# Multiplicative Topological Properties on Degree Based for Fourth Type of Hex-Derived Networks

Authors: Haidar Ali; Ghulam Dustigeer; Yong-Min Li; Muhammad Kashif Shafiq; Parvez Ali
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

Chemical graph theory is a subfield of graph theory that uses a molecular graph to describe a chemical compound. When there is at least one connection between the vertices of a graph, it is said to be connected. Topology of graph has been expressed by numerical quantity which is known as topological index. Cheminformatics is a product field that combines chemistry, mathematics, and computer science. The graph plays a key role in modelling and coming up with any chemical arrangement. In this paper, we computed the multiplicative degree-based indices like Randić, Zagreb, Harmonic, augmented Zagreb, atom-bond connectivity, and geometric-arithmetic indices for newly developed fourth type of hex-derived networks and also present the graphical representations of results.


---

## Body

## 1. Introduction

Graph theory has provided chemists with a number of useful methods, such as topological indices. A molecular graph is often used to represent molecules and molecular compounds. A molecular graph is a graph theory definition of a compound’s molecular formula, the vertices that correspond to the compound’s atoms, and hence the edges that correspond to chemical bonds. Cheminformatics may be a trendy subject that would be a mix of chemistry, arithmetic, and data science. Topological indices, given by graph theory, square measure a vital tool. The topology index could be a quantity associated with a graph that unambiguously characterizes that graph. The chemical graph theory could be a combination of chemistry and graph theory. It is the mathematical chemistry branch that applies the graph hypothesis for modelling chemical structure. A graph will acknowledge a network, a meeting of numbers, a numeric variety, and a polynomial that speaks to the structure of that chart. The vertices and the edges of any chart moreover speak to the topological records. Cheminformatics is a modern scholarly field that brings together the fields of chemistry, mathematics, and information science. It examines the relationships between QSAR and QSPR, which are used to estimate biological activities and chemical compound properties. Wiener is the pioneer of TIs; he developed this theory in 1947, when he was working on the boiling points of Paraffins. Wiener called it a path number but afterwards; it is introduced by Wiener [1]. It is the first distance-based topological index. Topological indices are valuable in QSAR and QSPR studies, as they can alter the chemical structure into numerical values. More than 100 topological descriptors evaluated to get the connection between the atoms. There are a few strategies for evaluating atomic structures, the topological index of which is the most common since it can be derived specifically from atomic structures and measured effectively for a huge number of atoms.


### 1.1. Method for Drawing HDN4 Networks

Step 1.
Let take a benzene network with dimensionr.Step 2.
Place benzene graph in eachK3 subgraph of hexagonal network.Step 3.
Connect alternating vertices of each benzene graph to every corner of triangle.Step 4.
At the end, the derived result of the graph is called the fourth type of hex-derived network HDN4 (see Figure1). In this way, we can also construct THDN4 (see Figure 2) and RHDN4 (see Figure 3).Figure 1
Fourth type of the hex-derived network (HDN4 (4)).Figure 2
Fourth type of the triangular hex-derived network (THDN4 (7)).Figure 3
Fourth type of rectangular hex-derived network (RHDN4 (4,4)).On the newly developed graphs, the degree-based TIs have been calculated in this paper. First of all, Randić index computes on the fourth type of hex-derived network.LetY be e simple graph. The general form of the Randić index RγY, where γ∈ℝ is the sum of κL´κM´γ over all edges e=c´d´∈EY, defined as follows:
(1)RγY=∑c´d´∈EYκL´κM´γforγ=1,12,−1,−12.The Zagreb index, denoted byM1Y, introduced by Gutman and Das [2] was familiar, and mathematically, it can be written as follows:
(2)M1Y=∑c´d´∈EYκL´+κM´.Zhong [3] calculated the harmonic index, and it can be written as follows:
(3)HY=∑c´d´∈EY2κL´+κM´.Furtula et al. [4] calculated the augmented Zagreb index, and this index is defined as follows:
(4)AZIY=∑c´d´∈EYκL´κM´κL´+κM´−23.The mathematical form ofABC-index has been computed by Estrada et al. [5] and defined as follows:
(5)ABCY=∑c´d´∈EYκL´+κM´−2κL´κM´.The mathematical form ofGA-index has been developed by Vukičević and Furtula [6] and defined as follows:
(6)GAY=∑c´d´∈EY2κL´κM´κL´+κM´.


