# Exact Values of Zagreb Indices for Generalized T-Sum Networks with Lexicographic Product

Authors: Jia-Bao Liu; Sana Akram; Muhammad Javaid; Zhi-Ba Peng
Journal: Journal of Mathematics
 (2021)

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## Abstract

The use of numerical numbers to represent molecular networks plays a crucial role in the study of physicochemical and structural properties of the chemical compounds. For some integerk and a network G, the networks SkG and RkG are its derived networks called as generalized subdivided and generalized semitotal point networks, where Sk and Rk are generalized subdivision and generalized semitotal point operations, respectively. Moreover, for two connected networks, G1 and G2, G1G2Sk and G1G2Rk are T-sum networks which are obtained by the lexicographic product of TG1 and G2, respectively, where TεSk,Rk. In this paper, for the integral value k≥1, we find exact values of the first and second Zagreb indices for generalized T-sum networks. Furthermore, the obtained findings are general extensions of some known results for only k=1. At the end, a comparison among the different generalized T-sum networks with respect to first and second Zagreb indices is also included.


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## Body

## 1. Introduction

Topological index (TI) being a molecular descriptor is a mathematical measure that associates a molecular network with a real number and predicts the underlying molecular network’s biological, chemical, and structural properties. Molecular descriptors were used by Wiener [1] and Trinjastic and Gutman and Trinajstić [2] to determine the boiling point of paraffin and the total π-electron energy of the molecules, respectively. TIs are also used in the study of cheminformatics to classify molecules in terms of quantitative structure behavior and property relationships. Most notably, all TIs are invariants under the networks’ isomorphism parameter [3, 4]. Many TIs exist in the literature for networks. Degree-based TIs, distance-based TIs, and polynomial-based TIs are the three major types of these. The degrees-based TIs are more familiar than the others [5].In the field of chemical network theory, networks operations are frequently used to discover the new families of networks. Yan et al. [6] explained the subdivision S1 and semitotal point R1 operations on a molecular network G and attain the Wiener indices of the consequent networks S1G and R1G. After that, Eliasi and Taeri [7] explained the T-sum network G1+GT2 by using Cartesian product of G1 and SG2, where TεS1,R1. Deng et al. [8] defined the T-sum network G1G2 with the help of lexicographic product and also calculated the first and second Zagreb indices, and Akhter and Imran [9] also calculated the forgotten index of the T-sum networks. For more studies of the TIs under the operations of networks, see [10, 11].Recently, for some integerk≥1, Liu et al. [12] defined the generalized subdivision Sk and generalized semitotal point Rk operations. They also computed the 1st and 2nd Zagreb indices for T-sum networks with the help of Cartesian product, i.e., G1+TkG2 for TkεSk,Rk [13, 14]. In this study, we find the exact values of the Zagreb indices for the generalized T-sum networks which are constructed with the help of generalized subdivision, generalized semitotal point operations, and lexicographic product. In the remaining paper, Section 1 has some previous knowledge related to our work and Section 2 has some basic definition. The key findings are presented in Section 3, and the conclusion and applications of these indices are presented in Section 4.


## 2. Preliminaries

We consider undirected, connected, and simple networks, whereVG is a vertex set and EG⊆VG×VG is an edge set. In addition, VG=n is order and EG=m is size of G. Each vertices of a molecular network is referred to as an atom, and edges reflect the bonding between atoms.The number of incident edges is called its degree. The networksVPn=bj:1≤j≤n and EPn=bjbj+1:1≤j≤n,VCn=bj:1≤j≤n&ECn=bibi+1:1≤j≤n−1∪bnb1 and VKn=bi:1≤j≤n, and EKn=bibj:1≤i,j≤n are called path Pn, cycle Cn, and complete Kn, respectively [3].Definition 1.
