# Forgotten Index of Generalized Operations on Graphs

Authors: Muhammad Javaid; Saira Javed; Saima Q. Memon; Abdulaziz Mohammed Alanazi
Journal: Journal of Chemistry
 (2021)

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## Abstract

In theoretical chemistry, several distance-based, degree-based, and counting polynomial-related topological indices (TIs) are used to investigate the different chemical and structural properties of the molecular graphs. Furtula and Gutman redefined theF-index as the sum of cubes of degrees of the vertices of the molecular graphs to study the different properties of their structure-dependency. In this paper, we compute F-index of generalized sum graphs in terms of various TIs of their factor graphs, where generalized sum graphs are obtained by using four generalized subdivision-related operations and the strong product of graphs. We have analyzed our results through the numerical tables and the graphical presentations for the particular generalized sum graphs constructed with the help of path (alkane) graphs.


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## Body

## 1. Introduction

Throughout the paper, we consider a simple and undirected graphH=VH,EH with vertex-set VH=v1,v2,v3,…,vn and edge-set E=e1,e2,…,em⊆VH×VH, where n=VG and m=EG. A molecular graph is a connected and undirected graph in which atoms are presented by vertices, and chemical bonds between these atoms are shown by edges (see Figure 1). For a finite set of graphs C and set of real numbers ℝ, the function I:C⟶ℝ defined by IH=∑u∈VHfu is called a degree-based topological index (TI), where the graph H belongs to C and fu is a degree-function from the vertex-set of H to the degree-set of its vertices. It is important to know that IH=IK if and only if H is isomorphic to K. For more details, see [1, 2].Figure 1
Clearly, structural formula of butane is isomorphic toP4.Graph-theoretic modeling of the molecular graphs plays a fundamental part in the analysis of the quantitative structures activity/property relationships (QSAR/QSPR). In chemistry, the study of structural relationships is used to characterize the various physicochemical properties of organic molecules such as surface tension, density, melting, freezing point, solubility, heat of evaporation, and heat of formation [3]. In last two decades, many TIs are introduced, but degree-based TIs got much more attention of the researchers, see the latest survey [4]. In 1947, Winer introduced the first distance-based TI to compute the boiling point of paraffin [5]. Also, we refer [6].In molecular graph theory, the different operations on a graph perform a fundamental role in the formation of different new classes of graphs. Yan et al. [7] introduced the four operations S1, R1, Q1, and T1 on a graph H and computed the Wiener indices for the graphs F1H, where F1∈S1,R1,Q1,T1. Eliasi and Taeri [8] defined the F1-sum graphs H1+F1H2 using the Cartesian product on graphs F1H1 and H2, where H1 and H2 are any two simple and connected graphs. They also computed the Wiener indices for these F1-sum graphs. Furthermore, Deng et al. [9] calculated the M1 and M2 Zagreb indices, Imran and Shehnaz [10] computed the F-index, Liu et al. [11] computed the first general Zagreb, Chu et al. [12] calculated the bounds of first general Zagreb index and general Randic index, and Sarala et al. [13] computed the F-index for these F1-sum graphs under the Cartesian and strong products. We also refer [14–20].Recently, fork≥1, Liu et al. [21] defined the generalized F-sum (Fk-sum) graphs using Cartesian product and computed their Zagreb indices. Moreover, Awais et al. computed the F-index [22], hyper-Zagreb [23], and FGZ index [24] for these generalized F-sum graphs. Recently, Javaid et al. [25] computed first and second Zagreb indices for the generalized F-sum graphs under the strong product.In this paper, we computeF-index of generalized sum graphs in terms of various TIs of their factor graphs, where generalized sum graphs are obtained by using four generalized subdivision-related operations and the strong product of graphs. We have analyzed our results through the numerical tables and the graphical presentations for the particular generalized sum graphs constructed with the help of path (alkane) graphs. The remaining paper is managed as follows: Section 2 consists of elementary definitions, Section 3 includes main results, and Section 4 covers the application and conclusion.


