# Comparative Study of Generalized Sum Graphs via Degree-Based Topological Indices

Authors: Muhammad Javaid; Saira Javed; Ebenezer Bonyah
Journal: Journal of Mathematics
 (2022)

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## Abstract

In theoretical chemistry, topological indices (TIs) have important role to predict various physical and structural properties of the study under molecular graphs. Among all topological indices, Zagreb-type indices have been used more effectively in the chemical literature. In this paper, we have computed first Zagreb, second Zagreb, forgotten, and hyper Zagreb indices of the generalizedQ-sum graph H1QαH2 in the form of different TIs of its basic graphs, where α≥1 is a positive integer. This family of graphs is obtained by the lexicographic product of the graph QαH1 and H2, where QαH1 is constructed with the help of the generalized line superposition operation Qα on H1. As a conclusion, we also checked the correlation between predefined graph H1H2 under the operation of lexicographic product H1 and H2 with newly defined generalized Q-sum graphs H1QαH2 using linear regression models of various degree-based TIs.


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## Body

## 1. Introduction and Preliminaries

In the computing analysis of chemical compounds, the chemical structural formulas are usually represented by graphs. Mathematically, for a graphH, the vertex and edge sets are denoted by VH and EH⊆VH×VH, respectively. The degree of the vertex z∈VH (denoted by dz) is the number of incident edges on it. The distance between any two vertices z and y in VH is denoted by dz,y and defined as the number of edges in a shortest path existing between the vertices z and y. Thoroughly, the graph H is considered as a finite and simple graph.The subject of chemical graph theory has many applications in chemistry. In a chemical graph, a vertex and an edge correspond to an atom and a chemical bond between them, respectively. A topological index is a function which presents a chemical graph in the form of a numerical number that is used to model the chemical and physical properties of molecules in quantitative structure property and activity relationships.For a molecular graphH, the first Zagreb index M1H and second Zagreb index M2H are firstly considered by Gutman and Trinajstic [1] in 1972 to study the total π electron energy of the molecular graph H which are defined as follows:(1)M1H=∑z∈VHd2z=∑zy∈EHdz+dy,M2H=∑zy∈EHdz×dy.In 2015, forgotten index (F index) is defined as follows [2]:(2)FH=∑z∈VHd3z=∑zy∈EHd2z+d2y.In 2005, the first general Zagreb index is defined as follows [3]:(3)M1β+1H=∑z∈VHdβ+1z=∑zy∈EHdβz+dβy.The concept of the general randic index (GRI) is defined by Bollobas and Erdos as(4)RβH=∑zy∈EHdz×dyβ.In 2013, hyper Zagreb index is defined by Shirdel et al. as follows [4]:(5)HMH=∑zy∈EHdz+dy2.For more studies, we refer to [5]. In particular, we can find the results of TIs for various families of graphs such as nanosheets & nanostars dendrimers [6, 7], hex-derived networks [8], benzene networks [9, 10], and cellulose networks [11]. In addition, for the studies of the complex graphs, operations on graphs play a key role, where the original graphs are called the factors of the newly constructed graph under operations [12].Yan et al. [13] defined the line superposition operation Q1 related to the subdivision of H and computed the Wiener index of this derived graph Q1H, where Q1H is obtained by inserting one new vertex in every edge of H and joining those pairs of new vertices by edges which have common adjacent (original) vertices (see Figure 1). Liu et al. extended this operation for any integral value of α≥1 and obtained the generalized line superposition graph QαH from the graph H by inserting α vertices in each edge and joining those pairs of new vertices by edges which have common adjacent (original) vertices [14] (see Figures 2 and 3).Figure 1