LetG1 and G2 be two networks with their vertex sets as VG1=y1,y2,…,yn and VG2=z1,z2,…,zn, respectively. The Cartesian product G1×G2 of these networks is defined as follows:(1)VG1×G2=VG1×VG2,EG1×G2=y1,z1y2,z2:y1,z1,y2,z2∈VG1×G2,with conditions either y1=y2 in VG1 and z1z2∈EG2 or y1y2∈EG1 and z1=z2 in VG2.Definition 2.
LetG1 and G2 be two networks with their vertex sets as VG1=y1,y2,…,yn and VG2=z1,z2,…,zn, respectively. Then, the lexicographic product G1G2 of these networks is defined as(2)VG1G2=VG1×VG2,EG1G2=y1,z1y2,z2:y1,z1,y2,z2∈VG1G2,with conditions either. y1=y2 in VG1 and z1z2∈EG2 or y1y2∈EG1 and z1,z2 in VG2.
For integerk≥1, Liu et al. [12] defined the following graphs using the generalized subdivision and semitotal point operations:(i)
SkG network is found out by inserting k-vertices in each edge of G(ii)
RkG is found out from SkG by connecting the old vertices which are adjacent in G
Eliasi and Taeri [7] proposed the generalized T-sum networks under the operation of two connected networks G1 and G2 based on the Cartesian product. Deng et al. [8] defined the generalized T-sum networks under the operation of two connected networks G1 and G2 based on the Lexicographic product as follows.Definition 3.
LetG1 and G2 be two networks and TkεSk,Rk and TkG1 be a network with vertex set VTkG1 and edge set ETkG1. Then, the generalized Tk-sum network G1G2Tk is a graph with the vertex set:(3)VG1G2Tk=VTkG1×VG2,EG1G2Tk=y1,z1y2,z2:y1,z1,y2,z2∈VG1G2Tk,with conditions either y1=y2 in VTkG1 and z1z2∈EG2 or y1y2∈ETkG1 and z1,z2 in VG2.
For more explanation, see Figures1–4 .Figure 1
(a)G≅P4, (b)S1G≅S1P4, and (c)R1G≅R1P4.
(a)(b)(c)Figure 2
(a)SG1≅SP3 and (b)G2≅P4.Figure 3
P3P4S4.Figure 4
P3P4R4.


## 3. Main Results

In this part, we will learn about the Zagreb indices of T-sum networks obtained by the operations of generalized subdivision and generalized semitotal point and lexicographic product.Theorem 1.
LetG1 and G2 be two networks; then, for some integer k, the first Zagreb index of Sk-sum network is(4)M1G1G2Sk=8n2e1e2+n1M1G2+n23M1G1+4e1n23+4e1n22k−1++8n2e1k−1∑i=1n2−1n2−i.Proof.
Letdu,v=dG1G2Sku,v be the degree of a vertex u,v in the network G1G2Sk:(5)M1G1G2Sk=∑u,vεVG1G2Skd2u,v=∑u1,v1u2,v2εEG1G2Skdu1,v1+du2,v2=∑u1=u2=uεVG1∑v1v2εEG2du,v1+du,v2+∑v1εVG2∑v2εVG2∑u1u2εESkG2du1,v1+du2,v2+∑v1=v2=vεVG2∑u1u2εEG1u1u2εVSkG1−VG1du1,v1+du2,v2∑v1εVG2∑v2εVG2∑u1u2εESkG2du1,v1+du2,v2=∑1+∑2+∑3+∑4.