## 1.1. Method for Drawing HDN4 Networks

Step 1.
Let take a benzene network with dimensionr.Step 2.
Place benzene graph in eachK3 subgraph of hexagonal network.Step 3.
Connect alternating vertices of each benzene graph to every corner of triangle.Step 4.
At the end, the derived result of the graph is called the fourth type of hex-derived network HDN4 (see Figure1). In this way, we can also construct THDN4 (see Figure 2) and RHDN4 (see Figure 3).Figure 1
Fourth type of the hex-derived network (HDN4 (4)).Figure 2
Fourth type of the triangular hex-derived network (THDN4 (7)).Figure 3
Fourth type of rectangular hex-derived network (RHDN4 (4,4)).On the newly developed graphs, the degree-based TIs have been calculated in this paper. First of all, Randić index computes on the fourth type of hex-derived network.LetY be e simple graph. The general form of the Randić index RγY, where γ∈ℝ is the sum of κL´κM´γ over all edges e=c´d´∈EY, defined as follows:
(1)RγY=∑c´d´∈EYκL´κM´γforγ=1,12,−1,−12.The Zagreb index, denoted byM1Y, introduced by Gutman and Das [2] was familiar, and mathematically, it can be written as follows:
(2)M1Y=∑c´d´∈EYκL´+κM´.Zhong [3] calculated the harmonic index, and it can be written as follows:
(3)HY=∑c´d´∈EY2κL´+κM´.Furtula et al. [4] calculated the augmented Zagreb index, and this index is defined as follows:
(4)AZIY=∑c´d´∈EYκL´κM´κL´+κM´−23.The mathematical form ofABC-index has been computed by Estrada et al. [5] and defined as follows:
(5)ABCY=∑c´d´∈EYκL´+κM´−2κL´κM´.The mathematical form ofGA-index has been developed by Vukičević and Furtula [6] and defined as follows:
(6)GAY=∑c´d´∈EY2κL´κM´κL´+κM´.


## 2. Main Results

In this research article, we introduced the different kinds of fourth type of hex-derived networks. The degree-based topological indices have been calculated in this research on the above networks. Currently, there is exhaustive study on the topological indices and with different kinds, see [7–9].For the basic definitions about graph theory and notations, see [10, 11].


### 2.1. Results for Fourth Type of the Hex-Derived NetworkHDN4r

For the first time, we compute the exact result on the above mentioned indices in Section1 for newly developed graphs in this section.Theorem 1.
Consider the fourth type of hex-derived networkHDN4r, the general form of the Randić index is equal to:
(7)RγHDN4r=11010+r−14610+5184r,γ=1,801+r−1262+535r,γ=12,16936r2−432355292r+81952646,γ=−1,12−31r+17r2,γ=−12.Proof.
TheY1≅HDN4r is shown in Figure 1, where condition r≥4. Thus, by using equation (1), it follows that
(8)RγY1=∑c´d´∈EY1κL´κM´γ.Forγ=1, the general form of the Randić index RγY1 has been computed as follows:
(9)R1Y1=∑k=16∑c´d´∈EkY1κL´·κM´.Using Table1, we get
