# Zagreb-Type Indices ofR-Vertex Join and R-Edge Join of Graphs

Authors: Jianxin Wei; Muhammad Imran; Muhamamd Azhar Iqbal; Muhammad Asad Zaighum
Journal: Journal of Chemistry
 (2020)

---

## Abstract

There are various methods available which are used to search large chemical databases and to predict the physicochemical properties of molecular structures. Using molecular descriptors for this purpose is the simplest of these methods. The Zagreb indices are amongst the oldest molecular descriptors, and their properties have been extensively studied and applied in QSAR/QSPR studies. The Zagreb coindices were recently introduced, attracting the attention of researchers in mathematical chemistry. In this paper, we study Zagreb indices and several other Zagreb-type indices including the general Randić index, sum-connectivity index, F-index, and Zagreb coindices ofR-vertex and edge join of two arbitrary graphs.


---

## Body

## 1. Introduction

The relation between the atoms and chemical bonds between atoms in a molecule can be represented by a molecular graph, commonly denoted byG. The atoms and their chemical bonds, respectively, play the role of vertices and edges of the corresponding molecular graph [1]. A graph G is a pair G=VG,EG with vertex set VG and edge set EG. The order and size of G are, respectively, the cardinalities VG and EG. The neighbourhood NGv of a vertex v in G is the set of vertices adjacent to v. The degree dGv of a vertex v in G is defined as dGv=NGv. Let dGv=NGv and Cn, respectively, denote the path and cycle graphs of order n. For simplicity, we use the notation du instead of dGu. Other important and undefined terminologies from graph theory can be found in [2].A scalar quantity, which is necessarily a graph invariant, which can be used to represent a certain topological feature of a molecular graph is known as a topological index [3, 4]. These indices are used as functions assigning numerical values to the qualitative properties of molecules. The structural properties of molecules have impact on their physicochemical properties. These properties are correlated by using topological indices in QSAR/QSPR studies [5]. In QSAR/QSPR studies, these indices are used in search for new chemicals which are then used in producing potent drugs [6–9]. These topological indices are also used for virtual screening of chemical libraries to search for new chemicals of specific needs in computational chemistry and pharmacology [10–13]. The first topological index was used as a numerical descriptor to correlate the boiling points of some paraffins by Wiener [14] in 1947. This descriptor was called Wiener number, but Hosoya [15] used the term topological index for the first time in 1971. He defined the Wiener index of a graph G as follows:(1)WG=∑u,v⊆VGdGu,v.Gutman and Trinajstć [16, 17] investigated the dependence of the total π-electron energy of an alternant hydrocarbon, where they encountered the terms denoted by M1 and M2 commonly known as Zagreb indices of the first and second kind and defined as follows:(2)M1G=∑u∈VGd2u,(3)M2G=∑uv∈EGdudv.Randić [18] introduced the Randić connectivity index in 1975 as follows:(4)RG=∑uv∈EG1dudv.Bollobás and Erdos [19] defined the general Randić index as follows:(5)RαG=∑uv∈EGdudvα.The general sum-connectivity index was defined by Zhou and Trinajstić [20] as follows:(6)χαG=∑uv∈EGdu+dvα.Furtula and Gutman [21] introduced the forgotten topological index as(7)FG=∑u∈VGd3u=∑uv∈EGd2u+d2v.Došlić [22] defined the first and second Zagreb coindices as follows:(8)M1¯G=∑uv∉EGdu+dv,(9)M2¯G=∑uv∉EGdudv.Some details and applications of these indices can be found in [23–27] and a survey paper on the Randić index in [28]. The zeroth-order general Randić index is a general case of the first Zagreb index [29, 30]. Zagreb indices of some tree-like graphs containing hydrocarbons like ethane, propane, and butane are studied in [31].In [32], the authors studied the first and second Zagreb indices of some graph operations including subdivision and total graphs. The Zagreb coindices were recently studied in [33–37], and details on the relations between Zagreb indices and coindices can be found in [38–44]. In [45], the upper and lower bounds of M1G and M2G are determined. An upper bound of M2G in k-apex trees is also studied, and the corresponding extremal k-apex trees are also characterized. Two distance-based indices of some graph operations were studied in [46].


## 2.R-Graphs

TheR-graph [47] of a graph G (denoted by RG) is the graph obtained from G by adding a vertex we∉VG corresponding to every edge e=uv∈EG and joining we to end vertices u and v for each edge e=uv∈EG. Let IG denote the set of all new vertices RG, that is,(10)IG=wee=uv∈EGwhereu,v∈VG.Similarly, letJG denote the new edges in RG, that is,(11)JG=weu,weve=uv∈EGandu,v∈VG.Then, we have(12)VRG=VG∪IG,(13)ERG=EG∪JG.The degree of vertexu∈VRG is given by(14)dRGu=2dGu,u∈VG,2,u∈IG.Based on graphRG, two new graph operations obtained from two arbitrary graphs G1 and G2 were defined in [48]. These operations are called R-vertex join and R-edge join of G1 and G2, respectively, denoted by G1vG2 and G1eG2. For two vertex disjoint graphs G1 and G2, the R-vertex join of G1 and G2 is obtained from RG1 and G2 by joining every vertex of VG1 and VG2 by an edge. Similarly, for two vertex disjoint graphs G1 and G2, the R-edge join of G1 and G2 is obtained from RG1 and G2 by joining every vertex of IG1 and VG2 by an edge. Graphs C6vC4 and C6eC4 are shown in Figure 1.Figure 1
TheR-vertex join (a) and R-edge join (b) of graphs C6 and C4.


## 3.R-Vertex Join and R-Edge Join of Graphs

LetG1 and G2 be two graphs. The order and size of Gi are denoted by ni and mi, i=1,2. Let Gv and Ge, respectively, denote graphs G1vG2 and G1eG2. The vertex and edge sets of graphs Gv and Ge are given as follows:(15)VG1vG2=VG1eG2=VG1∪VG2∪IG1,(16)EG1vG2=EG1∪EG2∪JG1∪Kv,where Kv=uvu∈VG1,v∈VG2. Similarly,(17)EG1eG2=EG1∪EG2∪JG1∪Ke,where Ke=weuwe∈IG1,u∈VG2.Then, the ordernv and size mv of graph Gv can be observed as VGv=nv=n1+n2+m1 and EGv=mv=m1+m2+n1n2+2m1. Similarly, the order ne and size me of graph Ge is given by VGe=ne=n1+n2+m1 and EGe=me=m1+m2+2m1+m1n2. The order and size of the complements of the sets EGv and EGe are given by(18)EGv¯=n1+n2+m12−3m1+m2+n1n2,EGe¯=n1+n2+m12−3m1+m2+m1n2.The degree of a vertex inGv and Ge can be obtained from the following expressions:(19)dGvw=2dG1w+n2,w∈VG1,dG2w+n1,w∈VG2,2,w∈IG1,dGew=2dG1w,w∈VG1,dG2w+m1,w∈VG2,2+n2,w∈IG1.The following lemma will be used for calculating the Zagreb coindices of raphsGv and Ge.Lemma 1 (see [33]).
The first and second Zagreb coindices of graphG of order n and size m can be computed as(20)M1¯G=2mn−1−M1G,2M2¯G=4mv2−2M2Gv−M1Gv.In the next section, we compute the Zagreb indices of the graphRG and R-vertex and R-edge join of graphs and the Zagreb coindices of RG and R-vertex and R-edge join of graphs.