Now, first, we calculate(6)∑1=∑u1=u2=uεVG1∑v1v2εEG2du,v1+du,v2=∑uεVG1∑v1v2εEG2n2dG1u+dG2v1+n2dG1u+dG2v2=∑uεVG1∑v1v2εEG22n2dG1u+dG2v1+dG2v2=∑uεVG1e2⋅2n2du+∑uεVG1M1G2=2e1e22n2+n1M1G2=4n2e1e2+n1M1G2,∑2=∑v1εVG2∑v2εVG2∑u1u2εESkG1du1,v1+du2,v2=∑v1εVG2∑v2εVG2∑uεvG1,aεEG1uandaare incidentdu,v1+da,v2=∑v1εVG2∑v2εVG2uεvG1,aεEG1n2dG1u+dG2v1+2n2=∑v1εVG2∑v2εVG2∑uεvG1dG1un2dG1u+dG2v1+2n2=n2n2n2∑uεvG1dG12u+∑v1εVG2∑v2εVG22e1dG2v1+2n2=n23M1G1+2e12e2n2+2n2n2n2=n23M1G1+4n2e1e2+4e1n23,∑3=∑v1=v2=vεVG2∑u1u2εEG1u1u2εVSkG1−VG1du1,v1+du2,v2=n2k−1e12n2+2n2=4e1n22k−1,∑4=∑v1εVG2∑v2εVG2∑u1u2εESkG1du1,v1+du2,v2=2n2+2n2∑i=1n2−12n2−2ik−2e1+2n2+2n2∑i=1n2−12n2−2ie1=8n2∑i=1n2−1n2−ik−2e1+8n2∑i=1n2−1n2−ie1=8n2e1k−1∑i=1n2−1n2−i.
So, the result is(7)=8n2e1e2+n1M1G2+n23M1G1+4e1n23+4e1n22k−1+8n2e1k−1∑i=1n2−1n2−i.Theorem 2.
LetG1 and G2 be two networks; then, for some integer k, the second Zagreb index of Sk-sum network is(8)M2G1G2Sk=n22e2M1G1+2n2e1M1G2+n1M2G2+2n24M1G1+8n22e1e2+4e1n32k−1+8n22e1k−1∑i=1n2−1n2−i.Proof.
Letdu,v=dG1G2Sku,v be the degree of a vertex u,v in the graph G1G2Sk:(9)M1G1G2Sk=∑u,vεVG1G2Skd2u,v=∑u1,v1u2,v2εEG1G2Skdu1,v1du2,v2=∑u1=u2=uεVG1∑v1v2εEG2du,v1du,v2+∑v1εVG2∑v2εVG2∑u1u2εESG1du1,v1du2,v2=∑u1=u2=uεVG1∑v1v2εEG2du,v1+du,v2+∑v1εVG2∑v2εVG2∑u1u2εESkG2du1,v1+du2,v2=∑1+∑2+∑3+∑4.
Now, first, we calculate(10)∑1=∑u1=u2=uεVG1∑v1v2εEG2du,v1du,v2=∑uεVG1∑v1v2εEG2n22dG12u+n2dG1udG2v1+dG2v2+dG2v1dG2v2=∑uεVG1n22e2dG12u+n2dG1uM1G2+M2G2=n22e2M1G1+2n2e1M1G2+n1M2G2,∑2=∑v1εVG2∑v2εVG2∑u1u2εESkG1du1,v1du2,v2=∑v1εVG2∑v2εVG2∑uεvG1,aεEG1uandaare incidentdu,v1da,v2=∑v1εVG2∑v2εVG2∑uεvG1,aεEG1uandaare incidentn2dG1u+dG2v12n2=n2n22n2∑uεvG1dG12u+∑v1εVG2∑v2εVG22e12n2dG2v1=2n24M1G1+2n2n22e12e2=2n24M1G1+8n22e1e2,∑3=∑v1=v2=vεVG2∑u1u2εEG1u1u2εVSkG1−VG1du1,v1du2,v2=n2k−1e12n22n2=4e1n32k−1,∑4=∑v1εVG2∑v2εVG2∑u1u2εESkG1du1,v1du2,v2=2n2×2n2∑i=1n2−12n2−2ik−2e1+2n2×2n2∑i=1n2−12n2−2ie1=8n22∑i=1n2−1n2−ik−2e1+8n22∑i=1n2−1n2−ie1=8n22e1k−1∑i=1n2−1n2−i.
