ℬCveMNm×n=8×15−254+8×19−290+2m+n−20×20−2100+8×23−2126+2m+n−16×25−2150+8×29−2210+2m+n−20×30−2225+2m+n−16×31−2240+2mn−9m+n+40×32−2256=308mn+325+22356+4715+29215−93016m+n+81336+817310−62+8332−162356+2470−4283−42915+5302=0.68mn−0.048m+n−0.138.


## 3.2.6. The Geometric-Arithmetic Index

Employing Table5, the above said index is calculated as follows:(18)GAveMNm×n=∑uv∈EMNm×n2ϕveu×ϕvevϕveu+ϕvev,GAveMNm×n=8×25415+8×29019+2m+n−20×210020+8×212623+2m+n−16×215025+8×221029+2m+n−20×222530+2m+n−16×224031+2mn−9m+n+40×225632=2mn+2+465+2+161531−9m+n+1665+481019−20+481423−3265+1621029−20−1281531+40=2mn−1.041m+n−0.037.


## 3.2.7. The Harmonic Index

Employing Table5, the harmonic index is calculated as follows:(19)ℋveMNm×n=∑uv∈EMNm×n2ϕveu+ϕvev,ℋveMNm×n=8×215+8×219+2m+n−20×220+8×223+2m+n−16×225+8×229+2m+n−20×230+2m+n−16×231+2mn−9m+n+40×232=18mn+15+425+215+431−916m+n+1615+1619−2+1623−3225+1629−43−3231+52=0.125mn+0.063m+n+0.0241.


## 3.2.8. The Sum-Connectivity Index

Employing Table5, the sum-connectivity index is calculated as follows:(20)χveMNm×n=∑uv∈EMNm×nϕveu+ϕvev−1/2,χveMNm×n=8×15−1/2+8×19−1/2+2m+n−20×20−1/2+8×23−1/2+2m+n−16×25−1/2+8×29−1/2+2m+n−20×30−1/2+2m+n−16×31−1/2+2mn−9m+n+40×32−1/2=122mn+15+25+230+231−942m+n+815+819−105+823−165+829−2030−1631+102=0.3533mn−0.0194m+n−0.071.


## 4. Conclusion

It’s crucial to explore the structure using graphs, and topological indicators are crucial for grasping the network’s core topology. This sort of analysis has a wide range of applications in computer science, where different indexes based on graph invariance are used to evaluate multiple stimulation summaries. Invariants stats are essential factors for analyzing and predicating the features of chemical structures in the quantitative structure-property relationship (QSPR) and quantitative tructure-activity relationship (QSAR) explorations. We offer several finished products for VE degree and EV degree-based indices, such as the indices depending on the vertex-edge degree are the 1st Zagreb alpha, 1st Zagreb beta, 2nd Zagreb, geometric-arithmetic, sum-connectivity, Randic, harmonic, and atom-bond connectivity indices, in this article, for the mesh networkMNm×n, and EV degree Randic and Zagreb indices. There will be some who are involved in designing new grids in the future, and we are examining their topological indices in order to grasp their core topology.



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