(10)R1Y1=9E1℧+21E2Y1+54E3Y1+49E4Y1+126E5Y1+324E6Y1,⇒R1Y1=11010+r−14610+5184r.Table 1
Degree-based edge partition ofHDN4r of the end vertices on each edge.
κL´κM´ where c´d´∈EY1Number of edgesκL´κM´ where c´d´∈EY1Number of edges3,336r2−72r+367,1812m−183,736r−4818,189r2−33r+303,1836r2−108r+847,76r−6Forγ=1/2, we apply the formula ofRγY1:
(11)R1/2Y1=∑k=16∑c´d´∈EkY1κL´·κM´‍.Using Table1, we get
(12)R1Y1=9E1℧+21E2Y1+54E3Y1+49E4Y1+126E5Y1+324E6Y1,⇒R1Y1=11010+r−14610+5184r.Forγ=−1, by using formula ofRγY1:
(13)R−1Y1=∑k=16∑c´d´∈EkY11κL´·κM´,R−1Y1=19E1Y1+121E2Y1+154E3Y1+149E4Y1+1126E5Y1+1324E6Y1,⇒R−1Y1=16936r2−432355292r+81952646.Forγ=−1/2, we apply the formula ofRγY1:
(14)R−1/2Y1=∑k=16∑c´d´∈EkY11κL´·κM´,R−1/2Y1=13E1Y1+121E2Y1+136E3Y1+17E4Y1+1314E5Y1+118E6Y1,⇒R−1/2Y1=12−31r+17r2.Theorem 2.
For fourth type of hex-derived networkY1, the first form of the Zagreb index is equal to:
(15)M1Y1=21023−1572r+648r2.Proof.
LetY1≅HDN4r be the hex-derived network. The below is the result of using the Table 1. Equation (2) can be used to calculate the Zagreb index as follows:
(16)M1Y1=∑c´d´∈EY1κL´+κM´=∑k=16∑c´d´∈EkY1κL´+κM´,M1Y1=6E1Y1+10E2Y1+21E3Y1+14E4Y1+25E5Y1+36E6Y1.
By doing some calculations, we get:(17)⇒M1Y1=21023−1572r+648r2.
TheH-index, AZI-index, ABC-index, and GA-index have been computed for the fourth type of hex-derived network Y1.Theorem 3.
LetY1 be the fourth form of the hex-derived network, then:
(i)
HY1=223/14r2−28961/1050r+5507/525(ii)
AZIY1=4865917013307/539172272r2−35453987024262256/1230020443767r+88794905734397152/3690061331301(iii)
ABCY1=8r2−92r+43(iv)
GAY1=70r2−131r+59Proof.
Using Table1 and equation (3) to calculate the Harmonic index.
(18)HY1=∑c´d´∈EY12κL´+κM´=∑k=16∑c´d´∈EkY12κL´+κM´,HY1=13E1Y1+15E2Y1+221E3Y1+17E4Y1+225E5Y1+118E6Y1.
By doing some calculations, we get(19)⇒HY1=22314r2−289611050r+5507525.
By using equation (4) to calculate the augmented Zagreb index is equal to:
(20)AZIY1=∑c´d´∈EY1κL´·κM´κL´+κM´−23=∑k=16∑c´d´∈EkY1κL´·κM´κL´+κM´−23,AZIY1=72964E1Y1+9261512E2Y1+1574646859E3Y1+1176491728E4Y1+200037612167E5Y1+3401222439304E6Y1.
By doing some calculations, we get(21)⇒AZIY1=4865917013307539172272r2−354539870242622561230020443767r+887949057343971523690061331301.
By using equation (5) to calculate the ABC-index:
(22)ABCY1=∑c´d´∈EY1κL´+κM´−2κL´·κM´=∑k=16∑c´d´∈EkY1κL´+κM´−2κL´·κM´,ABCY1=23E1Y1+24221E2Y1+11418E3Y1+237E4Y1+32242E5Y1+3418E6Y1.
By doing some calculations, we get(23)⇒ABCY1=48r2−92r+43.
By using equation (6) to calculate the geometric arithmetic index
(24)GAY1=∑c´d´∈EY12κL´κM´κL´+κM´=∑k=16∑c´d´∈EkY12κL´κM´κL´+κM´.
By doing some calculations, we get(25)GAY1=E1Y1+215E2Y1+267E3Y1+E4Y1+61425E5Y1+E6Y1,⇒GAY1=70r2−131r+59.