So, the final result is(11)=n22e2M1G1+2n2e1M1G2+n1M2G2+2n24M1G1+8n22e1e2+4e1n32k−1+8n22e1k−1∑i=1n2−1n2−i.Theorem 3.
LetG1 and G2 be two networks; then, for some integer k, the first Zagreb index of Rk-sum network is(12)M1G1G2Rk=16n2e1e2+n1M1G2+4n23M1G1+4e1n23+4e1n22k−1+8n2e1k−1∑i=1n2−1n2−i.Proof.
Letdu,v=dG1G2Rku,v be the degree of a vertex u,v in the graph G1G2Rk:(13)M1G1G2Rk∑u,vεVG1G2Rkd2u,v=∑u1,v1u2,v2εEG1G2Rkdu1,v1+du2,v2=∑u1=u2=uεVG1∑v1v2εEG2du,v1+du,v2+∑v1εVG2∑v2εVG2∑u1u2εERkG2du1,v1+du2,v2+∑v1=v2=vεVG2∑u1u2εERkG1u1u2ε∇RkG1−VG1du1,v1+du2,v2+∑v1εVG2∑v2εVG2∑u1u2εERkG2du1,v1+du2,v2=∑1+∑2+∑3+∑4.
Now, first, we calculate(14)∑1=∑u1=u2=uεVG1∑v1v2εEG2du,v1+du,v2=∑uεVG1∑v1v2εEG2n2dRkG1u+dG2v1+n2dRkG1u+dG2v2=∑uεVG1∑v1v2εEG22n2dRkG1u+dG2v1+dG2v2=∑uεVG1∑v1v2εEG24n2dG1u+dG2v1+dG2v2=4n2∑v1v2εEG2∑uεVG1dG1u+∑uεVG1∑v1v2εEG2dG2v1+dG2v2=4n2∑v1v2εEG2e1+∑uεVG1M1G2=4n22e2e1+n1M1G2=8n2e1e2+n1M1G2,∑2=∑v1εVG2∑v2εVG2∑u1u2εEG1u1εvG1u2εvRkG1−VG1du1,v1+du2,v2+∑v1εVG2∑v2εVG2∑u1u2εERkG1u1εvG1u2εvRkG1−VG1du1,v1+du2,v2+∑v1=v2=vεVG2∑u1u2εERkG1u1u2ε∇RkG1−VG1du1,v1+du2,v2,∑2=∑2′+∑2″,∑2′=∑v1εVG2∑v2εVG2∑u1u2εEG1u1εvG1u2εvRkG1−VG1du1,v1+du2,v2=∑v1εVG2∑v2εVG2∑u1u2εEG1n2dRkG1u1+dG2v1+n2dRkG1u2+dG2v2=∑v1εVG2∑v2εVG2∑u1u2εEG1n2dRkG1u1+dRkG1u2+dG2v1+dG2v2=∑v1εVG2∑v2εVG2∑u1u2εEG12n2dG1u1+dG1u2+dG2v1+dG2v2=∑v1εVG2∑v2εVG2∑u1u2εEG12n2M1G1+2e2e1n2+2e1e2n2=n2n22n2M1G1+4e1e2n2=2n23M1G1+4e1e2n2,∑2″=∑v1εVG2∑v2εVG2∑u1u2εERkG1u1εvG1u2εvRkG1−VG1du1,v1+du2,v2=∑v1εVG2∑v2εVG2∑u1u2εERkG1u1εvG1u2εvRkG1−VG1n2dRkG1u1+dG2v1+n2dRkG1u2=∑v1εVG2∑v2εVG2∑u1u2εERkG1u1εvG1u2εvRkG1−VG12n2dG1u1+dG2v1+2n2=n1n22n2∑u1u2εERkG1u1εvG1u2εvRkG1−VG1dG1u1+∑v1εVG2∑v2εVRk2∑u1u2εERkG1u1εvG1u2εvRkG1−VG1dG2v1+2n2=2n23∑u1εvG1dG12u1∑v1εVG2∑v2εVG2∑u1εvG1dG1u1dG2v1+2n2=2n23M1G1+2e1∑v1εVG2∑v2εVG2dG2v1+2n2=2n23M1G1+2e12e2n2+2n2n2n2=2n23M1G1+4e1e2n2+4e1n23,∑3=∑v1=v2=vεVG2∑u1u2εERkG1u1u2ε∇RkG1−VG1du1,v1+du2,v2=n2k−1e12n2+2n2=4e1n22k−1,∑4=∑v1εVG2∑v2εVG2∑u1u2εERkG1du1,v1+du2,v2=2n2+2n2∑i=1n2−12n2−2ik−2e1+2n2+2n2∑i=1n2−12n2−2ie1=8n2∑i=1n2−1n2−ik−2e1+8n2∑i=1n2−1n2−ie1=8n2e1k−1∑i=1n2−1n2−i.