Line superposition operation forP3 denoted by Q1P3.Figure 2
Generalized line superposition operation forP3 and α = 2 denoted by Q2P3.Figure 3
Generalized line superposition operation forP3 for α = 3 denoted by Q3P3.Taeri et al. [15] constructed the Q1-sum graph by Cartesian product of Q1H1 and H2, where Q1H1 is a line superposition graph, and H1 and H2 are assumed to be two connected graphs. Also, they computed the Wiener index of Q1-sum graph. Whereas, Chu et al. [16], Deng et al. [17], Akhter et al. [18], and Liu et al. [14] calculated the different indices of Q1-sum graph H1Q1+H2 based on the Cartesian product. Recently, Liu et al. [19] extended this operation for any integral value of k and obtained the generalized line superposition graph QkH1 of the graph H1. Moreover, they constructed the generalized Q-sum graph H1Qk+H2 and computed their Zagreb indices. Javaid et al. [20] constructed the generalized Q-sum graph based on strong product and computed its first and second Zagreb indices.The composition or lexicographic product of two connected graphsH1 and H2, which is denoted by H1H2, is a graph such that the set of vertices is VH1×VH2, and two vertices z1,y1 and z2,y2 will be adjacent in H1H2 if [z1 =  z2, and y1 is adjacent to y2] or [z1 is adjacent to z2, and y1 is adjacent to y2].Sarala et al. [21], De [22], Pattabiraman [23], Pattabiraman and Santhakumar [24], and Suresh and Devi [25] have computed different degree-related Zagreb indices for the graphs based online superposition operation and lexicographic product.(1)Qα-sum: let H1 and H2 be two graphs; first, we apply generalized line superposition operation on H1 to get QαH1 graph. Then, we take lexicographic product between QαH1 and H2 to get Qα-Sum graph denoted by H1QαH2. Qα-sum graph having vertex-set(6)VH1QαH2=VQαH1×VH2=VH1∪αEH1×VH2,such that two vertices z1,y1 and z2,y2 of VH1QαH2 are adjacent if [z1 =  z2 in VH1, and y1 is adjacent to y2 in EH2] or [z1 is adjacent to z2 in EQαH1, and y1 is adjacent to y2 in EH2], where α≥1 is a positive integer (see Figure 4).Figure 4
GeneralizedQα-sum graph H1QαH2 for P3Q3P2.To check the correlation between the predefined lexicographic product of two simple graphsH1H2 and newly defined Qα-sum graphs H1QαH2, one may use the linear regression when having a linear relationship between the dependent variable (X =H1H2) and the independent variable (Y =H1QαH2). For further studies related to operations and graph products, the readers are referred to [26–29]. In this paper, we have used the generalized line superposition operation to construct a new generalized line superposition graph QαH1 from the graph actual graph H1. Then, we have constructed a Qα-sum graph based on the lexicographic product of QαH1 and H2, denoted by H1QαH2. Moreover, we have computed the first Zagreb, second Zagreb, forgotten, and hyper Zagreb indices of H1QαH2 in terms of its factor graphs H1 and H2. We have extended this work by constructing linear regression models between H1H2 and Qα-sum graphs H1QαH2 via foresaid degree-based TIs.


