# Computing Analysis of Zagreb Indices for Generalized Sum Graphs under Strong Product

Authors: Muhammad Javaid; Saira Javed; Abdulaziz Mohammed Alanazi; Majdah R. Alotaibi
Journal: Journal of Chemistry
 (2021)

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## Abstract

Numerous studies based on mathematical models and tools indicate that there is a strong inherent relationship between the chemical properties of the chemical compounds and drugs with their molecular structures. In the last two decades, the graph-theoretic techniques are frequently used to analyse the various physicochemical and structural properties of the molecular graphs which play a vital role in chemical engineering and pharmaceutical industry. In this paper, we compute Zagreb indices of the generalized sum graphs in the form of the different indices of their factor graphs, where generalized sum graphs are obtained under the operations of subdivision and strong product of graphs. Moreover, the obtained results are illustrated with the help of particular classes of graphs and analysed to find the efficient subclass with dominant indices.


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## Body

## 1. Introduction

In many fields (chemistry, physics, computer science, and electrical networks) various physicochemical and structural properties such as melting point, boiling point, chemical bonds, bond energy, solubility, surface tension, critical temperature, connectivity, stability, density, and polarizability are studied with the help of various TIs (degree-based, distance-based, and polynomial-based). Moreover, degree-based TIs have been used as a powerful approach to discover many new drugs, such as anineoplastics, anticonvulsants, antiallergics, antimalarials, and silico generation (see [1]). Therefore, this practice has proven that the TIs and the quantitative structure-activity (or structure-property) relationships (QSAR or QSPR) have presented a foundation stone in chemical engineering and pharmaceutical industry for the process of the drug design and discovery (see [2, 3]).LetΩ be a collection of (molecular) graphs in which each graph is considered as a simple graph without multiedges and loops. A topological index (TI) is a function Top:Ω⟶ℝ that assigns a real number to each element (graph) of Ω, where ℝ is a set of real numbers. Moreover, for two graphs G1 and G2, TopG1=TopG2 if and only if G1 is isomorphic to G2. Mostly, TIs are computed for the hydrogen-suppressed molecular graphs in which the atoms are represented by nodes and bonds between them by edges. In 1947, Wiener index (path number), first distance-based TI, is utilized in the study of paraffin’s boiling point [4]. Gutman and Trinajstic [5] calculated total π-electrons energy of the molecules through a degree-based TI called as the first Zagreb index (FZI). They also studied the various properties of the second Zagreb index (SZI) in the same paper. In chemical graph theory, many more TIs are introduced in [6], but degree-based TIs are prominent than others. For more details, we refer to [7, 8].On the other hand, operations on graphs (addition, complement, deletion, switching, subdivision, union, intersection, and product) also play a very important role in the construction of new graphs and structures. Yan et al. [9] introduced the four operations S1,R1,Q1,andT1 on the subdivision of a graph and obtained the Wiener index of these resultant graphs S1G,R1G,Q1G,T1G. For Φ1∈S1,R1,Q1,T1, Taeri and Eliasi [10] defined the Φ1-sum graphs G1+Φ1G2 using the Cartesian product on graphs Φ1G1 and G2, where G1andG2 are connected graphs. They also studied the Wiener index of these Φ1-sum graphs. Liu et al. [11] constructed Φ-sum graphs with the help of the Cartesian product on the graphs ΦG1 and G2 and calculated the first general Zagreb index for these graphs, i.e., M1αG1+Φ1G2. Liu et al. [12] introduced the generalized Φ-sum (Φk-sum) graphs with the help of the Cartesian product on the graphs ΦkG1 and G2, where k represents some integral value. Moreover, they calculated the mathematical expressions of the Zagreb indices for these graphs, i.e., M1G1+Φ1G2 and M2G1+ΦkG2. Furthermore, Awais et al. [13, 14] computed the forgotten topological and hyper-Zagreb indices of generalized F-sum graphs based on Cartesian product in terms of its factor graphs. Recently, Awais et al. [15] computed the first general Zagreb index of Fk-sum graphs in terms of TIs of their factor graphs.In the current study, we study the generalizedΦ-sum graphs which are obtained under the operation of strong product on the graphs ΦkG1 and G2, where Φk∈Sk,Rk,Qk,Tk and k is some counting number. Mainly, we compute the Zagreb indices of these generalized Φ-sum graphs based on strong product such as M1G1⊠ΦkG2 and M2G1⊠ΦkG2. Moreover, a comparison is also organized of the generalized Φ-sum graphs G1⊠SkG2, G1⊠RkG2, G1⊠QkG2, and G1⊠TkG2 with respect to both the Zagreb indices (M1 and M2). The rest of the paper is settled as follows: Section 2 covers basic notions, Section 3 predicated on main results, and conclusively Section 4 included the application and conclusion.”


## 2. Preliminaries

A graphG1 is a structure consisting of two finite sets of vertices VG and edges EG in which pairs of vertices are connected by edges. In particular, a graph will refer to a simple undirected graph if each edge connects two distinct vertices and there are no parallel edges. Throughout the paper, the order of G1 is VG1=nG1, and the size of a G1 is EG1=eG1. Given two vertices p and z, if pz∈G1, then p and z are said to be adjacent. The strength of edges which are incident on any node p∈VG1 is known as its degree dG1p [16]. Here, we defined few topological indices.Definition 1.
