# Computing Correlation among the Graphs under Lexicographic Product via Zagreb Indices

Authors: Muhammad Javaid; Saira Javed; Yasmene F. Alanazi; Abdulaziz Mohammed Alanazi
Journal: Journal of Chemistry
 (2021)

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## Abstract

A topological index (TI) is a numerical descriptor of a molecule structure or graph that predicts its different physical, biological, and chemical properties in a theoretical way avoiding the difficult and costly procedures of chemical labs. In this paper, for two connected (molecular) graphsG1 and G2, we define the generalized total-sum graph consisting of various (molecular) polygonal chains by the lexicographic product of the graphs TkG1 and G2, where TkG1 is obtained by applying the generalized total operation Tk on G1 with k≥1 as some integral value. Moreover, we compute the different degree-based TIs such as first Zagreb, second Zagreb, forgotten Zagreb, and hyper-Zagreb. In the end, a comparison among all the aforesaid TIs is also conducted with the help of certain statistical tools for some particular families of generalized total-sum graphs under lexicographic product.


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## Body

## 1. Introduction

Chemical graph theory is a fascinating branch of mathematical chemistry in which the structural formulas of the different chemicals or chemical compounds are modelled as chemical structures or graphs such as the vertices correspond to the atoms and edges represent the chemical bonds between them. Topological indices (TIs) are graph-theoretic tools which are widely used in chemical graph theory to study the various structural and chemical characteristics (critical temperature, heat formation, density, extremal connectivity, unique classifications, and symmetric behaviours) of the chemical graphs. In the subject of cheminformatics, these indices also play a key role in the studies of the quantitative structure-property and quantitative structure-activity relationships which mathematically correlate the chemical properties with the physical structures of their chemical compounds. Moreover, TIs remain constant for the isomorphic structures.In 1947, the very first TI is introduced by Wiener to check the critical temperature of paraffin [1]. Gutman and Trinajsti (1972) [2] defined the 1st and 2nd Zagreb indices that are used to compute the different structure base characteristics of the molecular graphs. These Zagreb indices are also used to measure the extent of branching of the carbon-atom skeleton of the various underlying chemical structures. After that, many degree-, distance-, and polynomial-based TIs came into existence, but the degree-based indices received more attention from the researchers (see [3–5]). For various results of the TIs on different chemical graphs, we refer to [6–8].Li and Zheng and Bollobas and Erdos [9, 10] generalized the first and second Zagreb indices by the name of first general Zagreb index and general Randic index, respectively. In 2015, Furtula and Gutman redefined the concept of a F-index (forgotten index) with its basic properties [11]. It also measures the branching of the different atoms with predictive ability quite similar to the first Zagreb index. In the case of entropy and acentric factor, both first Zagreb and F-indices gain the correlation coefficients larger than 0.95. In addition, their linear combination yields a highly accurate mathematical model of certain physicochemical properties of alkanes. A significant improvement with the above model was obtained in the case of octanol-water partition coefficient. It is worth noting that this paper [11] has been cited more than 300 times so far. Moreover, in 2013, Shirdel et al. introduced another degree-based TI called by the hyper-Zagreb index [12]. For further study of different results on Zagreb indices, we refer to [13–15].On the other hand, in the studies of the complex graphs, the operations on graphs (subdivision, union, intersection, switching, sum, complement, and product) play an important role to dissolve this complexity by constructing the new graphs where old graphs are distinguished by the name of the factor graphs. Yan et al. [16] defined the total operation T1 and obtained the total graph T1G by applying the operation T1 on graph G. They also computed the Wiener indices of this newly derived graph. Liu et al. defined the extended version of this operation called generalized total operation for any integral value of k≥1 and obtained the generalized total graph TkG [17]. Eliasi and Taeri [18] defined the total-sum graph (T1-sum graph) with the help of the total operation and Cartesian product of graphs. They also computed the Wiener index of this total sum graph. Moreover, Deng et al. [19], Akhter and Imran [20], Chu et al. [21], and Liu et al. [22] computed the various indices of the T1-sum graph G1T1+G2 based on the Cartesian product. The results related to Zagreb indices for T1-sum graph under strong product can be found in [7, 23]. For further study, we refer to [24, 25].Moreover, Sarala et al. [26], De [27], Pattabiraman et al. [28, 29], and Suresh et al. [30] computed first, second, forgotten, and hyper- and reformulated Zagreb indices for the total sum (T1-sum) graph, respectively, where the T1-sum graph is obtained with the help of the total operation and lexicographic product of graphs.In this note, we defined the concept of the generalized total-sum graphs under the operation of generalized total operation and lexicographic product of two connected graphs. Moreover, we computed the different Zagreb indices such as first, second, forgotten, and hyper for the generalized total-sum graphs in terms of different TIs of factor graphs. A computing analysis among all the aforesaid TIs is also conducted with the help of the certain statistical tools for some particular families of lexicographic product and generalized total-sum graphs. The rest of the paper is distributed in Sections2, 3, 4, and 5 which cover the methodology, construction of graphs, main results, and regression modelling and conclusion, respectively.