## 2. Main Results

This section covers the main results.Theorem 1.
LetH1 and H2 be two connected graphs such that VH1, VH2≥4. For α≥1,(7)M1H1QαH2=8VH2EH1EH2+3VH23M1H1+VH1M1H2+2α−1VH22M1H1+αVH23M3H1+2M2H1−2M1H1.Proof.
(8)M1H1QαH2=∑z1,y1z2,y2∈EH1QαH2dz1,y1+dz2,y2=∑z∈VH1∑y1y2∈EH2dz,y1+dz,y2+∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH2dz1,y1+dz2,y2=∑A+∑B.
Now,(9)∑A=∑z∈VH1∑y1y2∈EH2dz,y1+dz,y2=∑z∈VH1∑y1y2∈EH22VH2dH1z+dH2y1+dH2y2=4VH2EH1EH2+VH1M1H2,∑B=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1dz1,y1+dz2,y2=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1dz1,y1+dz2,y2+∑y∈VH2∑z1z2∈EQαH1z1z2∈VQαH1−VH1dz1,y+dz2,y+∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1z2∈VQαH1−VH1dz1,y1+dz2,y1=∑B1+∑B2+∑B3,∑B1=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1dz1,y1+dz2,y2=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH2z1∈VH1z2∈VQαH1−VH1VH2dQαH1z1+dH2y1+VH2dQαH1z2=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH2z1∈VH1z2∈VQαH1−VH1VH2dH1z1+dH2y1+∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH2z1∈VH1z2∈VQαH1−VH1VH2dQαH1z2=4VH2EH1EH2+VH23M1H1+∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1VH2dQαH1z2.
Forz2∈VQαH1−VH1 which is inserted into the edge uv of H1, we have dQαH1z2=dH1u+dH1v. This gives(10)∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1dQαH1z2=2∑uv∈EH1dH1u+dH1v=2M1H1∑H2∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1dQαH1z2=2VH2M1H1.
So, we have(11)∑B1=4VH2EH1EH2+3VH23M1H1.
Now, we take(12)∑B2=∑y∈VH2∑z1z2∈EQαH1z1z2∈VQαH1−VH1dz1,y+dz2,y=∑y∈VH2∑z1z2∈EQαH1z1z2∈VQαH1−VH1VH2dQαH1z1+VH2dQαH1z2=2α−1VH22∑uv∈EH1dH1u+dH1v=2α−1VH2∣2M1H1,∑B3=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1z2∈VQαH1−VH1VH2dQαH1z1+VH2dQαH1z2=VH23α∑uv∈EH1vw∈EH2dH1u+dH1v+dH1v+dH1w,where z1 and z2 are the vertices that are inserted into the edges uv of H1 and vw of H1, respectively.(13)=αVH23∑v∈VH1dH13v−dH12v+∑v∈VH1dH1v−1∑u∈VH1uv∈EH1dH1u=αVH23M3H1+2M2H1−2M1H1.
Consequently, we have obtained our required result.Theorem 2.
LetH1 and H2 be two connected graphs such that VH1, VH2≥4. For α≥1,(14)M2H1QαH2=5EH2‖VH22M1H1+2EH1‖VH2M1H2+VH1M2H2+VH24M3H1+2M2H1+αVH2412M4H1−12M3H1+∑uv∈VH1rdudv+∑v∈VH1d2v∑u∈VH1uv∈EH1du−2M2H1+α−1VH232M2H1+M3H1,where r is the number of neighbours which are common between u and v in H1.Proof.
(15)M2H1QαH2=∑z1,y1z2,y2∈EH1QαH2dz1,y1dz2,y2=∑z∈VH1∑y1y2∈EH2dz,y1dz,y2+∑y1∈VH2∑y2∈VH2∑y2∈VH2∑z1z2∈EQαH1dz1,y1dz2,y2=∑A+∑B.
Now, we take(16)∑A=∑z∈VH1∑y1y2∈EH2dz,y1dz,y2=∑z∈VH1∑y1y2∈EH2VH2dH1z+dH2y1VH2dH1z+dH2y2=EH2‖VH22M1H1+2EH1‖VH2M1H2+VH1M2H2,∑B=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1dz1,y1dz2,y2=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH2dz1,y1dz2,y2∑y∈VH2∑z1z2∈EQαH1z1z2∈VQαH1−VH1dz1,ydz2,y+∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1z2∈VQαH1−VH1dz1,y1dz2,y2=∑B1+∑B2+∑B3.