LetG1 be a simple undirected graph. The first Zagreb index M1G1 and second Zagreb index M2G1 are:(1)M1G1=∑pεVG1dG1p2=∑pzεEG1dG1p+dG1z,M2G1=∑pzεEG1dG1p×dG1z.
In 1972, Trinajsti and Gutman [5] introduced these two TIs which are used in study of structure-based properties of (molecular) graphs (see [17–19]). In 1960, Sabidussi [6] introduced the strong product G1⊠G2 for two graphs G1 and G2 with vertex set as Cartesian product VG1⊠G2=VG1×VG2 such that p1,p2 and z1,z2 will be adjacent in G1⊠G2 iff p1=z1 and p2 is adjacent to z2 or p2=z2 and p1 is adjacent to z1 or p1 is adjacent to z1 and p2 is adjacent to z2. Strong product is union of tensor product and Cartesian product.Figure 1
(a)G1≅P4. (b) S3G1. (c) R3Gp1. (d) Q3G1. (e). T3G1.
(a)(b)(c)(d)(e)Figure 2
Right: =P3⊠S2P4. Left: = P3⊠R2P4.
(a)(b)Definition 2.
The four generalized operations related to the subdivision of graphs defined in [15] are given as follows:(i)
k-subdivision operation Sk=SkG1 can be made by adding k new node in each uvεEG1 of G1, where k≥1 is an integral value(ii)
k-semitotal-point graph Rk=RkG1 with node set VRk=VG1∪kEG1 and link set ERk=ESk∪EG1(iii)
k-semitotal line graph Qk=QkG1 with node set VQk=VG1∪kEG1 and link set EQk=ESk∪ELk.(iv)
k-total point graph Tk=TkG1 with node set VTk=VG1∪kEG1 and link set ETk=ESk∪ELk∪EG1 (for more details, see Figure 1).Definition 3.
LetG1andG2 be two graphs, Φk∈Sk,Rk,Qk,Tk is an operation, and ΦkH1 is obtained after applying Φk on G1 having edge-set EΦkG1 and node set VΦkG1. The generalized Φ-sum graph G1⊠ΦkG2 is a graph having node set:(2)VG1⊠ΦkG2=VΦkG1×VG2=VG1∪kEG1×VG2,such that two nodes p1,z1&p2,z2 of VG1⊠ΦkG2 are adjacent iff p1=p2∈VG1andz1,z2∈EG2 or z1=z2∈VG2andp1,p2∈EΦkG1 or [z1,z2∈EG2andp1,p2∈EΦkG1], where k≥1 is a positive number. We noticed that the generalized Φ-sum graphs G1⊠ΦkG2 contain VG2 copies of graphs ΦkG1 that are labeled with the nodes of G2. For more details, see Figures 2 and 3.Figure 3
Right: =P3⊠Q2P4. Left: = P3⊠T2P4.
(a)(b)Figure 4
(a) Graphical representation ofM1PnS1⊠Pm, M1PnQ1⊠Pm, M1PnR1⊠Pm, and M1PnT1⊠Pm by red, green, orange, and purple colour, respectively. (b) M2PnS1⊠Pm, M2PnQ1⊠Pm, M2PnR1⊠Pm, and M2PnT1⊠Pm by purple, orange, pink, and green colour, respectively.
(a)(b)


## 3. Main Results

Now, we will prove the key results ofM1G1⊠ΦkG2 and M2G1⊠ΦkG2 in terms of its factor graphs G1 and G2. We assume that G1andG2 be two simple, undirected, and connected graphs with order and size, VG1=nG1&EG1=eG1 and VG2=nG2&EG2=eG2 respectively.Theorem 1.
LetG1 and G2 be two connected graphs such that VG1≥3, VG2≥2. For k≥1,(3)aM1G1⊠SkG2=M1S1G1nG2+2eG2+82k−1eG1eG2+M1G1+4eG1M1G2+2eG2+M1G24eG1+nG1+4k−1eG1n2+M1G2,bM2G1⊠SkG2=M2G1+4eG15eG2+3M1G2+2M2G2+nG2+M1G1eG2+M1G2+M2G2+4k−1eG16eG2+3M1G2+M2G2+nG2+M2G212eG1+nG1+8eG1eG2+M1G214eG1.Proof.
Letdp,z=dG1⊠SkG2p,z be the degree of a vertex p,z in the graph G1⊠SkG2.(4)M1G1⊠SkG2=∑p,zεVG1⊠SkG2d2p,z=∑p1,z1p2,z2εEG1⊠SkG2dp1,z1+dp2,z2=∑pεVG1∑z1p2εEG2dp,z1+dp,z2+∑zεVG2∑p1p2εESkG1dp1,z+dp2,z+∑p1p2εESkG1∑z1z2εVG2dp1,z1+dp2,z2=∑1+∑2+∑3.
Consider(5)∑1=∑pεVG1∑z1z2εEG2dp,z1+dp,z2=∑pεVG∑z1z2εEH2dp+dz1+dz2+dpdz1+dz2=4eG1eG2+M1G2nG1+2M1G2eG1.∑2=∑zεVG2∑p1p2εESkG1dp1,y+dp2,z=∑zεVG2∑p1εVG1,p2εVSkG1−G1dp1,z+dp2,z+∑zεVG2∑p1,p2εVSkG1−G1dp1,z+dp2,z=∑2′+∑2″,∑2′=∑zεVG2∑p1p2εESkG1p1εVG1,p2εVSkG1−VG1dp1,z+dp2,z=∑zεVG2∑p1p2εESkG1p1εVG1,p2εVSkG1−VG1dp1+dp2+dz+dp1+dp2dz.