### 2.2. Results for Fourth Type of Triangular Hex-Derived NetworkTHDN4r

The degree-based TIs have been computed for the fourth form of the triangular hex-derived network in this portion. We calculate general form of Randić indexRγ with γ=1,−1,1/2,−1/2, M1-index, H-index, AZI-index, ABC-index, and GA-index in the coming theorems.Theorem 4.
Consider theTHDN4r, then general form of the Randić index is equal to:
(26)RγTHDN4r=6144r2−535r+53,γ=1,167−r236−89r,γ=12,169216r2−67935400r+611900,γ=−1,299100r2−4610r+2,γ=−12.Proof.
LetY2≅THDN4r, using Table 2 and equation (1), we have
(27)RγY2=∑c´d´∈EY2κL´κM´γ.Table 2
Edge partition of aTHDN4r based on degrees of end vertices of each edge.
κL´κM´ where c´d´∈EY2Number of edgesκL´κM´ where c´d´∈EY2Number of edges3,36r2−12r+64,1063,4610,103r−23,1092r−410,186m−33,1823r2−15r+1818,183r2−7r+122Forγ=1, the general form of the Randić indexRγY2can be computed as follows:
(28)R1Y2=∑k=18∑c´d´∈EkY2κL´·κL´.Using Table2, we get
(29)R1Y2=9E1Y2+12E2Y2+30E3Y2+54E4Y2+40E5Y2+100E6Y2+180E7Y2+324E8Y2,⇒R1Y2=6144r2−535r+537.Forγ=1/2, we apply the formula ofRγY2:
(30)R1/2Y2=∑k=18∑c´d´∈EkY2κL´·κM´.Using Table2, we get
(31)R1/2Y2=3E1Y2+23E2Y2+30E3Y2+36E4Y2+210E5Y2+10E6Y2+65E7Y2+18E8Y2,⇒R1/2Y2=89r2−236r+167.Forγ=−1, by using the formula ofRγY2:
(32)R−1Y2=∑k=18∑c´d´∈EkY21κL´·κM´,R−1Y2=19E1Y2+112E2Y2+130E3Y2+154E4Y2+140E5Y2+1100E6Y2+1180E7Y2+1324E8Y2,⇒R−1Y2=169216r2−67935400r+611900.Forγ=−1/2, by using the formula ofRγY2:
(33)R−1/2Y2=∑k=18∑c´d´∈EkY21κL´·κM´,R−1/2Y2=13E1Y2Y1+112E2Y2+130E3Y2+154E4Y2+140E5Y2+110E6Y2+1180E7Y2+118E8Y2,⇒R−1/2Y2=299100r2−4610r+2.Theorem 5.
The first form of the Zagreb index for fourth type of triangular hex-derived networkY2 is equivalent to:
(34)M1Y2=636r2−103r+79.Proof.
LetY2≅THDN3r. Using Table 2 and equation (2), the first Zagreb index is equal to
(35)M1Y2=∑c´d´∈EY2κL´+κM´=∑k=18∑c´d´∈EkY2κL´+κM´,M1Y2=6E1Y2+7E2Y2+13E3Y2+21E4Y2+14E5Y2+20E6Y2+28E5Y2+36E8Y2.
By doing some calculations, we get:(36)⇒M1Y2=636r2−103r+79.Theorem 6.
LetY2 be the THDN4, then:
(i)
HY2=223/84r2−21527/5460r+717/455(ii)
AZIY2=1203267/800r2−2816367/400r+325590119/36000(iii)
ABCY2=8r2−15r+7(iv)
GAY2=12r2−19r+7Proof.
Using Table2 to calculate the Harmonic index and using equation (3):
(37)HY2=∑c´d´∈EY22κL´+κM´=∑k=18∑c´d´∈EkY22κL´+κM´,HY2=13E1Y2+27E2Y2+213E3Y2+221E4Y2+17E5Y2+110E6Y2+114E7Y2+118E8Y2.
By doing some calculations, we get(38)⇒HY2=22384r2−215275460r+717455.
Using equation (5) to calculate the augmented Zagreb index.
(39)AZIY2=∑c´d´∈EY2κL´·κM´κL´+κM´−23=∑k=18∑c´d´∈EkY2κL´·κM´κL´+κM´−23,AZIY2=72964E1Y2+1728125E2Y2+2029100E3Y2+2295100E4Y2+100027E5Y2+17147100E6Y2+33182100E7Y2+86536100E8Y2.
By doing some calculations, we get(40)⇒AZIY2=1203267800r2−2816367400r+32559011936000.
By using equation (6) to calculate the atom bond connectivity index
(41)ABCY2=∑c´d´∈EY2κL´+κM´−2κL´·κM´=∑k=18∑c´d´∈EkY2κL´+κM´−2κL´·κM´,ABCY2=23E1Y2+156E2Y2+33030E3Y2+11418E4Y2+3010E5Y2+3210E6Y2+13030E7Y2+3418E8Y2.
By doing some calculations, we get(42)⇒ABCY2=8r2−15r+7.
By using equation (7) to calculate the geometric arithmetic index:
(43)GAY2=∑c´d´∈EY22κL´κM´κL´+κM´=∑k=18∑c´d´∈EkY22κL´κM´κL´+κM´.
By doing some calculations, we get(44)GAY2=E1Y2+437E2Y2+23013E3Y2+267E4Y2+2107E5Y2+E6Y2+357E7Y2+E8Y2,⇒GAY2=12r2−19r+7.


