# Sum-Connectivity Coindex of Graphs under Operations

Authors: Muhammad Ibraheem; Ebenezer Bonyah; Muhammad Javaid
Journal: Journal of Chemistry
 (2022)

---

## Abstract

Topological indices or coindices are mathematical parameters which are widely used to investigate different properties of graphs. The operations on graphs play vital roles in the formation of new molecular graphs from the old ones. LetΓ be a graph we perform four operations which are S, R, Q,andT and obtained subdivisions type graphs such that SΓ, RΓ, QΓ, and TΓ, respectively. Let Γ1 and Γ2 be two simple graphs; then, F-sum graph is defined by performing the Cartesian product on FΓ1 and Γ2; mathematically, it is denoted by Γ1+FΓ2, where F∈S,R,Q,T. In this article, we have calculated sum-connectivity coindex for F-sum graphs. At the end, we have illustrated the results for particular F-sum graphs with the help of a table consisting of numerical values.


---

## Body

## 1. Introduction

The field of science in which chemical graph theory and mathematical chemistry are studied is known as cheminformatics. In this field, computable properties of molecular graphs are investigated using different mathematical parameters, perhaps the best known as topological index (TI), which is defined as the mathematical formula that is applied to any graph which has a molecular structure. TIs play a significant role especially in quantitative structure activity relationshipQSAR and quantitative structure property relationship QSPR investigations, such as optimisation and physicochemical interpretation of molecules [1]. The role of bioinformatics and chemistry in drugs discovery is explained in [2], and Rucker calculated the boiling points of different cycloalkanes in [3]. Topological indices are categorized mainly in two types: one is known as degree-based and second is known as distance-based. According to a recent survey, it is found that degree-based TIs attracted a lot of attention in recent years [4].Wiener was pioneer that used distance-based TI and calculated accurate boiling point of paraffin [5]. Later on, Gutman and Trinajsti C (1972) worked on alternant hydrocarbons and calculated their total π-electron energy using two degree-based indices known as Zagreb indices [6]. The Randi C (1975) proposed a new index known as Randi c´ index and calculated theoretical characterization of molecular branching with the help of said index [7].Zhou and Trinajstic [8] introduced sum-connectivity index and calculated this index for different graphs. Xing et al. [9] computed bounds for trees graphs using pendant vertices, and Wang et al. [10] determined lower bound for triangle-free graphs. Ma and Deng calculated the sharp lower bounds for cacti [11] under the sum-connectivity indices. Du and Zhou computed minimum sum-connectivity index for bicyclic graphs [12]. Das et al. [13] derived relations among Randić index and sum-connectivity index. Later, Das et al. [14] proved that for trees graphs, sum-connectivity is smaller than zeroth-order Randić index. Zhou and Trinajstić derived different relations among the sum-connectivity indices and product [15].Farahani calculated sum-connectivity index and Randic´ of nanotubes [16] and calculated sum-connectivity index, Randi c´ connectivity index ABC index, and geometric-arithmetic index of a class of dendrimer [17], Jahanbani computed the sharp lower bound on the sum-connectivity index of two trees with the minimum and the second minimum sum-connectivity. Ramane et al. [18] derived the relationship among sum-connectivity index, Randic index, harmonic index, and π-electron energy for benzenoid hydrocarbons. Rodriguez et al. [19] computed sum-connectivity index and harmonic index of graphs. Su and Xu computed the general sum-connectivity coindex of different graphs such as cycle graph, path graph, complete graph, complete bipartite graph, and hypercube graph [20].We use different operations on a graph for the formation of new families of graphs such as joining, subtraction, union, intersection, and products. Ahmad et al. [21] computed exact values and improved bounds for graph operations. Akhter and Imran computed sharp bounds four operations on graphs [22] and computed bounds for four types of graphs operations involving R-graph using the general sum-connectivity index [23]. Yan et al. [24] gave the idea about new graphs with help of operations as DG, where D∈L,S,Q,R,T on G and computed Wiener index for said graphs. Deng et al. [25] defined new F-sum graphs by extending the work of Yan which are obtained with the help of Cartesian product of two different graphs FG1 and G2, where F∈S,R,Q,T and calculated Zagreb indices. Ibraheem et al. [26] computed F-coindex. Javaid et al. [27] investigated the bounds for first and second Zagreb coindex [28].In this study, we compute sum-connectivity coindex of sum graphs in terms of Zagreb indices and coindices of their factor graphs. We have illustrated results through table for the specific sum graphs obtained using path (alkane) graphs. The rest of article as follows: Section2 describes elementary definitions and notations. Section 3 contains main results of work, and Section 4 contains the application and conclusion.