## 2. Preliminaries

### 2.1. Degree-Based Topological Indices

In 1972, Trainajsi and Gutman defined first and second Zagreb indices that are utilized to find theπ-electron energy of molecular graphs [26]. Let H be any graph (molecular structure), then the first and second Zagreb indices are defined as M1H=∑v∈VHd2v and M2H=∑uv∈EHdudv.In 2015, Furtula and Gutman [22] redefined a TI called by forgotten TI [27]. The forgotten TI of a (molecular) graph H is defined as(1)FH=∑v∈VHd3v.They also verified that the different predictive abilities ofF-index and first Zagreb index are same. In particular, both the indices yield the values of correlation coefficients for entropy and acentric factor more than 0.95. For more results on its mathematical properties and chemical applications, see [23].


### 2.2. Four Generalized Operations

LetH1 be a (molecular) graph, then for the integral value k≥1, the graphs (FkH1 under the four generalized operations (Fk∈Sk,Rk,Qk,Tk) on H1 are defined as follows [21]:(i)
k-subdivision graph: we add k new vertices in every edge of H1 and obtain the new k-subdivision graph SkH1(ii)
k-semitotal point graph: the graph k-semitotal point RkH1 is defined from the graph SkH1 by joining the vertices of SkH1 which were adjacent in H1(iii)
k-semitotal line graph: the k-semitotal line graph QkH1 is defined from the graph SkH1 by joining the newly added k vertices for each incident pair of edges of H1(iv)
k-total point graph: the k-total point graph is defined from the graph SkH1 by applying both the operations Rk and Qk, respectively. For more details, see Figures 2–4.Figure 2
(a)H1≅C3. (b) S3H1.
(a)(b)Figure 3
(a)R3H1. (b) Q3H1.
(a)(b)Figure 4
T3H1.


### 2.3. Generalized Sum Operation for Strong Product

LetH1 and H2 be two graphs, Fk∈Sk,Rk,Qk,Tk presents generalized operations and FkH1 is obtained after applying Fk on H1 having edge-set EFkH1 and node-set VFkH1. The generalized F-sum graphs H1Fk⊠H2 under the operation of strong product is a graph having a vertex-set:(2)VH1Fk⊠H2=VFkH1×VH2=VH1∪kEH1×VH2,such that two vertices s1,t1 and s2,t2 of VH1Fk⊠H2 are adjacent iff s1=s2∈VH1 and t1,t2∈EH2 or t1=t2∈VH2 and s1,s2∈EFkH1 or t1,t2∈EH2 and s1,s2∈EFkH1, where k≥1 is natural number. Furthermore, the generalized F-sum graphs H1Fk⊠H2 contain VH2 copies of graphs FkH1 that are labeled with the vertices of H2. For more explanation, see Figures 5–7.Figure 5
(a)H1 = C3. (b) H2 = P2.
(a)(b)Figure 6
(a)H1S2⊠H2. (b) H1R2⊠H2.
(a)(b)Figure 7
(a)H1Q2⊠H2. (b) H1T2⊠H2.
(a)(b)


## 2.1. Degree-Based Topological Indices

In 1972, Trainajsi and Gutman defined first and second Zagreb indices that are utilized to find theπ-electron energy of molecular graphs [26]. Let H be any graph (molecular structure), then the first and second Zagreb indices are defined as M1H=∑v∈VHd2v and M2H=∑uv∈EHdudv.In 2015, Furtula and Gutman [22] redefined a TI called by forgotten TI [27]. The forgotten TI of a (molecular) graph H is defined as(1)FH=∑v∈VHd3v.They also verified that the different predictive abilities ofF-index and first Zagreb index are same. In particular, both the indices yield the values of correlation coefficients for entropy and acentric factor more than 0.95. For more results on its mathematical properties and chemical applications, see [23].