So, the final result is(15)=16n2e1e2+n1M1G2+4n23M1G1+4e1n23+4e1n22k−1+8n2e1k−1∑i=1n2−1.Theorem 4.
LetG1 and G2 be two networks; then, for some integer k, the second Zagreb index of Rk-sum network is(16)M2G1G2Rk=8n22e2M1G1+4e1n2M1G2+n1M2G2+4n24M2G1+4n24M1G2+8n22e1e2+4e1e22+4e1n32k−1+8n22e1k−1∑i=1n2−1n2−i.Proof.
Letdu,v=dG1G2Rku,v be the degree of a vertex u,v in the network G1G2Rk:(17)M1G1G2Rk∑u,vεVG1G2Rkd2u,v=∑u1,v1u2,v2εEG1G2Rkdu1,v1du2,v2=∑u1=u2=uεVG1∑v1v2εEG2du,v1du,v2+∑v1εVG2∑v2εVG2∑u1u2εERkG2du1,v1du2,v2+∑v1=v2=vεVG2∑u1u2εERkG1u1u2ε∇RkG1−VG1du1,v1+du2,v2+∑v1εVG2∑v2εVG2∑u1u2εERkG2du1,v1+du2,v2=∑1+∑2+∑3+∑4.
Now, first, we calculate(18)∑1=∑u1=u2=uεVG1∑v1v2εEG2du,v1du,v2=∑uεVG1∑v1v2εEG2n2dRkG1u+dG2v1n2dRkG1u+dG2v2=∑uεVG1∑v1v2εEG22n2dG1u+dG2v12n2dG1u+dG2v2=∑uεVG1∑v1v2εEG24n22dG12u+2n2dG1udG2v1+dG2v2=4e2n22M1G1+4e1n2M1G2+n1M2G2,∑2=∑v1εVG2∑v2εVG2∑u1u2εEG1u1εvG1u2εvRkG1−VG1du1,v1du2,v2+∑v1εVG2∑v2εVG2∑u1u2εERkG1u1εvG1u2εvRkG1−VG1du1v1du2v2,∑2=∑2′+∑2″,∑2′=∑v1εVG2∑v2εVG2∑u1u2εERkG1du1,v1⋅du2,v2=∑v1εVG2∑v2εVG2∑u1u2εEG1du1,v1du2,v2=∑v1εVG2∑v2εVG2∑u1u2εEG1n2dRkG1u1+dG2v1n2dRkG1u2+dG2v2=∑v1εVG2∑v2εVG2∑u1u2εEG12n2dG1u1+dG2v1n2dG1u2+dG2v2=∑v1εVG2∑v2εVG2∑u1u2εEG14n22dG1u1dG1u2+2n2dG1u1dG2v2+2n2dG1u2dG2v1+dG2v1dG2v2=4n24M2G1+4e2n22∑u1u2εEG1dG1u1+dG1u2+4e1e22=4n24M2G1+4e2n22M1G1+4e1e22,∑2″=∑v1εVG2∑v2εVG2∑u1u2εEG1u1εvG1u2εvRkG1−VG1du1,v1du2,v2=∑v1εVG2∑v2εVG2∑u1u2εEG1u1εvG1u2εvRkG1−VG1du1,v1du2,v2=∑v1εVG2∑v2εVG2∑u1u2εEG1u1εvG1u2εvRkG1−VG1n2dRkG1u1+dG2v1n2dRkG1u2=∑v1εVG2∑v2εVG2∑u1u2εEG1u1εvG1u2εvRkG1−VG1n2dRkG1u1+dG2v12n2=∑v1εVG2∑v2εVG2∑u1u2εERkG14n22dG1u1+2n2dG2v1=4n24∑u1u2εEN2G1dG1u1+∑v1εVG2∑v2εVG2∑u1u2εERkG12n2dG2v1=4n24∑u1εVG1dG12u1+4e2n22∑u1εVG1dG1u1=4n24M1G2+8e1e2n22,∑3=∑v1=v2=vεVG2∑u1u2εERkG1u1u2ε∇RkG1−VG1du1,v1du2,v2,∑v1=v2=vεVG2∑u1u2εEG2du1,v1du2,v2=n2k−1e12n2⋅2n2=4e1n23k−1,∑4=∑v1εVG2∑v2εVG2∑u1u2εERkG1du1,v1du2,v2=2n2×2n2∑i=1n2−12n2−2ik−2e1+2n2×2n2∑i=1n2−12n2−2ie1=8n22∑i=1n2−1n2−ik−2e1+8n22∑i=1n2−1n2−ie1=8n22e1k−1∑i=1n2−1n2−i.