## 2. Methodology

LetG=VG,EG be a simple, finite, and undirected graph with a vertex set VG and edge set EG⊆VG×VG, where VG and EG are called the order and size of the graph G, respectively. For a vertex or node v∈VG, the number of incident edges on v or the adjacent vertices with v is called its degree, where the set of these adjacent vertices is known by the neighborhood set of v in G (denoted by NGv). For more basic terminologies, we refer to [31]. Now, we defined some TIs which are frequently used in the main developments of the present study.Definition 1 (see [2, 11]).
The first, second, and forgotten Zagreb indices of a graphG are(1)M1G=∑z∈VGd2z=∑zt∈EGdz+dt,M2G=∑zt∈EGdz×dt,FG=∑z∈VGd3z=∑zt∈EGd2z+d2t.Definition 2 (see [9]).
The first general Zagreb index of a graphG is(2)M1α+1G=∑u∈VGdα+1u=∑uv∈EGdαu+dαv.Definition 3 (see [10]).
The general Randic´ index (GRI) of a graph G is(3)RαG=∑uv∈EGdu×dvα.
Forα = 2 and 3, the first Zagreb index and forgotten topological index are two special cases of the first general Zagreb index, respectively. Similarly, the first general Randic´ index becomes second Zagreb index for α = 1.Definition 4 (see [12]).
The hyper-Zagreb index of a graphG is(4)HMG=∑xy∈EGdx+dy2.
For further study of TIs, we refer [32, 33]. Now, if we assume X=G1G2 and Y=G1TkG2 as dependent and independent variables, respectively, where G1G2 is the simple lexicographic product graph of the graphs G1 and G2 and G1TkG2) is the generalized total-sum graph that is a lexicographic product graph of the generalized total graphs G1Tk and G2, then the simple linear regression modelling is described with H0:Y=b0 and H1:Y=b0+b1X.


## 3. Construction of Graphs

In this section, we define different families of graphs as follows:Definition 5 (see [16]).
For a connected graphG=VG,EG, a total graph T1G is defined with vertex set VG∪EG and two vertices m and n in T1G are adjacent if eitherm and n are in VG and m is adjacent to n in G orm and n are in EG and m and n are adjacent in G orm is in VG, n is in EG, and m and n are incident in G (for more details, see Figure 1)
Liu et al. [17] defined the generalized total graph TkG with vertex set VG∪kEG and edge set with same conditions on the vertices as defined in Definition 5, where k≥1 is some integral value. For more explanation, we refer to Figures 2 and 3.
The composition or lexicographic product of two connected graphsG1 and G2 (denoted by G1G2) is a graph such that the set of vertices is VG1×VG2 and two vertices x1,y1 and x2,y2 will be adjacent in G1G2 iff: [x1 =  x2 and y1 is adjacent to y2] or [x1 is adjacent to x2 and y1 is adjacent to y2] [31]. Now, we define the generalized total sum graph under the operations of generalized total operation and lexicographic product of graphs as follows.Figure 1
Total graphT1G, where G1≅P3.Figure 2
Generalized total graphTkG with G≅P3 and k=2.Figure 3
Generalized total graphTkG with G≅P3 and k=3.Definition 6.
LetG1 and G2 be two connected graphs; the generalized T-sum graph G1G2Tk is a graph having vertex set VG1G2Tk =  VTkG1×VG2 =  VG1∪kEG1×VG2 and edge set such as two vertices x1,y1 and x2,y2 of VG1G2Tk are adjacent iff [x1 =  x2 in VG1 and y1 is adjacent to y2 in EG2] or [x1 is adjacent to x2 in ETkG1 and y1 is adjacent to y2 in EG2], where k≥1 is a positive integer. For more explanation, we refer to Figure 4.Figure 4
GeneralizedT-sum graph G1G2Tk for G1≅P3, G2≅P2, and k=3.