Since in this caseES1G=2EG, we have(6)=M1S1G1nG2+4eG1eG2+2M1S1G1eG2,∑2″=∑zεVG2∑p1p2εESkG1p1,p2εVSkG1−VG1dp1,z+dp2,z=∑zεVG2∑p1p2εESkG1p1,p2εVSkG1−VG1dp1+dp1dz+dp2+dp2dz.
Since in this caseESkG1=k−1EG1, we have(7)=∑zεVG24k−1eG1+4dzk−1eG1=4k−1eG1nG2+8eG1k−1eG2,∑3=∑p1p2εESkG1∑z1z2εVG2dp1,z1+dp2,z2=∑z1z2εVG2∑p1p2εESkG1p1εVG1,p2εVSkG1−VG1dp1,z1+dp2,z2+∑z1z2εVG2∑p1p2εESkG1p1,p2εVSkG1−VG1dp1,z1+dp2,z2=∑3′+∑3″,∑3′=∑z1z2εVG2∑p1p2εESkG1p1εVG1,p2εVSkG1−VG1dp1,z1+dp2,z2=∑z1z2εVG2∑p1εVG1,p2εVSkG1−VG1dp1+dp2+dz1+dp1dz1+dp2dz2=2eG2M1G1+4eG1+2eG1M1G2+M1G2M1G1+4eG1,∑3″=∑z1z2εVG2∑p1p2εESkG1p1,p2εVSkG1−VG1dp1,z1+dp2,z2=∑z1z2εVG2∑p1p2εESkG1p1,p2εVSkG1−VG14+2dz1+dz2=8k−1eG1eG2+4k−1eG1M1G2.
Consequently, we get(8)M1G1⊠SkG2=M1S1G1nG2+2eG2+82k−1eG1eG2+M1G1+4eG1M1G2+2eG2+M1G24eG1+nG1+4k−1eG1n2+M1G2,bM2G1⊠SkG2=∑p,zεVG1⊠SkG2d2p,z=∑p1,z1p2,z2εEG1⊠SkG2dp1,z1dp2,z2=∑pεVG1∑z1z2εEG2dp,z1dp,z2+∑zεVG2∑p1p2εESkG1dp1,zdp2,z+∑p1p2εESkG1∑z1z2εVG2dp1,z1dp2,z2=∑1+∑2+∑3.
Consider(9)∑1=∑pεVG1∑z1z2εEG2dp,z1dp,z2=∑pεVG1∑z1z2εEG2dp+dz1dp+dz2=∑pεVG1∑z1z2εEG2d2p+d2pdz1+dz2+dpdz1+dz2+2dz1dz2+dz1dz2=M1G1eG2+M1G1M1G2+M2G2+2eG1M1G2+2M2G2+M2G2nG1,∑2=∑zεVG2∑p1p2εESkG1p1εVG1,p2εVSkG1−G1dp1,zdp2,z+∑zεVG2∑p1p2εESkG1p1,p2εVSkG1−G1dp1,zdp2,z=∑2′+∑2″,∑2′=∑zεVG2∑p1p2εESkG1p1εVG1,p2εVSkG1−VG1dp1,zdp2,z=∑zεVG2∑p1p2εESkG1p1εVG1,p2εVSkG1−VG1dp1+dz+dp1dzdp2+dp2dz=∑zεVG2∑p1p2εESkG1p1εVG1,p2εVSkG1−VG1dp1dp2+2dp1dp2dz+dp2dy+dp2d2z+dp1dp2d2z=M2G1+4eG1nG2+4eG2+M1G2+8eG1eG2+4eG1M1G2,∑2″=∑zεVG2∑p1p2εESkG1p1,p2εVSkG1−VG1dp1,zdp2,z=∑zεVG2∑p1p2εESkG1p1,p2εVSkG1−VG1dp1+dp1dzdp2+dp2dz=∑zεVG2∑p1p2εESkG1p1εVG1,p2εVSkG1−VG14+8dz+4d2z=4k−1eG1nG2+4eG2+M1G2,∑3=∑z1z2εVG2∑p1p2εESkG1p1εVG1,p2εVSkG1−VG1dp1,z1dp2,z2+∑z1z2εVG2∑p1p2εESkG1p1,p2εVSkG1−VG1dp1,z1dp2,z2=∑3′+∑3″,∑3′=∑z1z2εVG2∑p1p2εESkG1p1εVG1,p2εVSkG1−VG1dp1,z1dp2,z2=∑z1z2εVG2∑p1p2εESkG1p1εVG1,p2εVSkG1−VG1dp1+dz1+dp1dz1dp2+dp2dz2=∑zεVG2∑p1p2εESkG1p1εVG1,p2εVSkG1−VG1dp1dp2+dp1dp2dz2+dp2dz1+dp1dz1dz2+dp1dp2dz1+dp1dp2dz1dz2=M2G1+4eG1eG2+2M1G2+2M2G2+8eG1M1G2+8eG1M2G2,∑3″=∑zεVG2∑p1p2εESkG1z1,p2εVSkG1−VG1dp1,z1dp2,z2=∑zεVG2∑p1p2εESkG1p1,p2εVSkG1−VG1dp1+dp1dz1dp2+dp2dz2=∑zεVG2∑p1εVG1,p2εVSkG1−VG14+4dz2+dz1+4dz1dz2=8k−1eG1eG2+M1G2+M2G2.