## 2.2. Four Generalized Operations

LetH1 be a (molecular) graph, then for the integral value k≥1, the graphs (FkH1 under the four generalized operations (Fk∈Sk,Rk,Qk,Tk) on H1 are defined as follows [21]:(i)
k-subdivision graph: we add k new vertices in every edge of H1 and obtain the new k-subdivision graph SkH1(ii)
k-semitotal point graph: the graph k-semitotal point RkH1 is defined from the graph SkH1 by joining the vertices of SkH1 which were adjacent in H1(iii)
k-semitotal line graph: the k-semitotal line graph QkH1 is defined from the graph SkH1 by joining the newly added k vertices for each incident pair of edges of H1(iv)
k-total point graph: the k-total point graph is defined from the graph SkH1 by applying both the operations Rk and Qk, respectively. For more details, see Figures 2–4.Figure 2
(a)H1≅C3. (b) S3H1.
(a)(b)Figure 3
(a)R3H1. (b) Q3H1.
(a)(b)Figure 4
T3H1.


## 2.3. Generalized Sum Operation for Strong Product

LetH1 and H2 be two graphs, Fk∈Sk,Rk,Qk,Tk presents generalized operations and FkH1 is obtained after applying Fk on H1 having edge-set EFkH1 and node-set VFkH1. The generalized F-sum graphs H1Fk⊠H2 under the operation of strong product is a graph having a vertex-set:(2)VH1Fk⊠H2=VFkH1×VH2=VH1∪kEH1×VH2,such that two vertices s1,t1 and s2,t2 of VH1Fk⊠H2 are adjacent iff s1=s2∈VH1 and t1,t2∈EH2 or t1=t2∈VH2 and s1,s2∈EFkH1 or t1,t2∈EH2 and s1,s2∈EFkH1, where k≥1 is natural number. Furthermore, the generalized F-sum graphs H1Fk⊠H2 contain VH2 copies of graphs FkH1 that are labeled with the vertices of H2. For more explanation, see Figures 5–7.Figure 5
(a)H1 = C3. (b) H2 = P2.
(a)(b)Figure 6
(a)H1S2⊠H2. (b) H1R2⊠H2.
(a)(b)Figure 7
(a)H1Q2⊠H2. (b) H1T2⊠H2.
(a)(b)