## 4. Computational Results of Zagreb Indices

This section covers the main computational results.Theorem 1.
For two connected graphsG1 and G2 with VG1, VG2≥4, and k≥1, we have(5)M1G1G2Tk=VG1M1G2+8EG1EG2VG2+4EG1EG2VG2+2VG23M1G1+4EG1EG2VG2+4VG23M1G1+2k−1VG22M1G1+kVG23M3G1+2M2G1−2M1G1.Proof.
Consider(6)M1G1G2Tk=∑s1,t1s2,t2∈EG1TkG2ds1,t1+ds2,t2=∑s∈VG1∑t1t2∈EG2ds,t1+ds,t2+∑t1∈VG2∑t2∈VG2∑s1s2∈EG1ds1,t1+ds2,t2+∑t1∈VG2∑t2∈VG2∑s1s2∈EQkG1s1∈VG1s2∈VQkG1−VG1ds1,t1+ds2,t2+∑t∈VG2∑s1s2∈EQkG1s1s2∈VQkG1−VG1ds1,t+ds2,t+∑t1∈VG2∑t2∈VG2∑s1s2∈EQkG1s1s2∈VQkG1−VG1ds1,t1+ds2,t2=∑A+∑B+∑C+∑D+∑E.,∑A=∑s∈VG1∑t1t2∈EG2ds,t1+ds,t2,=∑s∈VG1∑t1t2∈EG22VG2dTkG1s+dG2t1+dG2t2=∑s∈VG1∑t1t2∈EG24VG2dG1s+dG2t1+dG2t2=VG1M1G2+8EG1EG2VG2,∑B=∑t1∈VG2∑t2∈VG2∑s1s2∈EG1ds1,t1+ds2,t2,=∑t1∈VG2∑t2∈VG2∑s1s2∈EG1dG2t1+dG2t2+VG2dTkG1s1+VG2dTkG1s2=4EG1EG2VG2+2VG23M1G1,∑C=∑t1∈VG2∑t2∈VG2∑s1s2∈ETkG1s1s2∈VG1ds1,t1+ds2,t2=∑t1∈VG2∑t2∈VG2∑s1s2∈EG1dG2t1+VG2dTkG1s1+VG2dTkG1s2=4EG1EG2VG2+4VG23M1G1.
Now, we take(7)∑D=∑t∫VG2∑s1s2∈ETkG1s1s2∈VTkG1−VG1ds1,t+ds2,t=∑t∈VG2∑s1s2∈ETkG1s1s2∈VTkG1−VG1VG2dTkG1s1+VG2dTkG1s2=2k−1VG22∑uvϵEG1dG1u+dG1v=2k−1VG22M1G1and(8)∑E=∑t1∫VG2∑t2∫VG2∑s1s2∈ETkG1s1s2∈VTkG1−VG1VG2dTkG1s1+VG2dTkG1s2=VG23k∑uv∈EG1vw∈EG2dG1u+dG1v+dG1v+dG1w,where s1 and s2 are the vertices that are inserted into the edges uv of G1 and vw of G1, respectively.(9)=kVG23∑v∈VG1dG13v−dG12v+∑v∈VG1dG1v−1∑u∈VG1uv∈EG1dG1u,=kVG23M3G1+2M2G1−2M1G1.
Consequently, we obtained our required result.Theorem 2.
For two connected graphsG1 and G2 with VG1, VG2≥4, and k≥1, we have(10)M2G1G2Tk=4EG2VG22M1G1+4EG1VG2M1G2+VG1M2G2+4VG24M2G1+4EG2VG22M1G1+4EG1EG22+2VG24M3G1+2M2G1+4EG2VG22M1G1+k−1VG232M2G1+M3G1+kVG2412M4G1−12M3G1+∑uv∈VG1rdG1udG1v+∑v∈VG1dG12v∑u∈VG1uv∈EG1dG1u−2M2G1.Proof.(11)M2G1G2Tk=∑s1,t1s2,t2∈EG1TkG2ds1,t1ds2,t2=∑s∈VG1∑t1t2∈EG2ds,t1ds,t2+∑t1∈VG2∑t2∈VG2∑s1s2∈EG1ds1,t1ds2,t2+∑t1∈VG2∑t2∈VG2∑s1s2∈EQkG1s1∈VG1s2∈VQkG1−VG1ds1,t1ds2,t2+∑t∈VG2∑s1s2∈EQkG1s1s2∈VQkG1−VG1ds1,tds2,t+∑t1∈VG2∑t2∈VG2∑s1s2∈EQkG1s1s2∈VQkG1−VG1ds1,t1ds2,t2=∑A+∑B+∑C+∑D+∑E.