So,(17)∑B1=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1dz1,y1dz2,y2=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1VH2dH1z1+dH2y1VH2dQαH1z2=∑y1∈VH2∑y2∈VH2∑z1z2∈EQH1z1∈VH1z2∈VQH1−VH1VH2dQαH1z2dH2y1+VH22dH1z1dQαH1z2=VH24∑z1z2∈EQH1z1∈VH1z2∈VQH1−VH1dQαH1z1dQαH1z2+2EH2‖VH22∑z1z2∈EQH1z1∈VH1z2∈VQH1−VH1dQαH1z2=VH24∑z1z2∈EQH1z1∈VH1z2∈VQH1−VH1dQαH1z1dQαH1z2+4EH2‖VH22M1H1,where z2 is the vertex inserted into the edge z1w1 of H1(18)=VH24∑z1∈VH1∑w1∈VNH1z1dQαH1z1dQαH1z1+dQαH1w1+4EH2‖VH22M1H1,where NH1z1 is the set of neighbor vertices of z1 in H1(19)=VH24M3H1+2M2H1+4EH2‖VH22M1H1,∑B2=∑y∈VH2∑z1z2∈EQαH1z1z2∈VQαH1−VH1dz1,ydz2,y∑B2=∑y∈VH2∑z1z2∈EQαH1z1z2∈VQαH1−VH1VH2dQαH1z1VH2dQαH1z2=α−1VH232M2H1+M3H1,∑B3=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1z2∈VQαH1−VH1VH2dQαH1z1VH2dQαH1z2=VH22α∑y1∈VH2∑y2∈VH2∑uv∈EH1vw∈EH2dH1u+dH1vdH1v+dH1w,where z1 is the added vertex in the edge uv, and z2 is the added vertex in the edges vw of H1(20)=αVH2412∑v∈VH1dH14v−dH13v+∑uv∈VH1rdH1udH1v+∑v∈VH1dH12v×∑u∈VH1uv∈EH1dH1u−2M2H1=αVH2412M4H1−12M3H1+∑uv∈VH1rdH1udH1v+∑v∈VH1dH12v∑u∈VH1uv∈EH1dH1u−2M2H1.where r is the number of neighbors which are common vertices of u and v in H1. Consequently, we have obtained our required result.Theorem 3.
LetH1 and H2 be two connected graphs such that VH1, VH2≥4. For α≥1,(21)FH1QαH2=3VH24FH1+VH1FH2+6VH22EH2M1H1+6VH2×EH1M1H2+4VH24M2H1+αVH24M4H1−FH1−4M2H1+∑u∈VH1∑v∈NH1udH1udH1v−1+∑uv∈EH1dH1udH1vdH1u+dH1v+2α−1VH23FH1.Proof.
Consider(22)FH1QαH2=∑z1,y1z2,y2∈EH1QαH2dz1,y12+dz2,y22=∑z∈VH1∑y1y2∈EH2dz,y12+dz,y22+∑y∈VH2∑z1z2∈EQαH2dz1,y2+dz2,y2=∑A+∑B.
Now,(23)∑A=∑z∈VH1∑y1y2∈EH2dH1z+dH2y12+dH1z+dH2y22=∑z∈VH1∑y1y2∈EH22VH22dH1z2+dH2y12+dH2y22+2dH1zVH1dH2y1+dH2y2=∑z∈VH1EH2‖VH2∣2×2dH1z2+FH2+2VH2M1H2dH1z=2EH2‖VH22M1H1+VH1FH2+4VH2‖EH1M1H2,∑B=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1dz1,y12+dz2,y22=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1dz1,y12+dz2,y22+∑y∈VH2∑z1z2∈EQkH1z1z2∈VQαH1−VH1dz1,y2+dz2,y2+∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1z2∈VQαH1−VH1dz1,y12+dz2,y22=∑B1+∑B2+∑B3.
Now, we take(24)∑B1=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1VH2dQαH1z1+dH2y12+VH22dQαH1z22=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1VH22dQαH1z12+dH2y12+2VH2dQαH1z1dH2y1+VH22dQαH1z22=∑y1∈VH2∑y2∈VH2∑z1∈VH1VH22dQαH1z12+dH2y12+2VH2dQαH1z1dH2y1+∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1VH22dQαH1z22=VH2FH1+2EH1M1H2+4EH2M1H1+VH22∑y∈VH2×∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1dQαH1z22.
Forz2∈VQαH1−VH1 that is introduced into the edges uv of H1, we have dQαH1z2=dH1u+dH1v. This gives(25)∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1dQαH1z22=2∑uv∈EH1dH1u+dH1v2=2HMH1.