## 3. Methodology

This section presents the main results.Theorem 1.
LetH1 and H2 be two connected graphs, then(3)FH1Sk⊠H2=VH2+3M1H2+6EH2FH1+VH1+3M1H1+14EH1FH2+6EH2M1H1+30EH1M1H2+6M1H1M1H2+48|EH1EH2+8k−1EH16EH2+VH2+FH2+3M1H2+FH1FH2+8|VH2EH1.Proof.
Letds,t=dH1Sk⊠H2s,t is the degree of the vertex s,t in H1Sk⊠H2, then(4)FH1Sk⊠H2=∑s,tϵVH1Sk⊠H2ds,t3=∑s1,t1s2,t2ϵEH1Sk⊠H2ds1,t12+ds2,t22=∑sϵVH1∑t1t2ϵEH2ds,t12+ds,t22+∑tϵVH2∑s1s2ϵESkH1ds1,t2+ds2,t2+∑t1t2ϵEH2∑s1s2ϵESkH1ds1,t12+ds2,t22=∑A+∑B+∑C.∑A=∑sϵVH1∑t1t2ϵEH2ds+dt1+dsdt12+ds+dt2+dsdt22=∑sϵVH1∑t1t2ϵEH22ds2+dt12+dt22+d2sd2t1+d2t2+2dsdt1+dt2+2d2sdt1+dt2+2dsd2t1+d2t2=2EH2M1H1+VH1FH2+4EH1M1H2+M1H1FH2+2M1H1M1H2+4EH1FH2,∑B=∑tϵVH2∑s1s2εESkH1d2s1,t+d2s2,t=∑tεVH2∑s1εVH1,s2εVSkH1−H1d2s1,t+d2s2,t+∑tεVH2∑s1,s2εVSkH1−H1d2s1,t+d2s2,t=∑B1+∑B2.∑B1=∑tεVH2∑s1s2ϵESkH1s1ϵVH1,s2ϵVSkH1−H1ds1+dt+ds1dt2+ds2+ds2dt2=∑tεVH2∑s1s2εESkH1s1εVH1,s2εVSkH1−H1ds12+ds22+dt2ds12+ds22+2dtds12+ds22+2dtds1+2d2tds1+d2t.
Sinces1ϵVH1 and s2ϵVSkH1−H1), therefore(5)=VH2FS1H1+2EH1M1H2+4EH2M1H1+4EH2FS1H1+M1H2FS1H1+2M1H1M1H2.
We know thatFS1H1=FH1+8EH1.(6)=VH2+4|EH2+M1H2FH1+8EH1+2EH1M1H2+4|EH2M1H1+4|EH2+2M1H1M1H2,∑B2=∑tϵVH2∑s1s2ϵESkH1s1,s2ϵVSkH1−H1ds1+ds1dt2+ds2+ds2dt2=∑tϵVH2∑s1s2ϵESkH1s1,s2ϵVSkH1−H1ds12+ds22+dt2ds12+ds22+2dtds12+ds22=∑tϵVH2∑s1s2ϵESkH1s1,s2ϵVSkH1−H122+22+8dH2t2+16dH2t.
Since in this caseESkH1=EH1k−1, so(7)=8k−1|VH2EH1|+8k−1M1H2|EH1|+32k−1|EH2EH1|,∑C=∑t1t2ϵEH2∑s1s2ϵESkH1d2s1,t1+d2s2,t2=∑t1t2ϵEH2∑s1ϵVH1,s2ϵVSkH1−H1d2s1,t1+d2s2,t2+∑t1t2ϵEH2∑s1,s2ϵVSkH1−H1d2s1,t1+d2s2,t2=∑C1+∑C1.∑C1=∑t1t2ϵVH2∑s1s2ϵESkH1s1ϵVH1,s2ϵVSkH1−H1ds1+dt1+ds1dt12+ds2+ds2dt22=∑tϵVH2∑s1s2ϵESkH1s1ϵVH1,s2ϵVSkH1−H1ds12+ds22+dt12ds12+dt22ds22+2dt1ds12+2dt2ds22+2dt1ds1+2d2t1ds1+dH22t1=2EH2FG1+16EH1EH2|+2EH1FG2+FG1FG2+2M1H1M1H2+2FG1M1H2+2M1H1FG2+8EH1FG2+16EH1M1H2,∑C2=∑t1t2ϵEH2∑s1s2ϵESkH1s1,s2ϵVSkH1−H1ds1+ds1dt12+ds2+ds2dt22=∑t1t2ϵEH2∑s1s2ϵESkH1s1,s2ϵVSkH1−H18+4d2t1+d2t2+8dt1+dt2=16k−1|EH2EH1|+8k−1FH2|EH1|+16k−1|EH1|M1H2.