### 2.3. Result for Fourth Type of Rectangular Hex-Derived NetworkRHDN4r,s

Some topological indices which based on the degree of theRHDN4r,s with condition r=s have been computed in this portion. We evaluate the general form Randić index RγRHDN4r for the γ=1,−1,1/2,−1/2, M1-index, H-index, AZI-index, ABC-index, and GA-index in the forward theorems of RHDN4r,s.Theorem 7.
For the fourth type of rectangular hex-derived networkRHDN4r, the general Randić index is equal to
(45)RγRHDN4r=4432r2−1358r+1093,γ=1,178r2−432r+267,γ=12,169108r2−55092025r+51533780,γ=−1,23r−2r−1,γ=−12.Proof.
LetY3≅RHDN4r be seen in Figure 3, with condition r=s≥4. The edge partition as seen in the table is as follows:
(46)RγY3=∑c´d´∈EY3κL´κM´γ.
Forγ=1, the general form of the Randić index RγY3 can be calculated as follows:
(47)R1Y3=∑k=111∑c´d´∈EkY3κL´·κM´.Using Table3, we get
(48)R1Y3=9E1Y3+12E2Y3+21E3Y3+30E4Y3+54E5Y3+40E6Y3+70E7Y3+126E8Y3+100E9Y3+180E10Y3+324E11Y3,⇒R1Y3=4432r2−1358r+1093.Table 3
The edge partition of fourth type of rectangular hex-derived network.
κL´κM´ where c´d´∈EY3Number of edgesκL´κM´ where c´d´∈EY3Number of edges3,312r2−24r+127,1043,447,1823,7810,1022r−53,1064r−810,1842r−53,1862r2−8r+818,183r2−16r+214,104Forγ=1/2, we apply the formula ofRγY3:
(49)R1/2Y3=∑k=111∑c´d´∈EkGκL´·κM´‍‍.Using Table3, we get
(50)R1/2Y3=3E1Y3+23E2Y3+21E3Y3+30E4Y3+36E5Y3+210E6Y3+70E7Y3+314E8Y3+10E9Y3+65E10Y3+18E11Y3,⇒R1/2Y3=178r2−432r+267.Forγ=−1, by using the formula ofRγY3:
(51)R−1Y3=∑k=111∑c´d´∈EkY31κL´·κM´,R−1Y3=19E1Y3+112E2Y3+121E3Y3+130E4Y3+154E5Y3+140E6Y3+170E7Y3+1126E8Y3+1100E9Y3+1180E10Y3+1324E11Y3,⇒R−1Y3=169108r2−55092025r+51533780.Forγ=−1/2, using formula ofRγY3:
(52)R−1/2Y3=∑k=111‍∑c´d´∈EkY3‍1κL´·κM´,R−1/2Y3=13E1Y3+123E2Y3+121E3Y3+130E4Y3+136E5Y3+1210E6Y3+170E7Y3+1314E8Y3+110E9Y3+165E10Y3+118E11Y3,⇒R−1/2Y3=23r−2r−1.Theorem 8.
The first Zagreb index for theRHDN4r is equivalent to
(53)M1Y3=2216r2−556r+367.Proof.
LetY3≅RHDN4r. Using Table 3 and equation (2) to calculate the Zagreb index:
(54)M1Y3=∑c´d´∈EY3κL´+κM´‍=∑k=112∑c´d´∈EkY3κL´+κM´,M1Y3=6E1Y3+7E2Y3+10E3Y3+13E4Y3+21E5Y3+14E6Y3+17E7Y3+25E8Y3+20E9Y3+28E10Y3+25E11Y3.
By doing some calculations, we get(55)⇒M1Y3=2216r2−556r+367.Now, we calculate theH-index, AZI-index, ABC-index, and GA-index for RHDN4r.Theorem 9.
LetY3 be the fourth type of rectangular hex-derived network, then:
(i)
HY3=439/84r2−4886/585r+1525241/464100(ii)
AZIY3=240663/80r2−2278779/200r+268079839/24000(iii)
ABCY3=28r−7r−1(iv)
GAY3=23r2−42r+18Proof.
Using Table3 and equation (3) to calculate the Harmonic index:
(56)HY3=∑c´d´∈EY32κL´+κM´=∑k=111∑c´d´∈EkY32κL´+κM´,HY3=13E1Y3+27E2Y3+15E3Y3+213E4Y3+221E5Y3+17E6Y3+217E7Y3+225E8Y3+110E9Y3+114E10Y3+136E11Y3.
By doing some calculations, we get(57)⇒HY3=43984r2−4886585r+1525241464100.
Using equation (4) to calculate the augmented Zagreb index:
(58)AZIY3=∑c´d´∈EY3κL´·κM´κL´+κM´−23=∑k=111∑c´d´∈EkY3κL´·κM´κL´+κM´−23,AZIY3=72964E1Y3+1728125E2Y3+9261512E3Y3+2029100E4Y3+2296100E5Y3+100027E6Y3+274427E7Y3+16441100E8Y3+17147100E9Y3+33181100E10Y3+86536100E11Y3.
By doing some calculations, we get(59)⇒AZIY3=24066380r2−2278779200r+26807983924000.
Using equation (5) to calculate the atom bond connectivity index:
(60)ABCY3=∑c´d´∈EY3κL´+κM´−2κL´·κM´=∑k=111∑c´d´∈EkY3κL´+κM´−2κL´·κM´,ABCY3=23E1Y3+156E2Y3+24221E3Y3+33030E4Y3+11418E5Y3+3010E6Y3+4214E7Y3+32242E8Y3+3210E9Y3+13030E10Y3+3418E11Y3.
By doing some calculations, we get(61)⇒ABCY3=28r−7r−1.
Using equation (6) to geometric arithmetic index:
(62)GAY3=∑c´d´∈EY32κL´κM´κL´+κM´=∑k=111∑c´d´∈EkY32κL´κM´κL´+κM´.
By doing some calculations, we get(63)GAY3=E1Y3+437E2Y3+215E3Y3+23013E4Y3+267E5Y3+2107E6Y3+27017E7Y3+61425E8Y3+E9Y3+357E10Y3+E11Y3,⇒GAY3=23r2−42r+18.For comparison through graphs, the comparison of the different topological indices for theHDN4, THDN4, and RHDN4, a newly developed fourth type of hex-derived networks has been evaluated for the different values. The graphical representation shows the correctness of the results as shown in Figures 4–9.Figure 4
Comparison of indices forHDN44.Figure 5
Comparison of indices forHDN44.Figure 6
Comparison of indices forTHDN47.Figure 7
Comparison of indices forTHDN47.Figure 8
Comparison of indices forRHDN44,4.Figure 9
Comparison of indices forRHDN44,4.