Consider(12)∑A=∑s∈VG1∑t1t2∈EG2ds,t1ds,t2,=∑s∈VG1∑t1t2∈EG2VG2dTkG1s+dG2t1VG2dTkG1s+dG2t2=∑s∈VG1∑t1t2∈EG22VG2dG1s+dG2t12VG2dG1s+dG2t2=∑s∈VG1∑t1t2∈EG24VG22dG12s+2VG2dG1sdG2t2+2VG2dG2t1dG1s+dG2t1dG2t2=4∣EG2VG2∣2M1G1+4∣EG1VG2∣M1G2+∣VG1∣M2G2,∑B=∑t1∈VG2∑t2∈VG2∑s1s2∈ETkG1s1s2∈VG1ds1,t1ds2,t2=∑t1∈VG2∑t2∈VG2∑s1s2∈EG1VG2dTkG1s1+dG2t1VG2dTkG1s2+dG2t2=∑t1∈VG2∑t2∈VG2∑s1s2∈EG12VG2dG1s1+dG2t12VG2dG1s2+dG2t2=4VG2∣4M2G1+4EG2VG2∣2M1G1+4EG1EG22,∑C=∑t1ϵVG2∑t2ϵVG2∑s1s2ϵETkG1s1ϵVG1s2ϵVTkG1−VG1ds1,t1ds2,t2,=∑t1∈VG2∑t2∈VG2∑s1s2∈ETkG1s1∈VG1s2∈VTkG1−VG12VG2dG1s1+dG2t1VG2dTkG1s2=∑t1∈VG2∑t2∈VG2∑s1s2∈ETG1s1∈VG1s2∈VTG1−VG12VG22dG1s1dTkG1s2+VG2dTkG1s2dG2t1=2VG24∑s1s2∈ETG1s1∈VG1s2∈VTG1−VG1dTkG1s1dTkG1s2+2EG2VG22∑s1s2∈ETG1s1∈VG1s2∈VTG1−VG1dTkG1s2=2VG24∑s1s2∈ETG1s1∈VG1s2∈VTG1−VG1dTkG1s1dTkG1s2+4EG2VG22M1G1=2VG24∑s1s2ϵETG1s1ϵVG1s2ϵVTG1−VG1dTkG1s1dG1s1+dG1w1+4EG2VG22M1G1,where s2 is the vertex inserted into the edge s1w1 of G1,(13)=2VG24∑s1∈VG1∑w1∈VNG1s1dTkG1s1dTkG1s1+dTkG1w1+4EG2VG22M1G1,where NG1s1 is the set of neighbor vertices of s1 in G1,(14)=2VG24M3G1+2M2G1+4EG2VG22M1G1,and(15)∑D=∑tϵVG2∑s1s2ϵETkG1s1s2ϵVTkG1−VG1ds1,tds2,t=∑t∫VG2∑s1s2∫ETkG1s1s2∫VTkG1−VG1VG2dTkG1s1VG2dTkG1s2=k−1VG22∑t∫VG2∑uv∫EG1dG1udG1v2=k−1VG232M2G1+M3G1,∑E=∑t1∫VG2∑t2∫VG2∑s1s2∫ETkG1s1s2∫VTkG1−VG1VG2dTkG1s1VG2dTkG1s2=VG22k∑t1∫VG2∑t2∫VG2∑uv∫EG1vw∫EG2dG1u+dG1vdG1v+dG1w,where s1 is the added vertex in the edge uv and s2 is the added vertex in the edges vw of G1. Also,(16)=k∣VG2∣412∑v∫VG1dG14v−dG13v+∑uv∫VG1rdG1udG1v+∑v∫VG1dG12v∑u∫VG1uv∫EG1dG1u−2M2G1=kVG2412M4G1−12M3G1+∑uvϵVG1rdG1udG1v+∑vϵVG1dG12v∑uϵVG1uvϵEG1dG1u−2M2G1,where r is the number of neighbors which are common vertices of u and v in G1.
Hence proved.Theorem 3.