Therefore,(10)M2G1⊠SkG2=M2G1+4eG15eG2+3M1G2+2M2G2+nG2+M1G1eG2+M1G2+M2G2+4k−1eG16eG2+3M1G2+M2G2+nG2+M2G212eG1+nG1+8eG1eG2+M1G214eG1.Theorem 2.
LetG1 and G2 be two connected graphs such that VG1, VG2≥4. For k≥1,(11)aM1G1⊠RkG2=M1G2nG1+12eG1+4M1G1+M1G112eG2+2nG2+24eG1eG2+M1R1G1−2M1G1nG2+2eG2+4k−1eG1nG2+4eG2+M1G2,bM2G1⊠RkG2=M1G18eG2+12M1G2+4M2G2+4M2G1nG2+6eG2+3M1G2+M2R1G1−4M2G1+4k−1eG16eG2+3M1G2+2M2G2+nG2+M2G2nG1+2eG1+4eG13M1G2+3M2G2+4eG2+eG1M1G2.Proof.
Letdp,z=dG1⊠RkG2p,z be the degree of a vertex p,z in the graph G1⊠RkG2.(12)M1G1⊠RkG2=∑p,zεVG1⊠RkG2d2x,z=∑p1,z1p2,z2εEG1⊠RkG2dp1,z1+dz2,z2=∑pεVG1∑z1z2εEG2dp,z1+dp,z2+∑zεVG2∑p1p2εERkG1dp1,z+dp2,z+∑p1p2εERkG1∑z1z2εVG2dp1,z1+dp2,z2=∑1+∑2+∑3,∑1=∑pεVG1∑z1z2εEG2dp,y1+dp,y2=∑pεVG1∑z1z2εEG22dp+dz1+2dpdz1+2dp+dz2+2dpdz2=∑pεVG1∑z1z2εEG24dp+dz1+dz2+2dpdz1+dz2=8eG1eG2+M1G2nG1+4M1G2eG1,∑2=∑zεVG2∑p1p2εERkG1dp1,z+dp2,z=∑zεVG2∑p1p2εERkG1p1,p2εVG1dp1,z+dp2,z+∑zεVG2∑p1p2εERkG1p1εVG1,p2εVRkG1−G1dp1,y+dp2,z+∑zεVG2∑p1p2εERkG1p1,p2εVRkG1−G1dp1,z+dp2,y=∑2′+∑2″+∑2‴,∑2′=∑zεVG2∑p1p2εERkG1p1,p2εVG1dp1,z+dp2,z=∑zεVG2∑p1p2εERkG1p1,p2εVG1dp1+dz+dp1dz+dp2+dz+dp2dz=∑zεVG2∑p1p2εERkG1p1,p2εVG1dp1+dp2+2dz+dydp1+dp2=2nG2M1G1+4eG1eG2+4eG2M1G1,∑2″=∑zεVG2∑p1p2εERkG1p1εVG1,p2εVSkG1−VG1dp1,z+dp2,y=∑zεVG2∑p1p2εERkG1p1εVG1,p2εVSkG1−VG1dp1+dp2+dz+dp1+dp2dz=M1R1G1−2M1G1nG2+2eG2+4eG1eG2,∑2‴=∑zεVG2∑p1p2εERkG1p1,p2εVRkG1−VG1dp1,z+dp2,z=∑zεVG2∑p1p2εERkG1p1,p2εVRkG1−VG1dp1+dp2+dp1+dp2dz=4k−1eG1nG2+8k−1eG1eG2,∑3=∑p1p2εERkG1∑z1z2εVG2dp1,z1+dp2,z2=∑z1z2εVG2∑p1p2εERkG1p1,p2εVG1dp1,z1+dp2,z2+∑z1z2εVG2∑p1p2εERkG1p1εVG1,p2εVRkG1−VG1dp1,z1+dp2,z2+∑z1z2εVG2∑p1p2εERkG1p1,p2εVRkG1−VG1dp1,z1+dp2,z2=∑3′+∑3″+∑3‴,∑3′=∑z1z2εVG2∑p1p2εERkG1p1,p2εVG1dp1,z1+dp2,z2=∑z1z2εVG2∑p1p2εERkG1p1,p2εVG1dp1+dz1+dp1dz1+dp2+dz2+dp2dz2=∑z1z2εVG2∑p1p2εERkG1p1,p2εVG1dp1+dp2+dz1+dz2+dp1dz1+dp2dz2=4eG2M1G1+2M1G2eG1+2M1G2M1G1,∑3″=∑z1z2εVG2∑p1p2εERkG1p1εVG1,p2εVRkG1−VG1dp1,z1+dp2,z2=∑z1z2εVG2∑p1p2εERkG1p1εVG1,p2εVRkG1−VG1dp1+dz1+dp1dz1+dp2+dp2z2=4eG2M1G1+2M1G2eG1+2M1G2M1G1+8eG1eG2+4eG1M1G2,∑3‴=∑z1z2εVG2∑p1p2εERkG1p1,p2εVRkG1−VG1dp1,z1+dp2,z2=∑z1z2εVG2∑p1p2εERkG1p1,p2εVRkG1−VG1dp1+dp2+dp1dz1+dp2dz2=8k−1eG1eG2+4k−1eG1M1G2.