We arrive at our desired result.Theorem 2.
LetH1 and H2 be two connected graphs, then(8)FH1Rk⊠H2=8VH2+6EH2|FH1+VH1+20|EH1FH2+8FH1FH2+24EH2M1H1+36EH1M1H2+24M1H1M1H2+24FH1M1H2+8|VH2EH1|+8k−1EH1VH2+FH2+4EH2+3M1H2+48|EH1EH2|+12FH2M1H1.Proof.
Letds,t=dH1Rk⊠H2s,t is the degree of the vertex s,t in H1Rk⊠H2, then(9)FH1Rk⊠H2=∑s,tεVH1Rk⊠H2ds,t3=∑s1,t1s2,t2εEH1Rk⊠H2ds1,t12+ds2,t22=∑sϵVH1∑t1t2ϵEH2ds,t12+ds,t22+∑tϵVH2∑s1s2ϵERkH1ds1,t2+ds2,t2+∑t1t2ϵEH2∑s1s2ϵERkH1ds1,t12+ds2,t22=∑A+∑B+∑C.∑A=∑sϵVH1∑t1t2ϵEH2ds,t12+ds,t22=8EH2M1H1+VH1FH2+4M1H1FH2+8EH1M1H2+8M1H1M1H2+8EH1FH2,∑B=∑tϵVH2∑s1s2ϵVH1,s1,s2ϵVH1d2s1,t+d2s2,t+∑tϵVH2∑s1ϵVH1,s2ϵVRkH1−H1d2s1,t+d2s2,t+∑tϵVH2∑s1,s2ϵVRkH1−H1d2s1,t+d2s2,t=∑B1+∑B2+∑B3.∑B1=∑tϵVH2∑s1s2ϵERkH1s1,s2ϵVH1ds1+dt+ds1dt2+ds2+dt+ds2dt2=∑tϵVH2∑s1s2ϵESkH1s1,s2ϵVH14d2s1+d2s2+2d2t+4d2td2s1+d2s2+4dtds1+ds2+4d2tds1+ds2+8dtd2s1+d2s2=4VH2FH1+2EH1M1H2+4M1H2FH1+8EH2M1H1+4M1H1M1H2+16EH2FH1,∑B2=∑tϵVH2∑s1s2ϵERkH1s1ϵVH1,s2ϵVRkH1−H1ds1+dt+ds1dt2+ds2+ds2dt2=∑tϵVH2∑s1s2ϵERkH1s1ϵVH1,s2ϵVRkH1−H14ds12+ds22+dt24ds12+ds22+8dtds12+4dtds1+4d2tds1+d2t+2ds22dt=4VH2FH1+10EH1M1H2+4M1H2FH1+8EH2M1H1a+4M1H1M1H2+16EH2FH1+8|VH2EH1|+32|EH1EH2|,∑B3=∑tϵVH2∑s1s2ϵERkH1s1,s2ϵVRkH1−H1ds1+ds1dt2+ds2+ds2dt2=∑tϵVH2∑s1s2ϵERkH1s1,s2ϵVRkH1−H1ds12+ds22+dt2ds12+ds22+2dtd2s1+d2s2=8k−1|VH2EH1|+8k−1M1H2EH1+32k−1|EH2EH1|,∑C=∑t1t2ϵEH2∑s1s2ϵVH1d2s1,t1+d2s2,t2+∑t1t2ϵEH2∑s1ϵVH1,s2ϵVRkH1−H1d2s1,t1+d2s2,t2+∑t1t2ϵEH2∑s1,s2ϵVRkH1−H1d2s1,t1+d2s2,t2=∑C1+∑C2+∑C3.∑C1=∑t1t2ϵVH2∑s1s2ϵERkH1s1,s2ϵVH1ds1+dt1+ds1dt12+ds2+dt2+ds2dt22=∑t1t2ϵVH2∑s1s2ϵERkH1s1,s2ϵVH1ds12+ds22+d2t1+d2t2+dt12ds12+d2t2ds22+2dt1ds1+2dt2ds2+2d2t1ds1+2d2t2ds2+2dt1d2s1+2d2t2ds2=8EH2FH1+2EH1FH2+4FH1FH2+4M1H1M1H2+4M1H1FH2+8M1H2FH1,∑C2=∑t1t2ϵVH2∑s1s2ϵERkH1s1ϵVH1,s2ϵVRkH1−H1ds1+dt1+ds1dt12+ds2+ds2dt22=∑tεVH2∑s1s2εERkH1s1εVH1,s2εVRkH1−H1ds12+ds22+d2t1+dt12ds12+2dt1ds1+2dt1ds12+2d2t1ds1+dt22ds22+2dt2ds22=8EH2FG1+16|EH1EH2|+4FG1FG2+4M1H1M1H2+8FG1M1H2+2EH1FG2+4M1H1FG2+8EH1FG2+16EH1M1H2,∑C3=∑t1t2ϵEH2∑s1s2ϵERkH1s1,s2ϵVRkH1−H1ds1+ds1dt12+ds2+ds2dt22=∑t1t2ϵEH2∑s1s2ϵERkH1s1,s2ϵVRkH1−H12+2dt12+2+2dt22=16k−1|EH2EH1|+8k−1FH2|EH1|+16k−1|EH1|M1H2.