## 4. Zagreb Indices ofR-Graphs and R-Vertex and Edge Join of Graphs

Theorem 1.
The first Zagreb index of graphRG is given by(21)M1RG=2M1G+4m.Proof.
Using the definition of the vertex set of graphRG and by using (2), we get(22)M1RG=∑w∈RGdRG2w=∑w∈VGdRG2w+∑w∈IGdRG2w=∑w∈VG2dG2w+∑w∈IG22=2∑w∈VGdG2dG2+∑w∈IG4=2M1G+4m.Theorem 2.
The second Zagreb index of graphRG is given by`(23)M2RG=4M1G+M2G.Proof.
Using the definition of the edge set of graphRG and by using (3), we get(24)M2RG=∑uv∈ERGdRGudRGv=∑uv∈EGdRGudRGv+∑uv∈JGdRGudRGv=4∑uv∈EGdGudRGv+∑u∈VG4dG2u=4M2G+4M1G=M1G+M2G.Theorem 3.
The first Zagreb index of graphGv is given by(25)M1Gv=4M1G1+n22n1+8n2m1+M2G2+n12n2+4n1m2+4m1.Proof.
The definition of the vertex set of graphGv and the use of (2) and (3) yield the following:(26)M1Gv=∑w∈VGvdGv2w=∑w∈VG1dGv2w+∑w∈VG2dGv2w+∑w∈IG1dGv2w=∑w∈VG12dG1w+n22+∑w∈VG2dG2w+n12+∑w∈IG122=∑w∈VG14dG12w+∑w∈VG1n22+∑w∈VG14n2dG1w+∑w∈VG2dG22w+∑w∈VG2n12+∑w∈VG22n1dG2w+∑w∈IG14=4M1G1+n22n1+4n22m1+M2G2+n12n2+2n2m2+4m1=4M1G1+n1n22+8n2m1+M2G2+n12n2+4n1m2+4m1.Theorem 4.
The second Zagreb index of graphGv is given as(27)M2Gv=4M2G1+2n2M1G1+n22m1+M2G2+n1M1G2+n12m2+4M1G1+8m1m2+2n1+n22n2m1+2n1m2+4n2m1+n12n22.Proof.
Using the definition of the edge set of graphGv and by using (2) and (3), we get the following:(28)M2Gv=∑uv∈VGvdGvudGvv=∑uv∈EG1dGvudGvv+∑uv∈EG2dGvu×dGvv+∑uv∈JG1dGvudGvv+∑uv∈KvdGvudGvv=∑uv∈EG12dG1u+n22dG1v+n2+∑uv∈EG2dG2u+n1dG2v+n1+∑u∈VG12dG12u×2+4n2m1+∑uv∈Kv2dG1u+n2dG2v+n1=∑uv∈EG14dG1udG1v+2dG1un2+2n2dG1v+n22+∑uv∈EG2dG2udG2v+n1dG1u+dG2u+n12+4∑u∈VG1dG12u+4n2m1+2∑uv∈KvdG1udG2v+2n1∑uv∈KvdG1u+n2∑uv∈KvdG2v+n1n22=4∑uv∈EG1dG1udG1v+2n2∑uv∈EGdG1u+dG1v+n22m1+∑uv∈EG2dG2udG2v+n1∑uv∈EG2dG2udG2v+n12m2+4M1G1+4n2m1+22m12m2+2n1+n2n22m1+n12m2+n12n22=4M2G1+2n2∑u∈VG1dG12u+n22m1+M2G2+n1∑u∈VG2dG22u+n12m2+4M1G1+4n2m1+8m1m2+2n1+n22n2m1+2n1m2+n12n22=4M2G1+2n2M1G1+n22m1+M2G2+n1M1G2+n12m2+4M1G1+8m1m2+2n1+n22n2m1+2n1m2+4n2m1+n12n22.Theorem 5.
The first Zagreb index of graphGe is calculated as(29)M1Ge=4M1G1+M1G2+4m1m2+m12n2+4m1+n22m1+4n2m1.Proof.
Using the vertex set of graphGe and by using (2) and (3), we get the following:(30)M1Ge=∑w∈VGedGe2w=∑w∈VG1dGe2w+∑w∈VG2dGe2w+∑w∈IG1dGe2w=∑w∈VG12dG1w2+∑w∈VG2dG2w+m12+∑w∈IG12+n22=4∑w∈VG1dG12w+∑w∈VG2dG22w+2m1∑w∈VG2dG2w+m12n2+4m1+n22m1+4n2m1=4M1G1+M1G2+2m12m2+m12n2+4m1+n22m1+4n2m1.Theorem 6.