## 2.1. Results for Fourth Type of the Hex-Derived NetworkHDN4r

For the first time, we compute the exact result on the above mentioned indices in Section1 for newly developed graphs in this section.Theorem 1.
Consider the fourth type of hex-derived networkHDN4r, the general form of the Randić index is equal to:
(7)RγHDN4r=11010+r−14610+5184r,γ=1,801+r−1262+535r,γ=12,16936r2−432355292r+81952646,γ=−1,12−31r+17r2,γ=−12.Proof.
TheY1≅HDN4r is shown in Figure 1, where condition r≥4. Thus, by using equation (1), it follows that
(8)RγY1=∑c´d´∈EY1κL´κM´γ.Forγ=1, the general form of the Randić index RγY1 has been computed as follows:
(9)R1Y1=∑k=16∑c´d´∈EkY1κL´·κM´.Using Table1, we get
(10)R1Y1=9E1℧+21E2Y1+54E3Y1+49E4Y1+126E5Y1+324E6Y1,⇒R1Y1=11010+r−14610+5184r.Table 1
Degree-based edge partition ofHDN4r of the end vertices on each edge.
κL´κM´ where c´d´∈EY1Number of edgesκL´κM´ where c´d´∈EY1Number of edges3,336r2−72r+367,1812m−183,736r−4818,189r2−33r+303,1836r2−108r+847,76r−6Forγ=1/2, we apply the formula ofRγY1:
(11)R1/2Y1=∑k=16∑c´d´∈EkY1κL´·κM´‍.Using Table1, we get
(12)R1Y1=9E1℧+21E2Y1+54E3Y1+49E4Y1+126E5Y1+324E6Y1,⇒R1Y1=11010+r−14610+5184r.Forγ=−1, by using formula ofRγY1:
(13)R−1Y1=∑k=16∑c´d´∈EkY11κL´·κM´,R−1Y1=19E1Y1+121E2Y1+154E3Y1+149E4Y1+1126E5Y1+1324E6Y1,⇒R−1Y1=16936r2−432355292r+81952646.Forγ=−1/2, we apply the formula ofRγY1:
(14)R−1/2Y1=∑k=16∑c´d´∈EkY11κL´·κM´,R−1/2Y1=13E1Y1+121E2Y1+136E3Y1+17E4Y1+1314E5Y1+118E6Y1,⇒R−1/2Y1=12−31r+17r2.Theorem 2.
For fourth type of hex-derived networkY1, the first form of the Zagreb index is equal to:
(15)M1Y1=21023−1572r+648r2.Proof.
LetY1≅HDN4r be the hex-derived network. The below is the result of using the Table 1. Equation (2) can be used to calculate the Zagreb index as follows:
(16)M1Y1=∑c´d´∈EY1κL´+κM´=∑k=16∑c´d´∈EkY1κL´+κM´,M1Y1=6E1Y1+10E2Y1+21E3Y1+14E4Y1+25E5Y1+36E6Y1.
By doing some calculations, we get:(17)⇒M1Y1=21023−1572r+648r2.
TheH-index, AZI-index, ABC-index, and GA-index have been computed for the fourth type of hex-derived network Y1.Theorem 3.
LetY1 be the fourth form of the hex-derived network, then:
(i)
HY1=223/14r2−28961/1050r+5507/525(ii)
AZIY1=4865917013307/539172272r2−35453987024262256/1230020443767r+88794905734397152/3690061331301(iii)
ABCY1=8r2−92r+43(iv)
GAY1=70r2−131r+59Proof.
Using Table1 and equation (3) to calculate the Harmonic index.
(18)HY1=∑c´d´∈EY12κL´+κM´=∑k=16∑c´d´∈EkY12κL´+κM´,HY1=13E1Y1+15E2Y1+221E3Y1+17E4Y1+225E5Y1+118E6Y1.
By doing some calculations, we get(19)⇒HY1=22314r2−289611050r+5507525.
By using equation (4) to calculate the augmented Zagreb index is equal to:
(20)AZIY1=∑c´d´∈EY1κL´·κM´κL´+κM´−23=∑k=16∑c´d´∈EkY1κL´·κM´κL´+κM´−23,AZIY1=72964E1Y1+9261512E2Y1+1574646859E3Y1+1176491728E4Y1+200037612167E5Y1+3401222439304E6Y1.
By doing some calculations, we get(21)⇒AZIY1=4865917013307539172272r2−354539870242622561230020443767r+887949057343971523690061331301.
By using equation (5) to calculate the ABC-index:
(22)ABCY1=∑c´d´∈EY1κL´+κM´−2κL´·κM´=∑k=16∑c´d´∈EkY1κL´+κM´−2κL´·κM´,ABCY1=23E1Y1+24221E2Y1+11418E3Y1+237E4Y1+32242E5Y1+3418E6Y1.
By doing some calculations, we get(23)⇒ABCY1=48r2−92r+43.
By using equation (6) to calculate the geometric arithmetic index
(24)GAY1=∑c´d´∈EY12κL´κM´κL´+κM´=∑k=16∑c´d´∈EkY12κL´κM´κL´+κM´.
By doing some calculations, we get(25)GAY1=E1Y1+215E2Y1+267E3Y1+E4Y1+61425E5Y1+E6Y1,⇒GAY1=70r2−131r+59.