We arrive at our desired result.Theorem 3.
LetH1 and H2 be two connected graphs, then(10)FH1Qk⊠H2=2k−1FH1+2M2H13nH2+5M1H2+14EH2+FH2+kVH2+6EH2+3M1H2+FH2M4H1−2FH1+2M2H1−4M2H1+∑uϵVH1d2u∑vϵNudv+6EH2M1H1+10EH2FH2+3FH1FH2+6M1H1M1H2+FH2VH1+3M1H1+6|EH2+4M2H1+FH1VH2+7M1H2+6EH2M2H1+8M2H1EH2+M1H2.Proof.
Letds,t=dH1Qk⊠H2s,t is the degree of the vertex s,t in H1Qk⊠H2, then(11)FH1Qk⊠H2=∑s,tϵVH1Qk⊠H2ds,t3=∑s1,t1s2,t2ϵEH1Qk⊠H2ds1,t12+ds2,t22=∑sϵVH1∑t1t2ϵEH2ds,t12+ds,t22+∑tϵVH2∑s1s2ϵEQkH1ds1,t2+ds2,t2+∑t1t2ϵEH2∑s1s2ϵEQkH1ds1,t12+ds2,t22=∑A+∑B+∑C.∑A=∑sϵVH1∑t1t2ϵEH2ds,t12+ds,t22=2EH2M1H1+VH1FH2+M1H1FH2+4EH1M1H2+2M1H1M1H2+4EH1FH2.∑B=∑tϵVH2∑s1s2ϵEQkH1d2s1,t+d2p2,t=∑tϵVH2∑s1ϵVH1,s2ϵVQkH1−H1d2s1,t+d2s2,t+∑tϵVH2∑s1,s2ϵVQkH1−H1d2s1,t+d2s2,t=∑B1+∑B2.∑B1=∑tϵVH2∑s1s2ϵEQkH1s1ϵVH1,s2ϵVQkH1−H1ds1+dt+ds1dt2+ds2+ds2dt2=∑tεVH2∑s1s2εEQkH1s1εVH1,s2εVQkH1−H1ds12+ds22+dt2ds12+ds22+2dtds12+2dtd2s22+2dtd2s22+2dtds1+d2t.
Considers1ϵVH1 and d2s1 occurs ds1 times. Thus,(12)D1=∑s1s2ϵEQH1,s1εVH1,s2εVQH1−VH1d3s1=FH1.
Let(13)D2=∑s1s2ϵEQH1,s1εVH1,s2ϵVQH1−VH1d2s2.as s2=uvϵEH1 and d2s2 occurs two times. Therefore,(14)D2=2∑s2=uvϵVQH1−VH1du+dv2=2∑uvϵEH1d2u+d2v+2dudv=2FH1+2M2H1=VH2FH1+2EH1M1H2+M1H2FH1+4EH2M1H1+FH1+2M1H1M1H2+2VH2FH1+2M2H1+2M1H2FH1+2M2H1+8EH2FH1+2M2H1.∑B2=∑tϵVH2∑s1s2ϵEQkH1s1,s2ϵVQkH1−H1ds1+ds1dt2+ds2+ds2dt2.
Now, we split sum in two parts,s1 and s2, where s1s2ϵVQkH1−VH1. Suppose that ∑B2=∑B21+∑B22, where ∑B21 covers the edges of QkH1 which are in the same edges of H1 and ∑B22 of QkH1 in two different adjacent edges of H1.(15)∑B21=∑tϵVH1∑s1,s2ϵEQH1ds1+ds1dt2+ds2+ds2dt2=∑tϵVH2∑s1,s2ϵEQH1d2s1+d2s2+d2td2s1+d2s2+2dtd2s1+d2s2=∑tϵVH22k−1FH1+2M2H1+d2tFH1+2M2H1+2dtFH1+2M2H1=2k−1FH1+2M2H1VH2+M1H2+4EH2,∑B22=∑tϵVH2∑s1s2ϵEQkH1s1,s2ϵVQkH1−H1ds12+ds22+dt2ds12+ds22+2dtds12+ds22D3=∑s1s2ϵEQH1s1,s2ϵVQH1−VH1d2s1+d2s2.