The second Zagreb index of graphGe is given as(31)M2Ge=4M2G1+M2G2+m1MG2+m12m2+m1m1m2n2+2+m22n2+4+4n2+8.Proof.
Using the edge set of graphGe and by using (2) and (3), we get(32)M2Ge=∑uv∈EGedGeudGev=∑uv∈EG1dGeudGev+∑uv∈EG2dGeudGev+∑uv∈JG1dGeudGev+∑uv∈KedGeudGev=∑uv∈EG12×2dG1udGev+∑uv∈EG2dG2u+m1dG2v+m1+∑u∈VG12dG12u2+n2+m1dGewe∑v∈VG2dGev=4∑uv∈EG1dG1udG1v+∑uv∈EG2dG2udG2v+m1dG2u+dG2v+m12+4+2n2∑u∈VG1dG12u+m12+n2∑v∈VG2dG2w+m1=4M2G1+M2G2+m1MG2+m12m2+4+2n22m1+m12+n22m2+m12+n2m1m2.It can be observed that, for graphG and v∈VG, we have dG¯v=VG−1−dGv. Using this relation and the definitions of Zagreb coindices, we obtain the following results.Theorem 7.
The first Zagreb coindex of graphGv is given by(33)M1¯Gv=2n1+n2+m12−3m1+m2+n1n2n1+n2+m1−1−4M1G1+n22n1+8n2m1+M2G2+n12n2+4nm2+4m1.Proof.
Using Lemma1, Theorem 3, and (8), we obtain the following:(34)M1¯Gv=2n1+n2+m12−m1+m2+n1n2+2m1×n1+n2+m1−1−M1Gv=2n1+n2+m12−m1+m2+n1n2+2m1×n1+n2+m1−1−4M1G1+n22n1+4n22m1+M2G2+n12n2+2n2m2+4m1.Theorem 8.
The first Zagreb coindex of graphGe is given as(35)M1¯Ge=2n1+n2+m12−3m1+m2+n1n2×n1+n2+m1−1−4M1G1+M1G2+4m1m2+m22n2+4m1+n22m1+4n2m1.Proof.
By substituting Theorem5 in (8) and using Lemma 1, we obtain(36)M1¯Ge=2n1+n2+m12−m1+m2+2m1+n1n2×n1+n2+m1−1−M1Ge=2n1+n2+m12−m1+m2+2m1+n1n2×n1+n2+m1−1−4M1G1+M1G2+2m12m2+m22n2+4m1+n22m1+4n2m1.Theorem 9.
The second Zagreb coindex of graphGv is as follows:(37)M2¯Gv=124mv2−24M2G1+2n2M1G1+n22m1+M2G2+n1M1G2+n12m2+4M1G1+8m1m2+2n1+n22n2m1+2n1m2+4n2m1+n12n22−4M1G1+n22n1+8n2m1+M2G2+n12n2+4nm2+4m1.Proof.
Using (9) and substituting Theorems 3 and 4 in Lemma 1, we obtain(38)M2¯Gv=124mv2−2M2Gv−M1Gv=124mv2−24M2G1+2n2M1G1+n22m1+M2G2+n1M1G2+n12m2+4M1G1+22m12m2+2n1+n2n22m1+n12m2+4n2m1+n12n22+8n2m1−4M1G1+n22n1+4n22m1+M2G2+n12n2+2n2m2+4m1.Theorem 10.
The second Zagreb coindex of graphGe is given as(39)M2¯Ge=124me2−24M2G1+M2G2+M1G2+m122m2+4+2n22m1+m12+n22m2+m12+n2m1m2−4M1G1+M1G2+4m1m2+m22n2+4m1+n22m1+4n2m1.Proof.
Using (9) and substituting Theorems 5 and 6 in Lemma 1, we obtain(40)M2¯Ge=124me2−2M2Ge−M1Ge=124me2−24M2G1+M2G2+M1G2+m122m2+4+2n22m1+m12+n22m2+m12+n2m1m2−4M1G1+M1G2+2m12m2+m22n2+4m1+n22m1+4n2m1.