## 2. Preliminaries

A graph consists of set of verticesVΓ and edges EΓ mathematically denoted as Γ=VΓ,EΓ. The total number of vertices is called order of graph and total number of edges is called size of graph. For any vertex r∈VΓ, then dΓr is called degree of r and defined as number of edges attached to r. Let Γ be a graph; then, its complement is denoted by Γ¯ and defined as r1r2∈EΓ¯ iff r1r2∉EΓ.The first Zagreb indexM1Γ and second Zagreb index M2Γ were introduced by Gutman and Trinajsti [6] which are defined as(1)M1Γ=∑r∈VΓdΓr2=∑r1r2∈EΓdΓr1+dΓr2,M2Γ=∑r1r2∈EΓdΓr1dΓr2.Ashrafi et al. [29] introduced Zagreb coindices M¯1ΓandM¯2Γ, which are defined as(2)M¯1Γ=∑r1r2∉EΓdΓr1+dΓr2,M¯2Γ=∑r1r2∉EΓdΓr1dΓr2.Zhou and Trinajstic [8] introduced sum-connectivity index χΓ, which is defined as(3)χΓ=∑r1r2∈EΓdΓr1+dΓr2−1/2=∑r1r2∈EΓ1dΓr1+dΓr2.Now, the sum-connectivity coindexχ¯Γ is defined as(4)χ¯Γ=∑r1r2∉EΓdΓr1+dΓr2−1/2=∑r1r2∉EΓ1dΓr1+dΓr2.Suppose thatΓ is a connected graph, then(i)
The graphSΓ is formed by replacing each edge of Γ with P3(ii)
The graphRΓ is obtained from SΓ by joining the vertices which are adjacent in Γ(iii)
The graphQΓ is a graph formed using SΓ by attaching the new pairs of vertices which are on the adjacent edges of Γ(iv)
If bothRΓ and QΓ operations are performed on SΓ, then TΓ is obtainedConsidered two graphsΓ1 and Γ2, then we defined their F-sum graphs which is denoted by Γ1+FΓ2 and defined as having vertex set VΓ1+SΓ2=VΓ1∪EΓ1×VΓ2 and edge set as s1,x2, and s1,x2 of Γ1+FΓ2 are joined iff s1=x1∈VΓ1 and s2∽x2∈Γ2s2=x2∈VΓ2 and s1∽x1∈FΓ1, where F∈S,R,Q,T.Figures1 and 2 show the explanation of F-sum graphs.Figure 1
(a)Γ≅C3, (b) SC3, (c) QC3, (d) RC3, and (e) TC3.
(a)(b)(c)(d)(e)Figure 2
GraphsΓ≅C3, H≅P2, and Γ+FH≅C3+FP2.