## 2.2. Results for Fourth Type of Triangular Hex-Derived NetworkTHDN4r

The degree-based TIs have been computed for the fourth form of the triangular hex-derived network in this portion. We calculate general form of Randić indexRγ with γ=1,−1,1/2,−1/2, M1-index, H-index, AZI-index, ABC-index, and GA-index in the coming theorems.Theorem 4.
Consider theTHDN4r, then general form of the Randić index is equal to:
(26)RγTHDN4r=6144r2−535r+53,γ=1,167−r236−89r,γ=12,169216r2−67935400r+611900,γ=−1,299100r2−4610r+2,γ=−12.Proof.
LetY2≅THDN4r, using Table 2 and equation (1), we have
(27)RγY2=∑c´d´∈EY2κL´κM´γ.Table 2
Edge partition of aTHDN4r based on degrees of end vertices of each edge.
κL´κM´ where c´d´∈EY2Number of edgesκL´κM´ where c´d´∈EY2Number of edges3,36r2−12r+64,1063,4610,103r−23,1092r−410,186m−33,1823r2−15r+1818,183r2−7r+122Forγ=1, the general form of the Randić indexRγY2can be computed as follows:
(28)R1Y2=∑k=18∑c´d´∈EkY2κL´·κL´.Using Table2, we get
(29)R1Y2=9E1Y2+12E2Y2+30E3Y2+54E4Y2+40E5Y2+100E6Y2+180E7Y2+324E8Y2,⇒R1Y2=6144r2−535r+537.Forγ=1/2, we apply the formula ofRγY2:
(30)R1/2Y2=∑k=18∑c´d´∈EkY2κL´·κM´.Using Table2, we get
(31)R1/2Y2=3E1Y2+23E2Y2+30E3Y2+36E4Y2+210E5Y2+10E6Y2+65E7Y2+18E8Y2,⇒R1/2Y2=89r2−236r+167.Forγ=−1, by using the formula ofRγY2:
(32)R−1Y2=∑k=18∑c´d´∈EkY21κL´·κM´,R−1Y2=19E1Y2+112E2Y2+130E3Y2+154E4Y2+140E5Y2+1100E6Y2+1180E7Y2+1324E8Y2,⇒R−1Y2=169216r2−67935400r+611900.Forγ=−1/2, by using the formula ofRγY2:
(33)R−1/2Y2=∑k=18∑c´d´∈EkY21κL´·κM´,R−1/2Y2=13E1Y2Y1+112E2Y2+130E3Y2+154E4Y2+140E5Y2+110E6Y2+1180E7Y2+118E8Y2,⇒R−1/2Y2=299100r2−4610r+2.Theorem 5.
The first form of the Zagreb index for fourth type of triangular hex-derived networkY2 is equivalent to:
(34)M1Y2=636r2−103r+79.Proof.
LetY2≅THDN3r. Using Table 2 and equation (2), the first Zagreb index is equal to
(35)M1Y2=∑c´d´∈EY2κL´+κM´=∑k=18∑c´d´∈EkY2κL´+κM´,M1Y2=6E1Y2+7E2Y2+13E3Y2+21E4Y2+14E5Y2+20E6Y2+28E5Y2+36E8Y2.
By doing some calculations, we get:(36)⇒M1Y2=636r2−103r+79.Theorem 6.
LetY2 be the THDN4, then:
(i)
HY2=223/84r2−21527/5460r+717/455(ii)
AZIY2=1203267/800r2−2816367/400r+325590119/36000(iii)
ABCY2=8r2−15r+7(iv)
GAY2=12r2−19r+7Proof.
Using Table2 to calculate the Harmonic index and using equation (3):
(37)HY2=∑c´d´∈EY22κL´+κM´=∑k=18∑c´d´∈EkY22κL´+κM´,HY2=13E1Y2+27E2Y2+213E3Y2+221E4Y2+17E5Y2+110E6Y2+114E7Y2+118E8Y2.
By doing some calculations, we get(38)⇒HY2=22384r2−215275460r+717455.
Using equation (5) to calculate the augmented Zagreb index.
(39)AZIY2=∑c´d´∈EY2κL´·κM´κL´+κM´−23=∑k=18∑c´d´∈EkY2κL´·κM´κL´+κM´−23,AZIY2=72964E1Y2+1728125E2Y2+2029100E3Y2+2295100E4Y2+100027E5Y2+17147100E6Y2+33182100E7Y2+86536100E8Y2.
By doing some calculations, we get(40)⇒AZIY2=1203267800r2−2816367400r+32559011936000.
By using equation (6) to calculate the atom bond connectivity index
(41)ABCY2=∑c´d´∈EY2κL´+κM´−2κL´·κM´=∑k=18∑c´d´∈EkY2κL´+κM´−2κL´·κM´,ABCY2=23E1Y2+156E2Y2+33030E3Y2+11418E4Y2+3010E5Y2+3210E6Y2+13030E7Y2+3418E8Y2.
By doing some calculations, we get(42)⇒ABCY2=8r2−15r+7.
By using equation (7) to calculate the geometric arithmetic index:
(43)GAY2=∑c´d´∈EY22κL´κM´κL´+κM´=∑k=18∑c´d´∈EkY22κL´κM´κL´+κM´.
By doing some calculations, we get(44)GAY2=E1Y2+437E2Y2+23013E3Y2+267E4Y2+2107E5Y2+E6Y2+357E7Y2+E8Y2,⇒GAY2=12r2−19r+7.