## 5. General Randić Index ofR-Vertex and Edge Join of Graphs

In this section, we compute the general Randić indexRα, for α=2, of the R-vertex join and R-edge join of graphs.Theorem 11.
The general Randić indexRα, for α=2, of Gv is given as(41)R2Gv=16R2G1+4n22FG1+4n23M1G1+16n2ReM3G1+4n22M2G1+n24+R2G2+n12FG2+2n12M1G2+2n1×ReM3G2+2n12M2G2+n14+16FG1+8n22m1+32n2M1G1+8m1M1G2+8m1n12n2+32m1m2n1+n22n1M1G1+n23n13+4n12n22m2+8m1n2M1G2+8n12n22m1+32m1m2n1n2.Proof.
Using (4) and the edge set of Gv, we obtain the following:(42)R2Gv=∑uv∈EGvdGvudGvv2=∑uv∈EG12dG1u+n222dG1v+n22+∑uv∈EG2dG2u+n12dG2v+n12+∑u∈VG1dG1u2dG1u+n22×22+∑uv∈Kv,u∈VG1,v∈VG22dG1u+n22dG2v+n12=∑uv∈EG116dG12udG12v+4n22dG12u+n22×4n2dG1v+n22×4n2dG1u+4n2dG1u×4dG12v+4n22dG1udG1v4n2dG1u×4dG12v+4n22dG1udG1v+4dG12u×4n2dG1v+4n22dG12v+n22n22+∑uv∈EG2dG22udG22v+n12dG22v+n12×dG22u+2n12×dG2u+2n1dG2udG22v+2n12×dG2udG2v+2n12dG2v+2n1dG2vdG22u+n12n12+∑u∈VG14dG1u4dG12u+n22+8dG1un2+∑u∈VG12dG1u+n22×∑v∈VG2dG2v+n12=∑uv∈EG116dG12udG12v+4n22∑uv∈EG1dG12u+dG12v+4n23∑uv∈EG1dG1u+dG1v+16n2∑uv∈EG1dG1udG12v+dG12udG1v+4n22∑uv∈EG1dG1udG12v+n24+∑uv∈EG2dG22u×dG22v+n12∑uv∈EG2dG22u+dG22v+2n12×∑uv∈EG2dG2u+dG2v+2n1∑uv∈EG2dG2udG22v+dG22udG2v+2n12∑uv∈EG2dG2udG22v+n14+16∑u∈VG1dG13u+8n22m1+32n2∑u∈VG1dG12u+∑u∈VG14dG1u+n22+4n2dG1u+∑u∈VG2dG22v+n12+2n1dG2v=16∑uv∈EG1dG1udG1v2+4n22FG1+4n23M1G1+16n2∑uv∈EG1dG1udG1vdG1u+dG1v+4n22M2G1+n24+R2G2+n12FG2+2n12M1G2+2n1ReM3G2+2n12M2G2+n14+16FG1+8n22m1+32n2M1G1+42m1+n22n1+4n22m1×M1G2+n12n2+2n12m2=16R2G1+4n22FG1+4n23M1G1+16n2ReM3G1+4n22M2G1+n24+R2G2+n12FG2+2n12M1G2+2n1ReM3G2+2n12M2G2+n14+16FG1+8n22m1+32n2M1G1+8m1M1G2+8m1n12n2+32m1m2n1+n22n1M1G1+n23n13+4n12n22m2+8m1n2M1G2+8n12n22m1+32m1m2n1n2.Theorem 12.
General Randić index of graphGe for α=2 is(43)R2Ge=16R2G1+R2G2+m12FG2+2m12M2G2+m12m22+4m1M2G2+2m12M1G2+8m1+4M1G1+4n22+16n2FG1+4M1G1M1G2+m12+n22×M1G2+m12n2+4m1m2.Proof.
Using (4) and the edge set of Ge, we obtain the following:(44)R2Ge=∑uv∈EGedGeudGev2=∑uv∈EG12dG1u×2dG1v2+∑uv∈EG2dG2u+m12dG2v+m12+∑u∈VG1dG1u+2+n222dG1u2+∑uv∈Ke,u∈IG1,v∈VG22+n22×dG2v+m12=∑uv∈EG116dG1udG1v2+∑uv∈EG2dG22u×dG22v+dG22um12+m12dG22v+m12m22+2m1dG2u×dG2v+2m12dG2u+2m1dG2u2m1dG2v+2m1dG2vdG2u+2m12dG2v+∑u∈VG14dG1u×4dG12u+n22dG1u×4dG12u+4n2dG1u×4dG12u+∑u∈IG12+n22∑v∈VG2dG2v+m12=16R2G1+R2G2+m12∑uv∈EG2dG22u+dG22v+2m12∑uv∈EG2dG2udG2v+m12m22+4m1×∑uv∈EG2dG2udG2v+2m12×∑uv∈EG2dG2u+dG2v+4×2m1+4×∑u∈VG1dG12u+4n22+16n2×∑u∈VG1dG13u+m12+n22×M1G2+m12n2+2m12m2=16R2G1+R2G2+m12FG2+2m12M2G2+m12m22+4m1M2G2+2m12M1G2+8m1+4M1G1+4n22+16n2FG1+4M1G1M1G2+m12+n22×M1G2+m12n2+4m1m2.