So, the final result is(19)=8n22e2M1G1+4e1n2M1G2+n1M2G2+4n24M2G1+4n24M1G2+8n22e1e2+4e1e22+4e1n32k−1+8n22e1k−1∑i=1n2−1n2−i.


## 4. Conclusion

LetG1=Pn and G2=Pm be two networks; then, we have n1=VG1=n, n2=VG2=m, e1=EG1=n−1, e2=EG1=m−1, M1G1=4n−6, M1G2=4m−6, M2G1=4n−2, and M2G2=4m−2.Applying these values on the above derived results in Theorems1–4, we obtain Tables 1–4 . The graphical representations of the abovementioned results in tables are shown in Figures 5–8 . In Figure 9, we summarize the results of generalized T-sum networks from k = 1 to k = 4.Table 1
Numerical comparison for generalized networkGPnPmTk for k=1.
[m, n]M1GPnPmSkM2GPnPmSkM1GPnPmRkM2GPnPmRk(5, 5)4460229201035080636(5, 6)56342931213184104952(5, 7)68083570416018129268(5, 8)79824209618852153584(5, 9)91564848821686177900(5, 10)103305488024520202216(6, 5)75304575217562162672(6, 6)95165856022380211844(6, 7)115027136827198261016(6, 8)1348884176732016310188(6, 9)154749698436834359360(6, 10)1746010979241652408532(7, 5)1174482364727494294900(7, 6)1484610548435048384216(7, 7)1794812860442602473532(7, 8)2105015172450156562848(7, 9)2415217484457710652164(7, 10)2725419796465264741480(8, 5)1728213740040578494408(8, 6)2185217604851740644388(8, 7)2642221469662902794368(8, 8)3099225334474064944348(8, 9)35562291992852261094328(8, 10)40132330640963881244308(9, 5)2432421617657246780780(9, 6)30762277080730081017944(9, 7)37200337984887701255108(9, 8)436383988881045321492272(9, 9)500764597921202941729436(9, 10)565145206961360561966600(10, 5)33050324680779301176096(10, 6)41804416272994041533732(10, 7)505585078641208781891368(10, 8)593125994561423522249004(10, 9)680666910481638262606640(10, 10)768207826401853002964276Table 2
Numerical comparison for generalized networksGPnPmNk for k=2.