Hence,(13)M1G1⊠RkG2=M1G2nG1+12eG1+4M1G1+M1G112eG2+2nG2+24eG1eG2+M1R1G1−2M1G1nG2+2eG2+4k−1eG1nG2+4eG2+M1G2,bM2G1⊠RkG2=∑x,zεVG1⊠RkG2d2p,z=∑p1,z1p2,z2εEG1⊠RkG2dp1,z1dp2,z2=∑pεVG1∑z1z2εEG2dp,z1dp,z2+∑zεVG2∑p1p2εERkG1dp1,zdp2,z+∑p1p2εERkG1∑z1z2εVG2dp1,z1dp2,z2=∑1+∑2+∑3.
Consider(14)∑1=∑pεVG1∑z1z2εEG2dp,z1dp,z2=∑pεVG1∑z1z2εEG2dp+dz1dp+dz2=∑pεVG1∑z1z2εEG2d2p+d2pdz1+dz2+d2pdz1+dz2+2dpdz1dz2+dz1dz2=∑z1z2εEG24M1G1+4M1G1dz1+dz2+dz1dz2+4eG1dz1+dz2+2dz1dz2+dz1dz2nG1=4M1G1eG2+M1G2+M2G2+4eG1M1G2+2M2G2+M2G2nG1,∑2=∑zεVG2∑p1p2εERkG1dp1,zdp2,z=∑zεVG2∑p1p2εERkG1p1,p2εVG1dp1,zdp2,z+∑zεVG2∑p1p2εERkG1p1εVG1,p2εVRkG1−G1dp1,zdp2,z+∑zεVG2∑p1p2εERkG1p1,p2εVRkG1−G1dp1,zdp2,z=∑2′+∑2″+∑2‴,∑2′=∑zεVG2∑p1p2εERkG1dp1,zdp2,z=∑zεVG2∑p1p2εERkG1dp1+dz+dp1dzdp2+dz+dp2dz=∑zεVG2∑p1p2εERkG14dp1dp2+2dp1dz+4dp1dp2dz+2dp2dz+d2z+2dp2d2z+4dp1dp2dz+4dp1dp2d2z=4M2G1nG2+4eG2+M1G2+2M1G1M1G2+2eG2+eG1M1G2,∑2″=∑zεVG2∑p1p2εERkG1p1εVG1,p2εVRkG1−VG1dp1,zdp2,z=∑zεVG2∑p1p2εERkG1p1εVG1,p2εVRkG1−VG1dp1+dz+dp1dzdp2+dp2dz=∑zεVG2∑p1p2εERkG1p1εVG1,p2εVRkG1−VG12dp11+dz+dy+d2z+2dz+2d2y=M2R1G1−4M2G1nG2+4eG2+M1G2+8eG1eG2+4eG1M1G2,∑2″′=∑zεVG2∑p1p2εERkG1p1,p2εVRkG1−VG1dp1,zdp2,z=∑zεVG2∑p1p2εERkG1p1,p2εVRkG1−VG1dp1+dp1dzdp2+dp2dz=∑zεVG2∑p1εVG1,p2εVRkG1−VG14+8dy+4d2z=4k−1eG1nG2+4eG2+M1G2,∑3=∑z1z2εVG2∑p1p2εERkG1p1,p2εVG1dp1,z1dp2,z2+∑z1z2εVG2∑p1p2εERkG1p1εVG1,p2εVRkG1−VG1dp1,z1dp2,z2+∑z1z2εVG2∑p1p2εERkG1p1,p2εVRkG1−VG1dp1,z1dp2,z2=∑3′+∑3″+∑3‴,∑3′=∑z1z2εVG2∑p1p2εERkG1p1,p2εVG1dp1,z1dp2,z2=∑z1z2εVG2∑p1p2εERkG1p1,p2εVG14dp1dp2+2dp1dz2+dp2dz1+dz1dz2+4dp1dp2dz1dz2+4dp1dp2dz1+dz2+2dp1+dp2dz1dz2=8M2G1eG2+2M1G1M1G2+2M2G2eG1+8M2G1M2G2+M1G2+4M1G1M2G2,∑3″=∑z1z2εVG2∑p1p2εERkG1p1εVG1,p2εVRkG1−VG1dp1,z1dp2,z2=∑z1z2εVG2∑p1p2εERkG1p1εVG1,p2εVRkG1−VG1dp1+dz1+dp1dz1dp2+dp2dz2=∑z1z2εVG2∑p1p2εERkG1p1εVG1,p2εVRkG1−VG14dp11+dz2+dz1+dz1dz2+2dz1+dz1dz2=M2R1G1−4M2G12eG2+2M1G2+2M2G2+4eG1M1G2+M2G2,∑3″′=∑z1z2εVG2∑p1p2εERkG1p1,p2εVRkG1−VG1dp1,z1dp2,z2=∑z1z2εVG2∑p1p2εERkG1p1,p2εVRkG1−VG1dp1+dp1dz1dp2+dp2dz2=∑z1z2εVG2∑p1p2εERkG1p1εVG1,p2εVRkG1−VG1dp1dp21+dy2+dz1+dz1dz2=8k−1eG1eG2+M1G2+M2G2.
Consequently, we have(15)M2G1⊠RkG2=M1G18eG2+12M1G2+4M2G2+4M2G1nG2+6eG2+3M1G2+M2R1G1−4M2G1+4k−1eG16eG2+3M1G2+2M2G2+nG2+M2G2nG1+2eG1+4eG13M1G2+3M2G2+4eG2+eG1M1G2.Theorem 3.