## 3. Main Results

This section contains main results of harmonic coindex forF-sum graphs. Here, we defined some useful supposition that will be used in theorems.(5)α1=∑p1p2∉EFΓ1p1∈VΓ1p2∈VFΓ1−Γ1dΓ1p1+dFΓ1p2,α2=∑p1p2∉EFΓ1p1∈VΓ1p2∈VFΓ1−Γ1dΓ1p1+dFΓ1p2.Theorem 1.
The sum-connectivity coindex forS-sum graph Γ1+SΓ2 is given as(6)χ¯Γ1+sΓ2=12n22e12−n2e1+14e1e2¯+n1M1¯Γ2+1M1Γ1n2+4e1e2+1M¯1Γ1n2+4e1¯e2+12e2M1Γ1+e1M1Γ2+12e2M1¯Γ1+e1¯M1Γ2+12e2¯M1¯Γ1+e1¯M1¯Γ2+12e¯2M1Γ1+e1M1¯Γ2+1n2α1+2e2e1n1−2+12e2α1+e2e1n1−2+12e¯2α1+e¯2e1n1−2+12e2M1SΓ1+2e2e1+12e¯2M1SΓ1+2e¯2e1.Proof.
Using equation (4), we have(7)χ¯Γ1+SΓ2=1∑p1,p2r1,r2∉EΓ1+SΓ2dp1,r1+dp2,r2,H¯Γ1+SΓ2=1∑p1,p2∈VΓ1−Γ1∑r1,r2∈VΓ2dp1,r1+dp2,r2+1∑p1,p2∈VΓ1∑r1,r2∈VΓ2dp1,r1+dp2,r2+1∑p1,p2∈VSΓ1p1∈VΓ1p2∈VSΓ1−Γ1∑r1,r2∈VΓ2dp1,r1+dp2,r2=∑A+∑B+∑C,(8)∑A=1∑p1,p2∈VSΓ1−Γ1∑r1,r2∈VΓ2dSΓ1p1+dSΓ1p2=1∑p1,p2∈VSΓ1−Γ1∑r1,r2∈VG22+2,(9)A=12n22e12−n2e1,(10)B=∑i=16Bi,(11)∑B1=1∑p∈VΓ1∑r1r2∉EΓ2dp,r1+dp,r2=1∑p∈VΓ1∑r1r2∉EΓ22dΓ1p+dG2r1+dΓ2r2=14e1e2¯+n1M1¯Γ2,(12)∑B2=1∑r∈VΓ2∑p1,p2∈VΓ1dp1,r+dp2,r=1∑r∈VΓ2∑p1,p2∈VΓ1dp1,r+dp2,r+1∑r∈VΓ2∑p1p2∉EΓ1dp1,r+dp2,r=1∑r∈VΓ2∑p1p2∈EG1dΓ1p1+dΓ1p2+2dΓ2r+1∑r∈VΓ2∑p1p2∈EG1dΓ1p1+dΓ1p2+2dΓ2r=1M1Γ1n2+4e1e2+1M¯1Γ1n2+4e1¯e2.(13)∑B3=1∑p1p2∈EΓ1∑r1r2∈EΓ2dp1,r1+p2,r2=12∑p1p2∈EΓ1∑r1r2∈EΓ2dΓ1p1+dΓ2r1+dΓ1p1+dΓ2r2=12e2M1Γ1+e1M1Γ2.(14)∑B4=1∑p1p2∉EΓ1∑r1r2∈EΓ2dp,r1+dp,r2=12∑p∈VΓ1∑r1r2∉EΓ2dΓ1p1+dΓ2r1+dΓ1p2+dΓ2r2=12e2M1¯Γ1+e1¯M1Γ2,(15)∑B5=1∑p1p2∉EΓ1∑r1r2∈EΓ2dp1,r1+dp2,r2=12∑p1p2∉EΓ1∑r1r2∉EΓ2dΓ1p1+dΓ2r1+dΓ1p2+dΓ2r2=12e2¯M1¯Γ1+e1¯M1¯Γ2,(16)∑B6=1∑p1p2∈EΓ1∑r1r2∉EΓ2dp1,r1+dp2,r2=12∑p1p2∈EΓ1∑r1r2∉EΓ2dG1p1+dΓ2r1+dΓ1p2+dΓ2r2=12e2¯M1¯Γ1+e1¯M1¯Γ2,(17)∑B=14e1e2¯+n1M1¯Γ2+1M1Γ1n2+4e1e2+1M¯1Γ1n2+4e1¯e2+12e2M1Γ1+e1M1Γ2+12e2M1¯Γ1+e1¯M1Γ2+12e2¯M1¯Γ1+e1¯M1¯Γ2+12e¯2M1Γ1+e1M1¯Γ2,(18)∑C=∑i=15Ci,(19)∑C1=1∑p1p2∉ESΓ1p1∈VΓ1p2∈VSΓ1−Γ1∑r∈VΓ2dp1,x+p2,x=1∑p1p2∉ESΓ1p1∈VΓ1p2∈VSΓ1−Γ1∑r∈VΓ2dΓ1p1+dx+dSΓ1p2=1n2α1+2e2e1n1−2,(20)∑C2=1∑p1p2∉ESΓ1p1∈VΓ1p2∈VSΓ1−Γ1∑r1,r2∈VΓ2dp1,r+dp2,r=1∑p1p2∉ESΓ1p1∈VΓ1p2∈VSΓ1−Γ1∑r1,r2∈VΓ2dΓ1p1+dr+dSΓ1p2=12e2α1+e2e1n1−2,(21)∑C3=1∑p1p2∉ESΓ1p1∈VΓ1p2∈VSΓ1−Γ1∑r1,r2∉VΓ2dp1,r+p2,r=1∑p1p2∉ESΓ1p1∈VΓ1p2∈VSΓ1−Γ1∑r1,r2∉VΓ2dΓ1p1+dr+dSΓ1p2=12e¯2α1+e¯2e1n1−2.(22)∑C4=1∑p1p2∈ESΓ1p1∈VΓ1p2∈VSΓ1−Γ1∑r1,r2∈VΓ2dp1,r+p2,r=1∑p1p2∈ESΓ1p1∈VΓ1p2∈VSΓ1−Γ1∑r1,r2∈VΓ2dΓ1p1+dr+dSΓ1p2=12e2M1SΓ1+2e2e1.(23)∑C5=1∑p1p2∈ESΓ1p1∈VΓ1p2∈VSΓ1−Γ1∑r1,r2∉VΓ2dp1,r+dp2,r=1∑p1p2∈ESΓ1p1∈VΓ1p2∈VSΓ1−Γ1∑r1,r2∉VΓ2dΓ1p1+dr+dSΓ1p2=12e¯2M1SΓ1+2e¯2e1.
We get required result by substituting all in equation (2).Theorem 2.
The sum-connectivity coindex forR-sum graph Γ1+RΓ2 is given as(24)χ¯Γ1+RΓ2=12n22e12−n2e1+18e1e2¯+n1M1¯Γ2+12M1¯Γ1n2+2e1¯e2+122e2M1Γ1+e1M1Γ2+122e2M1¯Γ1+e1¯M1Γ2+122e2¯M1¯Γ1+e1¯M1¯Γ2+122e¯2M1Γ1+e1M¯1Γ2+1n2α1+2e2e1n1−2+12e2α1+e2e1n1−2+12e¯2α1+e¯2e1n1−2+12e2M1RΓ1+2e2e1+12e¯2M1RΓ1+2e¯2e1.Proof.
We consider(25)H¯Γ1+RΓ2=1∑p1,p2∈VΓ1−Γ1∑r1r2∈EΓ2dp1,r1+dp2,r2+1∑p1,p2∈VΓ1∑r1,r2∈VΓ2dp1,r1+dp2,r2+1∑p1,p2∈VRΓ1p1∈VΓ1p2∈VRΓ1−Γ1∑r1,r2∈VΓ2dp1,r1+dp2,r2.(26)H¯G1+RG2=∑A+∑B+∑C.
The value of∑A follows by equation (9):(27)B=∑i=16Bi,(28)∑B1=1∑p∈VΓ1∑r1r2∉EΓ2ds,r1+ds,r2=1∑p∈VΓ1∑r1r2∉EΓ22dRΓ1s+dΓ2r1+dΓ2r2=1∑s∈VΓ1∑r1r2∉EΓ24dΓ1p+dΓ2r1+dΓ2r2=18e1e2¯+n1M1¯Γ2.(29)∑B2=1∑r∈VΓ2∑p1,p2∈VΓ1dp1,r1+dp2,r21∑r∈VΓ2∑p1,p2∉EΓ1dp1,r+dp2,r=1∑r∈VΓ2∑p1p2∉EΓ1dRΓ1p1+dRΓ1p2+2dΓ2r=1∑r∈VΓ2∑p1,p2∉EΓ12dΓ1p1+2dΓ1p2+2dΓ2r=12M1¯Γ1n2+2e1¯e2,(30)∑B3=1∑p1p2∈EΓ1∑r1r2∈EΓ2dp1,r1+dp2,r2=12∑p1p2∈EΓ1∑r1r2∈EΓ2dRΓ1p1+dΓ2r1+dRΓ1p2+2dΓ2r2=12∑p1p2∈EΓ1∑r1r2∈EG22dΓ1p1+dΓ2r1+2dΓ1p2+dΓ2r2=122e2M1Γ1+e1M1Γ2(31)∑B4=1∑p1p2∉EΓ1∑r1r2∈EΓ2dp1,r1+dp2,r2=12∑p1p2∉EΓ1∑r1r2∈EΓ2dRΓ1p1+dΓ2r1+dRΓ1p2+dΓ2r2=12∑p1p2∉EΓ1∑r1r2∈EΓ22dΓ1p1+dΓ2r1+2dΓ1p2+dΓ2r2=122e2M1¯Γ1+e1¯M1Γ2,(32)∑B5=1∑p1p2∉EΓ1∑r1r2∉EΓ2dp1,r1+dp2,r2=12∑p1p2∉EΓ1∑r1r2∉EΓ2dRΓ1p1+dΓ2r1+dRΓ1p2+dΓ2r2=12e2¯M1¯Γ1+e1¯M1¯Γ2.