## 6. Forgotten Index ofR-Vertex and Edge Join of Graphs

In this section, we calculate the forgotten index of theR-vertex and R-edge join of graphs.Theorem 13.
The forgotten index ofGv is given as(45)FGv=4FG1+2n12n2+4n2M1G1+FG2+2n12m2+2n1M1G2+4FG1+n22×2m1+4n2M1G1+8m1+4M1G1+n22n1+8n2m1+M1G2+n12n2+4n1m2.Proof.
Using (7) and the edge set of Gv, we get(46)FGv=∑uv∈EGvdGv2u+dGv2v=∑uv∈EG12dG1u+n22+2dG1v+n22+∑uv∈EG2dG2u+n12+dG2v+n12+∑u∈VG12dG1u+n22+4+∑uv∈Kv,u∈VG1v∈VG22dG1u+n22+dG2v+n12=∑uv∈EG14dG12u+n22+4n2dG1u+4dG12v+n22+4n2dG1v+∑uv∈EG2dG22u+n12+2n1dG2u+dG22v+n12+2n1dG2v+∑u∈VG1dG1u×4dG12u+dG1u×n22+4n2dG1udG1u+4dG1u+∑u∈VG14dG12u+n22+4n2dG1u+∑u∈VG1dG22v+n12+2n1dG2v=4∑uv∈EG1dG12u+dG12v+2n22m1+4n2×∑uv∈EG1dG1u+dG1v+∑uv∈EG2dG22u+dG22v+2n12m2+2n1∑uv∈EG2dG2u+dG2v+∑u∈VG14dG13u+n22×2m1+4n2×∑u∈VG1dG12u+4×2m1+4∑u∈VG1dG12u+n22n1+4n2×2m1+∑v∈VG2dG22v+n12n2+2n1×2m2=4FG1+2n12n2+4n2M1G1+FG2+2n12m2+2n1M1G2+4FG1+n22×2m1+4n2M1G1+8m1+4M1G1+n22n1+8n2m1+M1G2+n12n2+4n1m2.Theorem 14.
The forgotten index of graphGe is given as follows:(47)FGe=4FG1+FG2+2m12m2+2m1M1G2+4FG1+2+n22×2m1+m1×2m2+m13n2+2m12×2m2+2+n22m1n2.Proof.
Using (7) and the edge set of Ge, we get(48)FGe=∑uv∈EGedGe2u+dGe2v=∑uv∈EG12dG1u2+2dG1v2+∑uv∈EG2dG2u+m12+dG2v+m12+∑u∈VG12dG1u2+2+n22dG1u+∑u∈VG2dG2u+m12+2+n22m1=4∑uv∈EG1dG12u+dG12v+∑uv∈EG2dG22u+dG22v+∑uv∈EG2m12+m12+∑uv∈EG22m1×dG2u+dG2v+∑u∈VG14dG13u+2+n22×dG1u+∑u∈VG2m1dG2u+m13+2m12dG2u+2+n22m1=4FG1+FG2+2m12m2+2m1M1G2+4FG1+2+n22×2m1+m1×2m2+m13n2+2m12×2m2+2+n22m1n2.


## 7. General Sum-Connectivity Index ofR-Vertex and Edge Join of Graphs

In this section, we obtain the general sum-connectivity index of theR-vertex and R-edge join of graphs.Theorem 15.
The general sum-connectivity index of graphG is given by(49)HMG=FG+2M2G.Proof.
Using (6), we get(50)χ2G=∑uv∈EGdGu+dGv2=∑uv∈EGdG2u+dG2v+2dGudGv=∑uv∈EGdG2u+dG2v+2∑uv∈EGdGudGv=∑u∈EGdGv3u+2∑uv∈EGdGudGv=FG+2M2G.
WhenG=Gv, substituting expressions from Theorems 13 and 4, we get(51)χ2Gv=4FG1+2n12n2+4n2M1G1+FG2+2n12m2+2n1M1G2+4FG1+n22×2m1+4n2M1G1+8m1+4M1G1+n22n1+8n2m1+M1G2+n12n2+4n1m2+24M2G1+2n2M1G1+n22m1+M2G2+n1M1G2+n12m2+4M1G1+8m1m2+2n1+n22n2m1+2n1m2+4n2m1+n12n22.
Similarly, whenG=Ge, we obtain SCIGe by substituting expressions from Theorems 14 and 6 as follows:(52)χ2Ge=4FG1+FG2+2m12m2+2m1M1G2+4FG1+2+n22×2m1+m1×2m2+m13n2+2m12×2m2+2+n22m1n2+4M2G1+M2G2+m1MG2+m12m2+4+2n22m1+m12+n22m2+m12+n2m1m2.



---

*Source: 9767128-2020-08-06.xml*