LetG1 and G2 be two connected graphs such that VG1, VG2≥4. For k≥1,(16)aM1G1⊠QkG2=kM3G1+2M2G1−2M1G1nG2+4eG2+M1G2+8eG1eG2+M1G12k−1+3nG2+4eG2+M1G2+M1G2nG1+4eG1,bM2G1⊠QkG2=knG2+6eG2+3M1G2+2M2G212∑vεVG1dG14v−dG13v+∑uvεVG1rdG1udG1v+∑vεVG1dG12v∑uεVG1uvεEG1dG1u−2M2G1+M2G24eG1+nG1+2eG1M1G2+M1G15e2+5M1G2+5M2G2+kM3G1+2M2G16eG2+3M1G2+2M2G2+nG2.Proof.
Letdp,z=dG1⊠QkG2p,z be the degree of a vertex p,z in the graph G1⊠QkG2.(17)M1G1⊠QkG2=∑p,zεVG1⊠QkG2d2p,z=∑p1,z1p2,z2εEG1⊠QkG2dp1,z1+dp2,z2=∑pεVG1∑z1z2εEG2dp,z1+dp,z2+∑zεVG2∑p1p2εEQkG1dp1,z+dp2,z+∑p1p2εEQkG1∑z1z2εVG2dp1,z1+dp2,z2=∑1+∑2+∑3,∑1=∑pεVG1∑z1z2εEG2dp,z1+dp,z2=∑pεVG1∑z1z2εEG2dp+dz1+dpdz1+dp+dz2+dpdz2=∑pεVG∑z1z2εEG22dp+dz1+dz2+dpdz1+dz2=4eG1eG2+M1G2nG1+2M1G2eG1,∑2=∑zεVG2∑p1p2εEQkG1dp1,z+dp2,z=∑zεVG2∑p1p2εEQkG1p1εVG1,p2εVQkG1−G1dp1,z+dp2,z+∑zεVG2∑p1p2εEQkG1p1,p2εVQkG1−G1dp1,y+dp2,z=∑2′+∑2″,∑2′=∑zεVG2∑p1p2εEQkG1p1εVG1,p2εVQkG1−VG1dp1,z+dp2,z=∑zεVG2∑p1p2εEQkG1p1εVG1,p2εVQkG1−VG1dp1+dz+dp1dy+dp2+dp2dz=M1G1nG2+4eG1eG2+6M1G1eG2+2nG2M1G1,∑2″=∑zεVG2∑p1p2εEQkG1p1p2εVQkG1−VG1dp1,zdp2,z.
Now, we split this sum into two parts for the vertices,p1 and p2, where p1p2εVQkG1−VG1. Assume that ∑2″=∑2″a+∑2″b, where ∑2″a cover the edges of QkG1 which are in the same edges of G1 and ∑2″b of QkG1 in two different adjacent edges of G1.(18)∑2″a=∑zεVG2∑p1p2εEQkG1p1,p2εVQkG1−VG1dp1,z+dp2,z=∑zεVG2∑p1p2εEQkG1p1,p2εVQkG1−VG1dp1+dp1dz+dp2+dp2dz=2k−1nG2M1G1+4k−1eG2M1G1,∑2″b=∑zεVG2∑p1p2εEQkG1p1,p2εVQkG1−VG1dp1,y+dp2,z=∑zεVG2∑p1p2εEQkG1p1,p2εVQkG1−VG1dp1+dp1dz+dp2+dp2dz=∑zεVG2∑u,v,wεVG1uv,vwεEG1du+dv+dv+dw+du+dv+dv+dwdz=kM3G1+2M2G1−2M1G1nG2+2eG2,∑3=∑p1p2εEQkG1∑z1z2εVG2dp1,z1+dp2,z2=∑z1z2εVG2∑p1p2εEQkG1p1εVG1,p2εVQkG1−VG1dp1,z1+dp2,z2+∑z1z2εVG2∑p1p2εEQkG1p1,p2εVQkG1−VG1dp1,z1+dp2,z2=∑3′+∑3″,∑3′=∑z1z2εVG2∑p1p2εEQkG1p1,p2εVG1dp1,z1+dp2,z2=∑z1z2εVG2∑p1p2εEQkG1p1,p2εVG1dp1+dz1+dp1dz1+dp2+dz2+dp2z2=6eG2M1G1+2M1G2eG1+3M1G2M1G1.
Now, we split this sum into two parts for the vertices,p1 and p2, where p1p2ϵVQkΓ1−VΓ1. Assume that ∑3″=∑3″a+∑3″b, where ∑3″a cover the edges of QkG1 which are in the same edges of G1 and ∑3″b of QkG1 in two different adjacent edges of G1.(19)∑3″a=∑z1z2εVG2∑p1p2εEQkG1p1εVG1,p2εVQkG1−VG1dp1,z1+dp2,z2=∑z1z2εVG2∑p1p2εEQkG1p1εVG1,p2εVQkG1−VG1dp1+dp1dz1+dp2+dp2dz2=4k−1eG2M1G1+2k−1M1G2M1G1,∑3″b=∑z1z2εVG2∑p1p2εEQkG1p1,p2εVQkG1−VG1dp1,z1+dp2,z2=∑z1z2εVG2∑p1p2εEQkG1p1,p2εVQkG1−VG1dp1+dp2+dp1dz1+dp2dz2=kM3G1+2M2G1−2M1G12eG2+M1G2.
Consequently, we have(20)M1G1⊠QkG2=kM3G1+2M2G1−2M1G1nG2+4eG2+M1G2+8eG1eG2+M1G12k−1+3nG2+4eG2+M1G2+M1G2nG1+4eG1.