So,(26)∑B1=VH242HMH1+FH1+2VH2‖EH1M1H2+4VH22EH2M1H1,∑B2=∑y∈VH2∑z1z2∈EQαH1z1,z2∈VQαH1−VH1VH22dz1,y2+VH22dz2,y2=∑y∈VH2∑z1z2∈EQαH1z1z2∈VQαH1−VH1VH22dQαH1z12+VH22dQαH1z22=2α−1VH22∑y∈VH2∑uv∈EH1dH1u2+dH1v2=2α−1VH23FH1,∑B3=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1,z2∈VQαH1−VH1VH22dQαH1z12+VH22dQαH1z22=αVH24∑uv∈EH1vw∈EH2dH1u+dH1v2+dH1v+dH1w2,where z1 and z2 are the nodes that are introduced into the edge set of uv and vw of H1(27)=αVH24∑u∈VH1dH1v4−dH1v3+∑v∈VH1dH1v−1∑u∈VH1uv∈EH1dH1u2+2∑v∈VH1dH1vdH1v−1∑u∈VH1uv∈EH1dH1u=αVH24M4H1−FH1−4M2H1+∑u∈VH1∑v∈NH1udH1udH1v−1+∑uv∈EH1dH1udH1vdH1u+dH1v,
Consequently, we have obtained our required result.Theorem 4.
LetH1 and H2 be two connected graphs such that VH1, VH2≥4. For α≥1,(28)HZH1QαH2=αVH24HZLH1+8M1LH1+8M1H1−16EH1+4α−1VH23HZH1+12VH22EH2M1H1+VH1HZH2+10∣VH2‖EH1M1H2+VH242HZH1+3FH1+4M2H1.Proof.
(29)HZH1QαH2=∑z1,y1z2,y2∈EH1QαH2dz1,y1+dz2,y22∑z∈VH1∑y1y2∈EH2dz,y1+dz,y22+∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1dz1,y1+dz2,y22=∑A+∑B.
Now,(30)∑A=∑z∈VH1∑y1y2∈EH2dz,y1+dz,y22=∑z∈VH1∑y1y2∈EH2VH2dQαH1z+dH2y1+VH2dQαH1z+dH2y22=∑z∈VH1∑y1y2∈EH22VH2dQαH1z+dH2y1+dH2y22=∑z∈VH1∑y1y2∈EH24VH22dQαH1z2+dH2y12+dH2y22+4VH2dQαH1zdH2y1+4VH2dQαH1zdH2y2+2dH2y1dH2y2=4EH2‖VH22M1H1+VH1HZH2+∑z∈VH1∑y1y2∈EH24dH1zdH2y1+dH2y2=4EH2‖VH22M1H1+VH1HZH2+8EH1‖VH2M1H2,∑B=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1dz1,y+dz2,y2=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1dz1,y1+dz2,y22+∑y∈VH2∑z1z2∈EQαH1z1z2∈VQαH1−VH1dz1,y+dz2,y2+∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1z2∈VQαH1−VH1dz1,y1+dz2,y22=∑B1+∑B2+∑B3.
Now, we take(31)∑B1=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1dz1,y1+dz2,y22=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1VH2dQαH1z1+dH2y1+VH2dQαH1z22=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1VH22dQαH1z12+dH2y12+VH22dQαH1z22+2VH2dQαH1z1dH2y1+2VH2dH2y1dQαH1z2+2VH22dQαH1z1dQαH1z2=VH24FH1+2VH2‖EH1M1H2+4VH22EH2M1H1+∑y1∈VH2∑y2∈VH2×∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1VH22dQαH1z22+2VH22dH1z1+2VH22dQαH1z1dQαH1z2.
One can see that for a vertexz2∈VQαH1−VH1, dQαH1z2=dH1z+dH1w, where z2=xw∈EH1. Thus,(32)=VH24FH1+2VH2‖EH1M1H2+4VH22EH2M1H1+∑y1∈VH2∑y2∈VH2×∑z1z2∈EQαH1z1∈VH1z2∈VQαH1−VH1VH22dH1z+dH1w2+2VH22dH1z1+dQαH1z1dH1z+dH1w=VH24FH1+2VH2‖EH1M1H2+4VH22EH2M1H1+2VH24HZH1+2VH24FH1+2M2H1+8VH22EH1M1H2=2VH24HZH1+3VH24FH1+12VH22EH2M1H1+2VH2‖EH1M1H2+4VH24M2H1,∑B2=∑y∈VH2∑z1z2∈EQαH1z1,z2∈VQαH1−VH1VH2dQαH1z1+VH2dQαH1z22=4α−1VH22∑y∈VH2∑uv∈EH1dH1u+dH1v2=4α−1VH23HMH1,∑B3=∑y1∈VH2∑y2∈VH2∑z1z2∈EQαH1z1z2∈VQαH1−VH1VH2dQαH1z1+VH2dQαH1z22=αVH24∑uv∈EH1vw∈EH2dH1u+dH1v+dH1v+dH1w2.
W and X are the nodes of LH1, so we have(33)=αVH24∑WX∈ELH1dLH1W2+dLH1X2+16+2dLH1WdLH1X+8dLH1W+8dLH1X=αVH24HZLH1+8M1LH1+1612M1H1−EH1,∑B3=αVH24HZLH1+8M1LH1+8M1H1−16EH1.
Consequently, the result is over.