For two connected graphsG1 and G2 with VG1, VG2≥4, and k≥1, we have(17)FG1G2Tk=24VG22EG2M1G1+VG1FG2+8VG2EG1M1G2+8VG24FG1+4VG2EG1M1G2+2VG24HZG1+2k−1VH23FH1+kVH24M4H1−FH1−4M2H1+∑u∫VH1∑v∫NH1udH1udH1v−1+∑uv∫EH1dH1udH1vdH1u+dH1v.Proof.
Considerds,t=dG1TkG2s,t.Let be the degree of a node s,t in the molecular graph G1TkG2.(18)FG1G2Tk=∑s1,t1s2,t2ϵEG1TkG2ds1,t12+ds2,t22=∑s∫VG1∑t1t2∫EG2ds,t12+ds,t22+∑t1∫VG2∑t2∫VG2∑s1s2∫EG1ds1,t12+ds2,t22+∑t1∫VG2∑t2∫VG2∑s1s2∫ETkG1s1∫VG1s2∫VTkG1−VG1ds1,t12+ds2,t22+∑t∫VG2∑s1s2∫ETkG1s1s2∫VTkG1−VG1ds1,t2+ds2,t2+∑t1∫VG2∑t2∫VG2∑s1s2∫ETkG1s1s2∫VTkG1−VG1ds1,t12+ds2,t22=∑A+∑B+∑C+∑D+∑E.
Now,(19)∑A=∑s∫VG1∑t1t2∫EG2VG2dTkG1s+dG2t12+VG2dTkG1s+dG2t22=∑s∫VG1∑t1t2∫EG28VG22dTkG1s2+dG2t12+dG2t22+4VG2dTkG1sdG2t1+dG2t2=∑s∫VG1∑t1t2∫EG28VG22dG1s2+dG2t12+dG2t22+4dG1sdG2t1+dG2t2=∑s∫VG18EG2dG1s2+FG2+4M1G2dG1s1=8VG22EG2M1G1+VG1FG2+8VG2EG1M1G2,∑B=∑t1∫VG2∑t2∫VG2∑s1s2∫EG1ds1,t12+ds2,t22=∑t1∫VG2∑t2∫VG2∑s1s2∫EG1VG2dTkG1s1+dG2t12+VG2dTkG1s2+dG2t22=∑t1∫VG2∑t2∫VG2∑s1s2∫EG12VG2dTkG1s1+dG2t12+2VG2dTkG1s2+dG2t22=∑t1∫VG2∑t2∫VG2∑s1s2∫EG14VG22dG1s12+4VG22dG1s22+dG2t12+dG2t22+4VG2dG1s1dG2t1+dG1s2dG2t2=4VG24FG1+2VG2EG1M1G2+8VG22EG2M1G1,∑C=∑t1∫VG2∑t2∫VG2∑s1s2∫ETkG1s1∫VG1s2∫VTkG1−VG1VG2dTkG1s1+dG2t12+VG2dTkG1s22=∑t1ϵVG2∑t2ϵVG2∑s1s2ϵETkG1s1ϵVG1s2ϵVTkG1−VG14VG22dG1s12+dG2t2+4VG2dG1s1dG2t+VG22dG1s22=4VG24FG1+2VG2EG1M1G2+8VG22EG2M1G1+2VG24HZG1,∑D=∑t∫VG2∑s1s2∫ETkG1s1,s2∫VTkG1−VG1VG22ds1,t2+VG22ds2,t2=∑tϵVG2∑s1s2ϵETkG1s1s2ϵVTkG1−VG1VG22dTkG1s12+VG22dTkG1s22=2k−1VG22∑tϵVG2∑uvϵEG1dG1u2+dG1v2=2k−1VG23FG1,and(20)∑E=∑t1∫VG2∑t2∫VG2∑s1s2∫ETkG1s1,s2∫VTkG1−VG1VG22dTkG1s12+VG22dTkG1s22=kVG24∑uv∫EG1vw∫EG2dG1u+dG1v2+dG1v+dG1w2,where s1 and s2 are the nodes that are introduced into the edge set of uv and vw of G1.(21)=kVG24∑u∫VG1dG1v4−dG1v3+∑v∫VG1dG1v−1∑u∫VG1uv∫EG1dG1u2+2∑v∈VG1dG1vdG1v−1∑u∈VG1uv∈EG1dG1u=kVG24M4G1−FG1−4M2G1+∑u∫VG1∑v∫NG1udG1udG1v−1+∑uv∫EG1dG1udG1vdG1u+dG1v.Hence proved.Theorem 4.