Next,(21)bM2G1⊠QkG2=∑p,zεVG1⊠QkG2d2p,z=∑p1,1p2,z2εEG1⊠QkG2dp1,z1dp2,z2=∑pεVG1∑z1z2εEG2dp,z1dp,z2+∑zεVG2∑p1p2εEQkG1dp1,zdp2,z+∑p1p2εEQkG1∑z1z2εVG2dp1,z1dp2,z2=∑1+∑2+∑3.
Consider(22)∑1=∑pεVG1∑z1z2εEG2dp,z1dp,z2=∑pεVG1∑z1z2εEG2dp+dz1dp+dz2=M1G1eG2+M1G1M1G2+M2G2+2eG1M1G2+2M2G2+M2G2nG1,∑2=∑zεVG2∑p1p2εEQkG1dp1,zdp2,z=∑zεVG2∑p1p2εEQkG1p1εVG1p2εVQkG1−VG2dp1,zdp2,z+∑zεVG2∑p1p2εEQkG1p1p2εVQkG1−VG1dp1,zdp2,z=∑′2+∑″2,∑′2=∑zεVG2∑p1p2εEQkG1p1εVG1p2εVQkG1−VG1dp1,zdp2,z=∑zεVG2∑p1p2εEQkG1p1εVG1p2εVQkG1−VG1dp1+dz+dp1dzdp2+dz+dp2dz=∑zεVG2∑p1p2εEQkG1dp1dp21+2dz+d2z+dp2dz+d2z=M3G1+2M2G1nG2+4eG2+M1G2+2M1G12eG2+M1G2,∑2″=∑zεVG2∑p1p2εEQkG1p1p2εVQkG1−VG1dp1,zdp2,z.
Now, we split this sum into two parts for the vertices,p1 and p2, where p1p2εVQkG1−VG1. Assume that ∑2″a which are in the same edges of G1 and ∑2″b of QkG1 in two different adjacent edges of G1.(23)∑2″a=∑zεVG2∑p1p2εEQkG1p1p2εVQkG1−VG1dp1+dp1dzdp2+dp2dz=∑zεVG2∑p1p2εEQkG1p1p2εVQkG1−VG1dp1dp2+2dp1dp2dz+dp1dp2d2z=∑zεVG2∑p1p2εEQkG1p1p2εVQkG1−VG1dp1dp21+2dz+d2z=k−1M3G1+2M2G1nG2+4eG2+M1G2,∑2″b=∑zεVG2∑p1p2εEQkG1p1p2εVQkG1−VG1dp1+dp1dzdp2+dp2dz=k∑zεVG21+2dz+d2z∑p1p2εEQkG1p1p2εVQkG1−VG1dp1dp2=knG2+4eG2+M1G2∑uvεEG1vwεEG2dG1u+dG1vdG1v+dG1w,where p1 is the added vertex in the edge uv and p2 is added vertex in the edges vw of G1:(24)=knG2+4eG2+M1G212∑vεVG1dG14v−dG13v+∑uvεVG1rdG1udG1v+∑vεVG1dG12v∑uεVG1uvεEG1dG1u−2M2G1,where r is the number of neighbors which are common vertices of u and v in Γ1.(25)∑3=∑z1z2εVG2∑p1p2εEQkG1p1εVG1,p2εVQkG1−VG1dp1,z1dp2,z2+∑y1z2εVG2∑p1ps2εEQkG1p1,p2εVQkG1−VG1dp1,z1dp2,z2=∑3′+∑3″,∑3′=∑z1z2εVG2∑p1p2εEQkG1p1εVG1,p2εVQkG1−VG1dp1,z1dp2,z2=∑z1z2εVG2∑p1p2εEQkG1p1εVG1,p2εVQkG1−VG1dp1+dz1+dp1dz1dp2+dp2dz2=M3G1+2M2G12eG2+2M1G2+2M2G2+2M1G12M2G2+M1G2,∑3″=∑z1z2εVG2∑p1p2εEQkG1p1p2εVQkG1−VG1dp1,zdp2,z.
Now, we split this sum into two parts for the vertices,p1 and p2, where p1p2εVQkG1−VG1. Assume that ∑3″a which are in the same edges of G1 and ∑3″b of QkG1 in two different adjacent edges of G1.(26)∑3″a=∑z1z2εVG2∑p1p2εEQkG1p1,p2εVQkG1−VG1dp1,z1dp2,z2=∑z1z2εVG2∑p1p2εEQkG1p1,p2εVQkG1−VG1dp1+dp1dz1dp2+dp2dz2=∑z1z2εVG21+dz2+dz1+dz1dz2∑p1p2εEQkG1p1εVG1,p2εVQkG1−VG1dp1dp2=k−12eG2+2M1G2+2M2G2M3G1+2M2G1,∑3″b=∑z1z2εVG2∑p1p2εEQkG1p1,p2εVQkG1−VG1dp1,z1dp2,z2=∑z1z2εVG2∑p1p2εEQkG1p1,p2εVQkG1−VG1dp1+dp1dz1dp2+dp2dz2=∑z1z2εVG21+dz2+dz1+dz1dz2∑p1p2εEQkG1p1εVG1,p2εVQkG1−VG1dp1dp2=k2eG2+2M1G2+2M2G2∑uvεEG1vwεEG2dG1u+dG1vdG1v+dG1w,where p1 is the added vertex in the edge uv and p2 is added vertex in the edges vw of G1:(27)=k2eG2+2M1G2+2M2G212∑vεVG1dG14v−dG13v+∑uvεVG1rdG1udG1v+∑vεVG1dG12v∑uεVG1uvεEG1dG1u−2M2G1,where r is the number of neighbors which are common vertices of u and v in G1.