## 3. Applications and Conclusion

For any two simple path graphsPn and Pm with n,m>3, we construct their lexicographic product graphs PnPm and compute various degree-related TIs, as shown in Table 1 with certain values of m and n. Furthermore, we construct QαPn by applying generalized line superposition operation on Pn and then construct Qα-sum graph PnQαPm based on lexicographic product of QαPn and Pm. Finally, we compute certain degree-related TIs using Theorem 1–Theorem 4, as given in Table 2. We note that by the addition of vertices α≥1, the values of obtained TIs are increasing for the newly constructed graphs comparatively to the graphs obtained by the ordinary lexicographic product of graphs.Table 1
PnPm.
[m, n]VEM1M2FHHH(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
PnQαPm.
[m, n, k]VEM1M2FHHH(3,3,3)27189636653 701114 786222 188(4,4,4)6494469 0241256 1602745 7605258 080(5,5,5)1253175392 0301.219 569 2×1072.581 541×1075.020 679 4×107(6,6,6)21684241561 9927.308 936 4×1071.522 399 68×1082.984 186 96×108(7,7,7)34319 0614939 9983.229 925 64×1086.663 437 62×1081.312 328 89×109


### 3.1. Linear Regression Model

To check the correlation between the predefined lexicographic product of two simple graphsPnPm and newly defined Qα-sum graphs PnQαPm, we have made the linear regression model for all the obtained indices as follows:


### 3.2. First Zagreb Index

(34)M1PnPm=b0+b1M1PnQαPm,where Y =  M1PnPm index values, b0 is the constant, b1 is the regression coefficient, and X =  M1PnQαPm index values (see Tables 1 and 2).(35)Y=1207.7339+0.001805X.


### 3.3. Second Zagreb Index

(36)M2PnPm=b0+b1M2PnQαPm,where Y =  M2PnPm index values, b0 is the constant, b1 is the regression coefficient, and X =  M2PnQαPm index values (see Tables 1 and 2).(37)Y=1548.9242+0.00002654X.


### 3.4. Forgotten Topological Index

(38)FPnPm=b0+b1FPnQαPm,where Y =  FPnPm index values, b0 is the constant, b1 is the regression coefficient, and X =  FPnQαPm index values (see Tables 1 and 2).(39)Y=11263.1338+0.00008912X.