For two connected graphsG1 and G2 with VG1, VG2≥4, and k≥1, we have(22)HZG1G2Tk=16VG22EG2M1G1+VG1HMG2+18VG2EG1M1G2+2EG1M1G2+4VG24HZG1+4EG2M1G1+8EG1EG22+2VG222FG1+3HZG1+16VG2EG2M1G1+4k−1VG23×HZG1+kVG24HZLG1+8M1LG1+8M1G1−16EG1.Proof.
Letds,t=dG1TkG2s,t be the degree of a node s,t in the molecular graph G1TkG2.(23)HZG1G2Tk=∑s1,t1s2,t2∫EG1TkG2ds1,t1+ds2,t22=∑s∫VG1∑t1t2∫EG2ds,t1+ds,t2+∑t1∫VG2∑t2∫VG2∑s1s2∫EG1ds1,t1+ds2,t2+∑t1∫VG2∑t2∫VG2∑s1s2∫ETkG1s1∫VG1s2∫VTkG1−VG1ds1,t1+ds2,t22+∑t∫VG1∑s1s2∫ETkG1s1s2∫VTkG1−VG1ds1,t+ds2,t2+∑t1∫VG2∑t2∫VG2∑s1s2∫ETkG1s1s2∫VTkG1−VG1ds1,t1+ds2,t22=∑A+∑B+∑C+∑D+∑E.
Now, consider(24)∑A=∑s∫VG1∑t1t2∫EG2ds,t1+ds,t22=∑s∫VG1∑t1t2∫EG2VG2dTkG1s+dG2t1+VG2dTkG1s+dG2t22=∑s∫VG1∑t1t2∫EG22VG2dTkG1s+dG2t1+dG2t22=∑s∫VG1∑t1t2∫EG24VG2dTkG1s2+dG2t12+dG2t22+4VG2dTkG1sdG2t1+4VG2dTkG1sdG2t2+2dG2t1dG2t2=∑s∫VG1∑t1t2∫EG216VG2dG1s2+dG2t12+dG2t22+8VG2dG1sdG2t1+8VG2dG1sdG2t2+2dG2t1dG2t2=16VG22EG2M1G1+VG1FG2+16VG2EG1M1G2+2VG1M2G2=16VG22EG2M1G1+VG1HZG2+16VG2EG1M1G2,(25)∑B=∑t1∫VG2∑t2∫VG2∑s1,s2∫VG1VG2dTkG1s1+dG2t1+VG2dTkG1s2+dG2t22=∑t1∫VG2∑t2∫VG2∑s1,s2∫VG12VG2dG1s1+dG2t1+2VG2dG1s2+dG2t22=∑t1∫VG2∑t2∫VG2∑s1,s2∫VG14VG22dG1s1+dG1s22+dG2t12+dG2t22+4VG2dG1s1+dG1s2dG2t1+4VG2dG1s1+dG1s2dG2t2+2dG2t1dG2t2=2EG1M1G2+4VG24HZG1+16VG24EG2M1G1+8EG1EG22,∑C=∑t1∫VG2∑t2∫VG2∑s1,s2∫VG1dTkG1s1+dTkG1s2+dG2t22=∑t1∫VG2∑t2∫VG2∑s1,s2∫VG14dG1s12+dG2t22+dQG1s22+4dG1s1dG2t2+2dG2t2dQG1s2+2dRG1s1dQG1s22=4VG22FG1+2VG2EG1M1G2+6VG22HZG1+16VG2EG2M1G1,∑D=∑t∫VG2∑s1s2∫EQkG1s1,s2∫VQkG1−VG1ds1,t+ds2,t2=∑t∫VG2∑s1s2∫EQkG1s1,s2∫VQkG1−VG1VG2dQkG1s1+VG2dQkG1s22=4k−1VG22∑t∫VG2∑uv∫EG1dG1u+dG1v2=4k−1VG23HMG1,and(26)∑E=∑t1∫VG2∑t2∫VG2∑s1s2∫EQkG1s1s2∫VQkG1−VG1VG2dQkG1s1+VG2dQkG1s22=kVG24∑uv∫EG1vw∫EG2dG1u+dG1v+dG1v+dG1w2.Here, W and X are the nodes of LG1, so we have(27)=kVG24∑WX∫ELG1dLG1W+dLG1X+42=kVG24∑WX∫ELG1dLG1W2+dLG1X2+16+2dLG1WdLG1X+8dLG1W+8dLG1X=kVG24HZLG1+8M1LG1+8M1G1−16EG1.Hence, we obtain our required result.