Consequently, we have(28)M2G1⊠QkG2=knG2+6eG2+3M1G2+2M2G212∑vεVG1dG14v−dG13v+∑uvεVG1rdG1udG1v+∑vεVG1dG12v∑uεVG1uvεEG1dG1u−2M2G1+M2G24eG1+nG1+2eG1M1G2+M1G15e2+5M1G2+5M2G2+kM3G1+2M2G16eG2+3M1G2+2M2G2+nG2.Theorem 4.
LetG1 and G2 be two connected graphs such that VG1, VG2≥4. For k≥1,(29)aM1G1⊠TkG2=M1G28eG1+nG1+6M1G1+2k−1M1G12eG2+M1G2+kM3G1+2M2G1−2M1G14eG2+nG2+M1G2+16eG1eG2+M1G1211+keG2+2k+2nG2,bM2G1⊠TkG2=knG2+6eG2+3M1G2+2M2G212∑vεVG1dG14v−dG13v+∑uvεVG1rdG1udG1v+∑vεVG1dG12v∑uεVG1uvεEG1dG1u−2M2G1+M2G14nG2+24eG2+8M2G2+12M1G2+kM3G1+2M2G16eG2+nG2+3M1G2+2M2G2+5M1G2eG1+M2G210eG1+nG1+M1G110eG2+11M1G2+10M2G2.


## 4. Applications and Conclusion

In this section, we have computed the first and second Zagreb indices of generalizedΦ-sum graphs based on strong product as application of Theorem 1 to Theorem 4 for k=1 given as follows. We also find the subclass with better Zagreb indices as presented in Tables 1 and 2 and Figure 4(i)
S1-sum:(30)aM1G1⊠S1G2=M1S1G1nG2+2eG2+8eG1eG2+M1G1+4eG1M1G2+2eG2+M1G24eG1+nG1,bM2G1⊠S1G2=M2G1+4eG15eG2+3M1G2+2M2G2+nG2+14eG1M1G2+M2G212eG1+nG1+M1G1eG2+M1G2+M2G2+8eG1eG2.(ii)
R1-sum:(31)aM1G1⊠R1G2=24eG1eG2+M1G2nG1+12eG1+M1G112eG2+2nG2+M1R1G1−2M1G1nG2+2eG2,bM2G1⊠R1G2=2M1G14eG2+6M1G2+2M2G2+4eG13M1G2+3M2G2+4eG2+M2R1G1−4M2G16eG2+3M1G2+2M2G2+nG2+eG1M1G2+4M2G1nG2+6eG2+3M1G2+M2G2nG1+2eG1.(iii)
Q1-sum:(32)aM1G1⊠Q1G2=M3G1+2M2G1−2M1G1nG2+4eG2+M1G2+8eG1eG2+3M1G1nG2+4eG2+M1G2+M1G2nG1+4eG1,bM2G1⊠Q1G2=nG2+6eG2+3M1G2+2M2G212∑vεVG1dG14v−dG13v+∑uvεVG1rdG1udG1v+∑vεVG1dG12v∑uεVG1uvεEG1dG1u−2M2G1+M2G24eG1+nG1+2eG1M1G2+M1G15e2+5M1G2+5M2G2+M3G1+2M2G16eG2+3M1G2+2M2G2+nG2,(iv)
T1-sum:(33)aM1G1⊠T1G2=M1G28eG1+nG1+6M1G1+M1G124eG2+6nG2+M3G1+2M2G1−2M1G14eG2+nG2+M1G2+16eG1eG2,bM2G1⊠T1G2=nG2+6eG2+3M1G2+2M2G212∑vεVG1dG14v−dG13v+∑uvεVG1rdG1udG1v+∑vεVG1dG12v∑uεVG1uvεEG1dG1u−2M2G1+M3G1+2M2G16eG2+nG2+3M1G2+2M2G2+5M1G2eG1+M2G210eG1+nG1+M1G110eG2+11M1G2+10M2G2+M2G14nG2+24eG2+8M2G2+12M1G2.Table 1
M1Pn⊠Φ1Pm for path graph.
[m, n]M1Pn⊠S1PmM1Pn⊠R1PmM1Pn⊠Q1PmM1Pn⊠T1Pm(3, 3)336722472892(4, 4)804177612202204(5, 5)1086329422964108(6, 6)1854704037008368(7, 7)26069878543211848Table 2
M2Pn⊠Φ1Pm for path graph.
[m, n]M2Pn⊠S1PmM2Pn⊠R1PmM2Pn⊠Q1PmM2Pn⊠T1Pm(3, 3)824238415352899(4, 4)2262523050029700(5, 5)440411636967417818(6, 6)6738175461581828696(7, 7)10740258162302640942Now, we close our discussion with the conclusion that both the Zagreb indices of the generalized T-sum graph are dominant among the Zagreb indices of all the generalized sum graphs as shown in Figure4. We also conclude that, in the generalized T-sum graph, the number of vertices (atoms) and edges (bonds) between them are more than the other graphs of this family for each integral values of k. Thus, the role of Zagreb indices for generalized T-sum graph remains dominant for each integral values of k. However, the problem is still open to find different degree- and distance-based TIs for the generalized sum graphs obtained under the various operations of product of graphs.



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