[m, n]M1GPnPmSkM2GPnPmSkM1GPnPmRkM2GPnPmRk(5, 5)66603112012350103136(5, 6)83743957215684129952(5, 7)100884802419018156768(5, 8)118025647622352183584(5, 9)135166492825686210400(5, 10)152307338029020237216(6, 5)112266336821018209328(6, 6)141248059226700263684(6, 7)170229781632382318040(6, 8)1992011504038064372396(6, 9)2281813226443746426752(6, 10)2571614948849428481108(7, 5)1751211579632982381336(7, 6)2204214728841908480256(7, 7)2657217878050834579176(7, 8)3110221027259760678096(7, 9)3563224176468686777016(7, 10)4016227325677612875936(8, 5)2579419544848770641864(8, 6)3247624862461980808228(8, 7)3915830180075190974592(8, 8)45840354976884001140956(8, 9)525224081521016101307320(8, 10)592044613281148201473684(9, 5)36348310424689101016976(9, 6)45774394908875881280384(9, 7)552004793921062661543792(9, 8)646265638761249441807200(9, 9)740526483601436222070608(9, 10)834787328441623002334016(10, 5)49450469880939301536096(10, 6)622845977921194041933732(10, 7)751187257041448782331368(10, 8)879528536161703522729004(10, 9)1007869815281958263126640(10, 10)11362011094402213003524276Table 3
Numerical comparison for generalized networksGPnPmNk for k=3.
[m, n]M1GPnPmSkM2GPnPmSkM1GPnPmRkM2GPnPmRk(5, 5)84603952014750100636(5, 6)106345007218684129952(5, 7)128086062422618159268(5, 8)149827117626552188584(5, 9)171568172830486217900(5, 10)193309228034420247216(6, 5)144428122425050204144(6, 6)1815610291231740263684(6, 7)2187012460038430323224(6, 8)2558414628845120382764(6, 9)2929816797651810442304(6, 10)3301218966458500501844(7, 5)2272014950839254371732(7, 6)2856618942849748480256(7, 7)3441222934860242588780(7, 8)4025826926870736697304(7, 9)4610430918881230805828(7, 10)5195034910891724914352(8, 5)3366625381657986625480(8, 6)4233232158473500808228(8, 7)5099838935289014990976(8, 8)596644571201045281173724(8, 9)683305248881200421356472(8, 10)769965926561355561539220(9, 5)4765240503281870990732(9, 6)599225131681037881280384(9, 7)721926213041257061570036(9, 8)844627294401476241859688(9, 9)967328375761695422149340(9, 10)1090029457121914602438992(10, 5)650506154801115301496096(10, 6)818047797921414041933732(10, 7)985589441041712782371368(10, 8)11531211084162011522809004(10, 9)13206612727282310263246640(10, 10)14882014370402609003684276Table 4
Numerical comparison for generalized networksGPnPmNk for k=4.