=12∑p1p2∉EΓ1∑r1r2∉EΓ2dRΓ1p1+dΓ2r1+dΓ1p2+dΓ2r2=122e2¯M1¯Γ1+e1¯M1¯Γ2.(33)∑B6=1∑p1p2∈EΓ1∑r1r2∉EΓ2dp1,r1+dp2,r2=12∑p1p2∈EΓ1∑r1r2∉EΓ2dRΓ1p1+dΓ2r1+dRΓ1p2+dΓ2r2=12∑p1p2∉EΓ1∑r1r2∉EΓ22dΓ1p1+dΓ2r1+2dΓ1p2+dΓ2r2=122e¯2M1Γ1+e1M¯1Γ2.(34)∑B=18e1e2¯+n1M1¯Γ2+12M1¯Γ1n2+2e1¯e2+122e2M1Γ1+e1M1Γ2+122e2M1¯Γ1+e1¯M1Γ2+122e2¯M1¯Γ1+e1¯M1¯Γ2+122e¯2M1Γ1+e1M¯1Γ2,(35)∑C=∑i=15Ci,(36)∑C1=1∑p1p2∉ERΓ1p1∈VΓ1p2∈VRΓ1−Γ1∑r∈VΓ2dp1,r+dp2,r=1∑p1p2∉ERΓ1p1∈VΓ1p2∈VRΓ1−Γ1∑r∈VΓ2dΓ1p1+dr+dRΓ1p2=1n2α1+2e2e1n1−2,(37)∑C2=1∑p1p2∉ERΓ1p1∈VΓ1p2∈VRΓ1−Γ1∑r1,r2∈VΓ2dp1,x+dp2,x=1∑p1p2∉ERΓ1p1∈VΓ1p2∈VRΓ1−Γ1∑r1,r2∈VΓ2dG1p1+dx+dRΓ1p2=12e2α1+e2e1n1−2.(38)∑C3=1∑p1p2∉ERΓ1p1∈VR1p2∈VRΓ1−Γ1∑r1,r2∉VΓ2dp1,x+dp2,x=1∑p1p2∉ERΓ1p1∈VΓ1p2∈VRΓ1−Γ1∑r1,r2∉VΓ2dΓ1p1+dx+dRΓ1p2=12e¯2α1+e¯2e1n1−2,(39)∑C4=1∑p1p2∉ERΓ1p1∈VΓ1p2∈VRΓ1−Γ1∑r1,r2∈VΓ2dp1,x+dp2,x=1∑p1p2∉ERΓ1p1∈VΓ1p2∈VRΓ1−Γ1∑r1,r2∈VΓ2dΓ1p1+dx+dRΓ1p2=12e2M1RΓ1+2e2e1,(40)∑C5=1∑p1p2∉ERΓ1p1∈VR1p2∈VRΓ1−Γ1∑r1,r2∉VΓ2dp1,x+dp2,x=1∑p1p2∉ERΓ1p1∈VΓ1p2∈VRΓ1−Γ1∑r1,r2∉VΓ2dΓ1p1+dx+dRΓ1p2=12e¯2M1RΓ1+2e¯2e1.
We get the required result by substituting all in equation (10).Theorem 3.
The sum-connectivity coindex forQ-sum graph G1+QG2 is given as(41)χ¯Γ1+QΓ2=1α+14e1e2¯+n1M1¯Γ2+1M1Γ1n2+4e1e2+1M¯1Γ1n2+4e1¯e2+12e2M1Γ1+e1M1Γ2+12e2M1¯Γ1+e1¯M1Γ2+12e2¯M1¯Γ1+e1¯M1¯Γ2+12e¯2M1Γ1+e1M1¯Γ2+.1n2α1+2e2e1n1−2+12e2α1+e2e1n1−2+12e¯2α1+e¯2e1n1−2+12e2α2+2e2e1+12e¯2α1+2e¯2e1.Proof.
We considered(42)H¯Γ1+QΓ2=1∑p1,p2∈VΓ1−Γ1∑r1,r2∈VΓ2dp1,r1+dp2,r2+1∑p1,p2∈VΓ1∑r1,r2∈VΓ2dp1,r1+dp2,r2+1∑p1p2∉EQΓ1p1∈VΓ1p2∈VQΓ1−Γ1∑r1,r2∈VΓ2dp1,r1+dp2,r2.(43)H¯Γ1+QΓ2=∑A+∑B+∑C,(44)∑A=1∑s1,s2∈VQΓ1−Γ1∑x1,x2∈VΓ2ds1,x1+ds2,x2=1α.