## 2.3. Result for Fourth Type of Rectangular Hex-Derived NetworkRHDN4r,s

Some topological indices which based on the degree of theRHDN4r,s with condition r=s have been computed in this portion. We evaluate the general form Randić index RγRHDN4r for the γ=1,−1,1/2,−1/2, M1-index, H-index, AZI-index, ABC-index, and GA-index in the forward theorems of RHDN4r,s.Theorem 7.
For the fourth type of rectangular hex-derived networkRHDN4r, the general Randić index is equal to
(45)RγRHDN4r=4432r2−1358r+1093,γ=1,178r2−432r+267,γ=12,169108r2−55092025r+51533780,γ=−1,23r−2r−1,γ=−12.Proof.
LetY3≅RHDN4r be seen in Figure 3, with condition r=s≥4. The edge partition as seen in the table is as follows:
(46)RγY3=∑c´d´∈EY3κL´κM´γ.
Forγ=1, the general form of the Randić index RγY3 can be calculated as follows:
(47)R1Y3=∑k=111∑c´d´∈EkY3κL´·κM´.Using Table3, we get
(48)R1Y3=9E1Y3+12E2Y3+21E3Y3+30E4Y3+54E5Y3+40E6Y3+70E7Y3+126E8Y3+100E9Y3+180E10Y3+324E11Y3,⇒R1Y3=4432r2−1358r+1093.Table 3
The edge partition of fourth type of rectangular hex-derived network.
κL´κM´ where c´d´∈EY3Number of edgesκL´κM´ where c´d´∈EY3Number of edges3,312r2−24r+127,1043,447,1823,7810,1022r−53,1064r−810,1842r−53,1862r2−8r+818,183r2−16r+214,104Forγ=1/2, we apply the formula ofRγY3:
(49)R1/2Y3=∑k=111∑c´d´∈EkGκL´·κM´‍‍.Using Table3, we get
(50)R1/2Y3=3E1Y3+23E2Y3+21E3Y3+30E4Y3+36E5Y3+210E6Y3+70E7Y3+314E8Y3+10E9Y3+65E10Y3+18E11Y3,⇒R1/2Y3=178r2−432r+267.Forγ=−1, by using the formula ofRγY3:
(51)R−1Y3=∑k=111∑c´d´∈EkY31κL´·κM´,R−1Y3=19E1Y3+112E2Y3+121E3Y3+130E4Y3+154E5Y3+140E6Y3+170E7Y3+1126E8Y3+1100E9Y3+1180E10Y3+1324E11Y3,⇒R−1Y3=169108r2−55092025r+51533780.Forγ=−1/2, using formula ofRγY3:
(52)R−1/2Y3=∑k=111‍∑c´d´∈EkY3‍1κL´·κM´,R−1/2Y3=13E1Y3+123E2Y3+121E3Y3+130E4Y3+136E5Y3+1210E6Y3+170E7Y3+1314E8Y3+110E9Y3+165E10Y3+118E11Y3,⇒R−1/2Y3=23r−2r−1.Theorem 8.
The first Zagreb index for theRHDN4r is equivalent to
(53)M1Y3=2216r2−556r+367.Proof.
LetY3≅RHDN4r. Using Table 3 and equation (2) to calculate the Zagreb index:
(54)M1Y3=∑c´d´∈EY3κL´+κM´‍=∑k=112∑c´d´∈EkY3κL´+κM´,M1Y3=6E1Y3+7E2Y3+10E3Y3+13E4Y3+21E5Y3+14E6Y3+17E7Y3+25E8Y3+20E9Y3+28E10Y3+25E11Y3.
By doing some calculations, we get(55)⇒M1Y3=2216r2−556r+367.Now, we calculate theH-index, AZI-index, ABC-index, and GA-index for RHDN4r.Theorem 9.
LetY3 be the fourth type of rectangular hex-derived network, then:
(i)
HY3=439/84r2−4886/585r+1525241/464100(ii)
AZIY3=240663/80r2−2278779/200r+268079839/24000(iii)
ABCY3=28r−7r−1(iv)
GAY3=23r2−42r+18Proof.
Using Table3 and equation (3) to calculate the Harmonic index:
(56)HY3=∑c´d´∈EY32κL´+κM´=∑k=111∑c´d´∈EkY32κL´+κM´,HY3=13E1Y3+27E2Y3+15E3Y3+213E4Y3+221E5Y3+17E6Y3+217E7Y3+225E8Y3+110E9Y3+114E10Y3+136E11Y3.
By doing some calculations, we get(57)⇒HY3=43984r2−4886585r+1525241464100.
Using equation (4) to calculate the augmented Zagreb index:
(58)AZIY3=∑c´d´∈EY3κL´·κM´κL´+κM´−23=∑k=111∑c´d´∈EkY3κL´·κM´κL´+κM´−23,AZIY3=72964E1Y3+1728125E2Y3+9261512E3Y3+2029100E4Y3+2296100E5Y3+100027E6Y3+274427E7Y3+16441100E8Y3+17147100E9Y3+33181100E10Y3+86536100E11Y3.
By doing some calculations, we get(59)⇒AZIY3=24066380r2−2278779200r+26807983924000.
Using equation (5) to calculate the atom bond connectivity index:
(60)ABCY3=∑c´d´∈EY3κL´+κM´−2κL´·κM´=∑k=111∑c´d´∈EkY3κL´+κM´−2κL´·κM´,ABCY3=23E1Y3+156E2Y3+24221E3Y3+33030E4Y3+11418E5Y3+3010E6Y3+4214E7Y3+32242E8Y3+3210E9Y3+13030E10Y3+3418E11Y3.
By doing some calculations, we get(61)⇒ABCY3=28r−7r−1.
Using equation (6) to geometric arithmetic index:
(62)GAY3=∑c´d´∈EY32κL´κM´κL´+κM´=∑k=111∑c´d´∈EkY32κL´κM´κL´+κM´.
By doing some calculations, we get(63)GAY3=E1Y3+437E2Y3+215E3Y3+23013E4Y3+267E5Y3+2107E6Y3+27017E7Y3+61425E8Y3+E9Y3+357E10Y3+E11Y3,⇒GAY3=23r2−42r+18.For comparison through graphs, the comparison of the different topological indices for theHDN4, THDN4, and RHDN4, a newly developed fourth type of hex-derived networks has been evaluated for the different values. The graphical representation shows the correctness of the results as shown in Figures 4–9.Figure 4
Comparison of indices forHDN44.Figure 5
Comparison of indices forHDN44.Figure 6
Comparison of indices forTHDN47.Figure 7
Comparison of indices forTHDN47.Figure 8
Comparison of indices forRHDN44,4.Figure 9
Comparison of indices forRHDN44,4.


## 3. Conclusion

In this paper, certain degree-based topological indices, namely, the Randić, Zagreb, Harmonic, augmented Zagreb, atom-bond connectivity, and geometric-arithmetic indices for theHDN4, THDN4, and RHDN4 networks, were studied for the first time, and analytical closed formulas for these networks were determined that will help the people working in network science to understand the underlying topologies of these networks. In future, we are interested in designing some new architectures/networks and then studying their topological indices, which will be quite helpful in understanding their underlying topologies.



---

*Source: 2376289-2022-05-10.xml*