[m, n]M1GPnPmSkM2GPnPmSkM1GPnPmRkM2GPnPmRk(5, 5)104604792016750110636(5, 6)131346057221184142452(5, 7)158087322425618174268(5, 8)184828587630052206084(5, 9)211569852834486237900(5, 10)2383011118038920269716(6, 5)178989908028506224880(6, 6)2247612523236060289604(6, 7)2705415138443614354328(6, 8)3163217753651168419052(6, 9)3621020368858722483776(6, 10)4078822984066276548500(7, 5)2820818322044742410148(7, 6)3542623156856608528276(7, 7)4264427991668474646404(7, 8)4986232826480340764532(7, 9)5708037661292206882660(7, 10)642984249601040721000788(8, 5)4185831218466178691016(8, 6)5257239454483740890148(8, 7)632864769041013021089280(8, 8)740005592641188641288412(8, 9)847146416241364261487544(8, 10)954287239841539881686676(9, 5)59316499640935341095708(9, 6)745026314281183681411604(9, 7)896887632161432021727500(9, 8)1048748950041680362043396(9, 9)12006010267921928702359292(9, 10)13524611585802177042675188(10, 5)810507610801275301656096(10, 6)1018049617921614042133732(10, 7)12255811625041952782611368(10, 8)14331213632162291523089004(10, 9)16406615639282630263566640(10, 10)18482017646402969004044276Figure 5
Numerical comparison of Zagreb indices of generalized subdivision and semitotal point operations with lexicographic product of the networks fork=1.Figure 6
Numerical comparison of Zagreb indices of generalized subdivision and semitotal point operations with lexicographic product of the networks fork=2.Figure 7
Numerical comparison of Zagreb indices of generalized subdivision and semitotal point operations with lexicographic product of the networks fork=3.Figure 8
Numerical comparison of Zagreb indices of generalized subdivision and semitotal point operations with lexicographic product of the networks fork=4.Figure 9
Numerical comparison of Zagreb indices of generalized subdivision and semitotal point operations with lexicographic product of the networks forM2PnPmRk, where k∈1,2,3,4.Fork=1,(a)
G1PnPmS=8m3n−10m3+8m2n−8m2−4mn+8m−6n(b)
G2PnPmS=8m4n−12m4+12m3n−14m3−4m2n+6m2+12m−8n(c)
G1PnPmR=20m3n−28m3+16m2n−16m2−12mn+16m−6n(d)
G2PnPmR=32m4n−56m4+40m3n−56m3−20m2n+36m2−36mn+32m−4n−4Fork=2,(a)
G1PnPmS=8m3n−10m3+12m2n−12m2+4mn+8m−6n+8mn−8m∑i=1m−1m−i(b)
G2PnPmS=8m4n−12m4+12m3n−14m3+2m2−8mn+12m−8n+8m2n−8m2∑i=1m−1m−i(c)
G1PnPmR=20m3n−28m3+20m2n−20m2−12mn+16m−6n+8mn−8m∑i=1m−1m−i(d)
G2PnPmR=32m4n−56m4+44m3n−60m3−20m2n+36m2−36mn+32m−4n−4+8m2n−8m2∑i=1m−1m−iFork=3,(a)
G1PnPmS=8m3n−10m3+16m2n−16m2−4mn+8m−6n+16mn−16m∑i=1m−1m−i(b)
G2PnPmS=8m4n−12m4+12m3n−14m3+4m2n−2m2−8mn+12m−8n+16m2n−16m∑i=1m−1m−i(c)
G1PnPmR=20m3n−28m3+28m2n−28m2−12mn+16m−6n+16mn−16m∑i=1m−1m−i(d)
G2PnPmR=32m4n−56m4+48m3n−64m3−20m2n+36m2−36mn+32m−4n−4+16m2n−16m∑i=1m−1m−iFork=4,(a)
G1PnPmS=8m3n−10m3+20m2n−20m2−4mn+8m−6n+24mn−24m∑i=1m−1m−i(b)
G2PnPmS=8m4n−12m4+12m3n−14m3+8m2n−6m2−8mn+12m−8n+24m2n−24m2summ−i(c)
G1PnPmR=20m3n−28m3+32m2n−32m2−12mn+16m−6n+24mn−24msumm−i(d)
G2PnPmR=32m4n−56m4+52m3n−68m3−20m2n+36m2−36mn+32m−4n−4+24m2n−24m2summ−iIn this paper, we proved the result of 1st and 2nd Zagreb indices of networksG1G2Tk, by using generalized subdivision and generalized semitotal point operation and lexicographic product. The first and second Zagreb indices are in the form MiG1G2Tk, where (i = 1, 2). We plot the network of these derived exact values individually and after that plot a combine network for k∈1,2,3,4 to compare these results. In this result, it is clearly shown that the 2nd Zagreb indices M2G1G2TR shows better result as compared to others’ exact values.



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*Source: 4041290-2021-08-31.xml*