The values of∑A and ∑B follow by equations (9) and (43).(45)∑C=∑i=15Ci,(46)∑C1=1∑p1p2∉EQΓ1p1∈VΓ1p2∈VQΓ1−Γ1∑r∈VΓ2dp1,r+p2,r=1∑p1p2∉EQΓ1p1∈VΓ1p2∈VQΓ1−Γ1∑r∈VΓ2dΓ1p1+dr+dQΓ1p2=1n2α1+2e2e1n1−2,(47)∑C2=1∑p1p2∉EQΓ1p1∈VΓ1p2∈VQΓ1−Γ1∑r1,r2∈VΓ2dp1,r+p2,r=1∑p1p2∉EQΓ1p1∈VΓ1p2∈VQΓ1−Γ1∑r1,r2∈VΓ2dΓ1p1+dr+dQΓ1p2=12e2α1+e2e1n1−2,(48)∑C3=1∑p1p2∉EQΓ1p1∈VΓ1p2∈VQΓ1−Γ1∑r1,r2∉VΓ2dp1,r+dp2,r=1∑p1p2∉EQΓ1p1∈VΓ1p2∈VQΓ1−Γ1∑r1,r2∉VΓ2dΓ1p1+dr+dQΓ1p2=12e¯2α1+e¯2e1n1−2.(49)∑C4=1∑p1p2∉EQΓ1p1∈VG1p2∈VQΓ1−Γ1∑r1,r2∈VΓ2dp1,r+dp2,r=1∑p1p2∉EQΓ1p1∈VΓ1p2∈VQΓ1−Γ1∑r1,r2∈VΓ2dΓ1p1+dr+dQΓ1p2=12e2α2+2e2e1,(50)∑C5=1∑p1p2∉EQΓ1p1∈VQ1p2∈VQΓ1−Γ1∑r1,r2∉VΓ2dp1,r+dp2,r=1∑p1p2∉EQΓ1p1∈VΓ1p2∈VQΓ1−Γ1∑r1,r2∉VΓ2dΓ1p1+dr+dQΓ1p2=12e¯2α1+2e¯2e1.
We get required result by substituting all in equation (6).Theorem 4.
The sum-connectivity coindex forT-sum graph Γ1+TΓ2 is given as(51)χ¯Γ1+TΓ2=18e1e2¯+n1M1¯Γ2+12[M1¯Γ1n2+2e1¯e2]+12[2e2M1Γ1+e1M1Γ2]]+122e2M1¯Γ1+e1¯M1Γ2+122e2¯M1¯Γ1+e1¯M1¯Γ2+122e¯2M1Γ1+e1M¯1Γ2+1n2α1+2e2e1n1−2+12e2α1+e2e1n1−2+12e¯2α1+e¯2e1n1−2+12e2α2+2e2e1+12e¯2α1+2e¯2e1.Proof.
It follows from Theorem2 and Theorem 3.Table 1
Sum-connectivity coindex for certainF-sum graphs.
Cn,Pmχ¯Cn+SPmχ¯Cn+RPmχ¯Cn+QPmχ¯Cn+TPm3,20.18220.15670.15630.14144,20.14120.13290.12210.11675,20.12440.11600.10700.10186,20.11260.10430.09650.09157,20.10360.11100.08730.08398,20.09640.08870.08240.07799,20.09060.09380.07730.073010,20.08570.08090.07310.068911,20.08120.07460.06950.0655


## 4. Conclusion

(i)
In this study, we have computed sum-connectivity coindex in the form of first and second Zagreb indices and coincides forF-sum graphs such as χ¯Γ1+SΓ2, χ¯Γ1+RΓ2, χ¯Γ1+QΓ2, and χ¯Γ1+TΓ2.(ii)
Table1 consisting of numerical values present that sum-connectivity coindex of χ¯Γ1+SΓ2 is dominant than χ¯Γ1+RΓ2, χ¯Γ1+QΓ2, and χ¯Γ1+TΓ2.



---

*Source: 4523223-2022-04-14.xml*