## 5. Comparison and Conclusion

To check the correlation between the predefined lexicographic product of two simple graphs and newly defined generalized total-sum graphs via understudy Zagreb indices, we apply the linear regression model on the obtained formulas (Theorem1–Theorem 4) of the first, second, forgotten, and hyper-Zagreb indices. For the purpose, we consider two simple path graphs Pn and Pm and construct their lexicographic product graph PnPm and generalized total-sum graph PnTkPm based on the lexicographic product, where m,n≥1. By using Theorem 1 to Theorem 4, the first, second, forgotten, and hyper-Zagreb indices of these graphs for different values of m and n are shown in Tables 1 and 2.Table 1
Zagreb indices ofPnPm.
[m, n]VEM1M2FGHG(3, 3)92427676417043232(4, 4)16609683868828016 016(5, 5)25120246012 57626 46051 612(6, 6)36210519632 13266 948131 212(7, 7)49336971670 276145 488286 040Table 2
Zagreb indices ofPnTkPm.
[m, n, k]VEM1M2FGHG(3, 3, 3)27207694858989122616240594(4, 4, 4)6499271232129359627873765374568(5, 5, 5)12532753979201.235160 8×1072.595604×1075.0659256×107(6, 6, 6)216860415748567.357629 6×1071.52609976×1082.99762568×108(7, 7, 7)343193554964523.24253988×1086.67171624×1081.315679 6×109By using Tables1 and 2, we construct the following linear regression lines:(i)
First Zagreb Index. For X=M1PnTkPm and Y=M1PnPm, we have Y=2260.3506+0.002871X with a value of correlation coefficient 0.471 1(ii)
Second Zagreb Index. For X=M2PnTkPm and Y=M2PnPm, we have Y=1548.9242+0.00002654X with a value of correlation coefficient 0.953(iii)
Forgotten Topological Index. For X=FPnTkPm and Y=FPnPm, we have Y=7355.407+0.0002013X with a value of correlation coefficient 0.9738 (see Table 3).(iv)
Hyper-Zagreb Index. For X=HZPnTkPm and Y=HZPn⌊Pm⌋, we have Y=30154.5419+0.0002018X with a value of correlation coefficient 0.973 7Now, we present these values in Table3 more precisely.Table 3
Correlation coefficient betweenPnTkPm and PnPm.
TI’sValue of correlation coefficientRelationship betweenX and YFirst Zagreb index0.471 1Very strong direct relationshipSecond Zagreb index0.953Very strong direct relationshipForgotten Zagreb index0.9738Very strong direct relationshipHyper-Zagreb index0.973 7Very strong direct relationshipIn this paper, we computed different Zagreb indices for the generalized total-sum graphs. To compute the correlation values, we considered certain families of graphs that belong to lexicographic product graphs and the generalized total-sum graphs under the lexicographic product for their regression lines and values of the correlation coefficients as given in Table3. From Table 3, we obtain the following two conclusions.Firstly, there exists a strong correlation between the lexicographic product graphs and the generalized total-sum graphs under lexicographic product via all the understudy Zagreb indices. More clearly, we can state that by increasing the number of atoms and bonds in lexicographic product graphs to obtain the generalized total-sum graphs under the lexicographic product, the chemical properties do not change their strength. Secondly, hyper-Zagreb index shows the strong correlation among all the understudy Zagreb indices by attaining value 0.9738 as shown in Table3.Now, we close this section with the comment that the problem is still open to investigate the comparison among the other TIs for different operations on graphs.



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