InD3, the coefficient of(16)d2u=22dH1u+∑vϵNudv−du=d2u−2du+∑vϵNudv.
Therefore,(17)∑uϵVH1d2u=M4H1−2FH1+∑uϵVH1d2u∑vϵNudv.
For the coefficient ofdudv, let s1s2ϵEQH1 with s1=uv and s2=wz. As s1s2ϵEQH1, we have either v = w or z or u = w or z. So, uv is adjacent to all those vertices in H1 which are adjacent to u and v. So, the number of such dudv is du+dv−2. Therefore,(18)2∑uvϵEH1dudv=2∑uvϵEH1du+dv−2dudv=2∑uvϵEH1du+dvdudv−4∑uvϵEH1dudv=2M21H1−4M2H1.
So,(19)D3=M4H1−2FH1+∑uϵVH1d2u∑vϵNudv+2M21H1−4M2H1∑B22=kVH2+4EH2+M1H2M4H1−2FH1+2M21H1−4M2H1+∑uϵVH1d2u∑vϵNudv,∑C=∑t1t2ϵEH2∑s1s2ϵEQkH1d2s1,t1+d2s2,t2=∑t1t2ϵEH2∑s1ϵVH1,s2ϵVQkH1−H1d2s1,t1+d2s2,t2+∑t1t2ϵEH2∑s1,s2ϵVQkH1−H1d2s1,t1+d2s2,t2=∑C1+∑C2.∑C1=∑t1t2ϵVH2∑s1s2ϵEQkH1s1ϵVH1,s2ϵVQkH1−H1ds1+dt1+ds1dt12+ds2+ds2dt22=∑tϵVH2∑s1s2ϵEQkH1s1ϵVH1,s2ϵVQkH1−H1ds12+dt12ds12+2dt1ds1+2d2t1ds1+2dt1ds12+d2t1+ds22+dt22ds22+2dt2ds22=6EH2+M1H2FH1+3FH1FH2+2M1H1M1H2+2EH1+M1H1+2M2H1FH2+8M2H1EH2+M1H2.|∑C2=∑t1t2ϵEH2∑s1s2ϵEQkH1s1,s2ϵVQkH1−H1ds1+ds1dt12+ds2+ds2dt22.
Now, we split this sum into two parts for the vertices,t1 and t2, where s1s2ϵVQkG−VG. Assume that Tex translation failed, where Tex translation failed cover the edges of QkH1 which are in the same edges of H1 and Tex translation failed of QkH1 in two different adjacent edges of H1.(20)∑C21=∑t1t2ϵEH2∑s1s2ϵEQkH1s1,s2ϵVQkH1−H1ds1+ds1dt12+ds2+ds2dt22=∑t1t2ϵEH2∑s1s2ϵEQkH1s1,s2ϵVQkH1−H1d2s1+d2s2+d2s1d2t1+d2s2d2t2+2d2s1dt1+d2s2dt2=2k−1FH1+2M2H12EH2+FH2+2M1H2,∑C22=∑t1t2ϵEH2∑s1s2ϵEQkH1s1,s2ϵVQkH1−H1ds1+ds1dt12+ds2+ds2dt22=k2EH2+FH2+2M1H2M4H1−2FH1+2M2H1−4M2H1+∑uϵVH1d2u∑vϵNudv.
We arrive at our desired result.Theorem 4.
LetH1 and H2 be two connected graphs, then(21)FH1Tk⊠H2=2k−1FH1+2M2H1nH2+3M1H2+6EH2+FH2+kVH2+6EH2+3M1H2+FH2M4H1−2FH1+2M2H1−4M2H1+∑uϵVH1d2u∑vϵNudv+FH1+2M2H12nH2+6M1H2+12EH2+2FH2+4FH12nH2+6M1H2+12EH2+2FH2+FH2nH1+12M1H1+12EH2+12EH1M1H2+16EH2M1H1+20M1H1M1H2.Proof.
It follows from Theorems2 and 3.