### 3.5. Hyper Zagreb Index

(40)HMPnPm=b0+b1HMPnQαPm,where Y =  HMPnPm index values, b0 is the constant, b1 is the regression coefficient, and X =  HMPnQαPm index values (see Tables 1 and 2).(41)Y=30210.7664+0.0002023X.Now, we represent a tabular representation of the correlation and square of correlation coefficient values related to the TI’s of certain type of graphsPnQαPm and PnPm (see Tables 3 and 4). The highest value of R and R2 are indicated in bold.Table 3
Correlation coefficient betweenPnQαPm and PnPm.
TI’sValue ofRRelationship betweenX and YFirstZagrebindex0.9756VerystrongdirectrelationshipSecond Zagreb index0.953Very strong direct relationshipForgotten Zagreb index0.9123Very strong direct relationshipHyper Zagreb index0.9736Very strong direct relationshipTable 4
Square of correlation coefficient betweenPnQαPm and PnPm.
TI’sValue ofR2Variability ofY is explained by XFirstZagrebindex0.951895.1percentSecond Zagreb index0.908 090.8 percentForgotten Zagreb index0.832 383.23 percentHyper Zagreb index0.947 894.78 percentFor more explanation, see the graphical representation of correlation and square of correlation coefficient in Figures5 and 6.Figure 5
Value ofR for number of vertices, edges, first, second, forgotten, and hyper Zagreb indices.Figure 6
Value ofR2 for number of vertices, edges, first, second, forgotten, and hyper Zagreb indices.


## 3.1. Linear Regression Model

To check the correlation between the predefined lexicographic product of two simple graphsPnPm and newly defined Qα-sum graphs PnQαPm, we have made the linear regression model for all the obtained indices as follows:


## 3.2. First Zagreb Index

(34)M1PnPm=b0+b1M1PnQαPm,where Y =  M1PnPm index values, b0 is the constant, b1 is the regression coefficient, and X =  M1PnQαPm index values (see Tables 1 and 2).(35)Y=1207.7339+0.001805X.


## 3.3. Second Zagreb Index

(36)M2PnPm=b0+b1M2PnQαPm,where Y =  M2PnPm index values, b0 is the constant, b1 is the regression coefficient, and X =  M2PnQαPm index values (see Tables 1 and 2).(37)Y=1548.9242+0.00002654X.


## 3.4. Forgotten Topological Index

(38)FPnPm=b0+b1FPnQαPm,where Y =  FPnPm index values, b0 is the constant, b1 is the regression coefficient, and X =  FPnQαPm index values (see Tables 1 and 2).(39)Y=11263.1338+0.00008912X.


## 3.5. Hyper Zagreb Index

(40)HMPnPm=b0+b1HMPnQαPm,where Y =  HMPnPm index values, b0 is the constant, b1 is the regression coefficient, and X =  HMPnQαPm index values (see Tables 1 and 2).(41)Y=30210.7664+0.0002023X.Now, we represent a tabular representation of the correlation and square of correlation coefficient values related to the TI’s of certain type of graphsPnQαPm and PnPm (see Tables 3 and 4). The highest value of R and R2 are indicated in bold.Table 3
Correlation coefficient betweenPnQαPm and PnPm.
TI’sValue ofRRelationship betweenX and YFirstZagrebindex0.9756VerystrongdirectrelationshipSecond Zagreb index0.953Very strong direct relationshipForgotten Zagreb index0.9123Very strong direct relationshipHyper Zagreb index0.9736Very strong direct relationshipTable 4
Square of correlation coefficient betweenPnQαPm and PnPm.
TI’sValue ofR2Variability ofY is explained by XFirstZagrebindex0.951895.1percentSecond Zagreb index0.908 090.8 percentForgotten Zagreb index0.832 383.23 percentHyper Zagreb index0.947 894.78 percentFor more explanation, see the graphical representation of correlation and square of correlation coefficient in Figures5 and 6.Figure 5
Value ofR for number of vertices, edges, first, second, forgotten, and hyper Zagreb indices.Figure 6
Value ofR2 for number of vertices, edges, first, second, forgotten, and hyper Zagreb indices.



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*Source: 9001167-2022-02-14.xml*