## 4. Applications and Discussion

In this paper, we computed the F-index of the generalized sum graphs. Assume thatH1=Pn1 and H2=Pn2 are particular alkane called as paths of orders n1, n2≥3, and k≥1, respectively. Then, the following outcomes are the direct consequences of the achieved results:(i))
FPn1Sk⊠Pn2=128kn1−1+250n2+n1+512n2−2n1−2+216kn1−1n2−2−892(ii)
FPn1Rk⊠Pn2=128kn1−1+216kn1−1n2−2+2744n2−2n1−2+1458n1+1024n2−4464(iii)
FPn1Qk⊠Pn2=432kn1−1+729kn1−1n2−2+512n2−2n1−2+250n1+n2−892(iv)
FPn1Tk⊠Pn2=432kn1−1+729kn1−1n2−2+2744n2−2n1−2+1458n1+1024n2−4464(1)
Four subdivision operationsS1G,R1G,Q1G,T1G are restricted in case of newly addition of vertices upto k = 1, while four subdivision operations SkG,RkG,QkG,TkG are not restricted in case of newly addition of vertices as k can be any natural number.(2)
For a certain graphG, newly constructed graphs S1G,R1G,Q1G,T1G will have specific number of edges. While, in SkG,RkG,QkG,TkG, the number of edges will be varying according to k.(3)
Molecular structure ofSkG,RkG,QkG,TkG is more general as compared to the S1G,R1G,Q1G,T1G.(4)
Molecular structure ofFk-sum is more complex and general as compared to the molecular structure of F1-sum.(5)
In this paper, main resultsFH1Sk⊠H2, FH1Qk⊠H2, FH1Rk⊠H2, and FH1Tk⊠H2 based on strong product are generalization of FH1S1⊠H2, FH1Q1⊠H2, FH1R1⊠H2, and FH1T1⊠H2 based on strong product.Now, we present its tabular form, Table1 and Table 2 and graphical representations in Figures 8 and 9 for k=1 and k=2.Table 1
F-index of F1-sum path graphs.
n1,n2FPn1S1⊠Pn2FPn1R1⊠Pn2FPn1Q1⊠Pn2FPn1T1⊠Pn2(3, 3)1808641434428048(4, 4)483618120882622,110(5, 5)932035,74616,69243,118(6, 6)15,26059,29227,04071,072(7, 7)22,65688,75839,870105,972Table 2
F-index of F2-sum path graphs.
n1,n2FPn1S2⊠Pn2FPn1R2⊠Pn2FPn1Q2⊠Pn2FPn1T2⊠Pn2(3, 3)24967102576410,370(4, 4)651619,80014,49627,780(5, 5)12,42438,85027,16853,594(6, 6)20,22064,25243,78087,812(7, 7)29,90496,00664,332130,434Figure 8
(a)FPn1S1⊠Pn2, FPn1Q1⊠Pn2, FPn1R1⊠Pn2, and FPn1T1⊠Pn2 denoted by red, purple, green, and pink colours, respectively. (b) FPn1S2⊠Pn2, FPn1Q2⊠Pn2, FPn1R2⊠Pn2, and FPn1T2⊠Pn2 denoted by purple, green, pink, and red colours, respectively.
(a)(b)Figure 9
Compression betweenF-index of F1-sum and F2-sum graphs. (a) FPn1S1⊠Pn2 and FPn1S2⊠Pn2 are denoted by light green dotted line and dark green solid line, and FPn1R1⊠Pn2 and FPn1R2⊠Pn2 are denoted by sky blue dotted line and blue solid line, respectively. (b) FPn1Q1⊠Pn2 and FPn1Q2⊠Pn2 are denoted by pink dotted line and red solid line, and FPn1T1⊠Pn2 and FPn1T2⊠Pn2 by grey dotted line and black solid line, respectively.
(a)(b)In this paper, we computed theF-index of the subdivision-related generalized F-sum graphs based on strong product. However, the problem is still open for other topological indices on the generalized F-sum graphs.



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*Source: 9971277-2021-05-03.xml*