# The Sigma Coindex of Graph Operations

Authors: Yasar Nacaroglu
Journal: Journal of Mathematics
 (2021)

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## Abstract

The sigma coindex is defined as the sum of the squares of the differences between the degrees of all nonadjacent vertex pairs. In this paper, we propose some mathematical properties of the sigma coindex. Later, we present precise results for the sigma coindices of various graph operations such as tensor product, Cartesian product, lexicographic product, disjunction, strong product, union, join, and corona product.


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## Body

## 1. Introduction

LetG be a simple graph with a vertex set VG and edge set EG, where VG=nG and EG=mG. The degree of a vertex g in G, denoted by degGg, is defined as the number of incident edges to it. The complement of a G graph, denoted by G¯, is the graph with the same vertex set. Here, any two vertices g1 and g2 are adjacent if and only if they are not adjacent in G. The number of edges of the graph G¯ is denoted by mG¯, where mG¯=nG2−mG. For other undefined notations and terminology from graph theory, the readers are referred to [1].Chemical graph theory is the field of study of mathematical chemistry in relation to chemical graphs. The basic idea here is to reveal the properties of molecules using the information corresponding to chemical graphics. For this, topological indices are among the most used tools. Topological indices are constant numbers that reveal the structure of the graph. These constant numbers are used in the modeling of molecules in chemistry and biology. Until today, many topology indices have been defined and used as a tool in QSAR/QSPR studies.The oldest degree-based topological indices are the first and second Zagreb indices being considered in [2]. These indices are defined as follows:(1)M1G=∑g∈VGdG2g=∑g1g2∈EGdGg1+dGg2,M2G=∑g1g2∈EGdGg1dGg2.The first and second Zagreb coindices are defined as [3](2)M¯1G=∑g1g2∉EGdGg1+dGg2,M¯2G=∑g1g2∉EGdGg1dGg2,respectively.The forgotten topological index was introduced by Furtula and Gutman [4], F(G), as the sum of cubes of vertex degrees:(3)FG=∑g∈VGdG3g=∑g1g2∈EGdG2g1+dG2g2.The forgotten coindex of a graphG is introduced as follows [5]:(4)F¯G=∑g1g2∉EGdG2g1+dG2g2.The hyper-Zagreb index was first introduced by [6]. This index is defined as follows:(5)HMG=∑g1g2∈EGdGg1+dGg22.The hyper-Zagreb coindex was introduced by Veylaki et al. [7], as are defined as follows:(6)HM¯G=∑g1g2∉EGdGg1+dGg22.The sigma index of a graph is defined as [8](7)σG=∑g1g2∈EGdGg1−dGg22=FG−2M2G.In [8], the authors studied sigma index and its properties, especially the inverse problem for it. After that Jahanbani and Ediz [9] presented the properties of this index under various graph products.With this motivation, we define the sigma coindex and the total sigma index as, respectively,(8)σ¯G=∑g1g2∉EGdGg1−dGg22,σtG=12∑g1,g2∈VGdGg1−dGg22.Graph operations are an important subject of graph theory. Many complex graphs can be obtained by applying graph operations to simpler graphs. Until today, many studies have been done on graph operations. In [10–12], the algebraic properties of tensor, lexicographic, and Cartesian products of monogenic semigroup graphs were presented. Azari [13] put forward some results on the eccentric connectivity coindex of several graph operations. In [14], the upper bounds on the multiplicative Zagreb indices of graph operations were given. Nacaroglu et al. [15] gave some bounds on the multiplicative Zagreb coindices of graph operations. Ascioglu et al. [16] presented formulae for omega invariant of some graph operations. In [17], F index of different corona products of two given graphs was calculated. Das et al. [18] examined the Harary index of graph operations. We refer the reader to [19] for more properties and applications of graph products.In this study, we will calculate the sigma coindex of two graphs under some graph products as corona, join, union, lexicographic product, disjunction, tensor product, Cartesian product, and strong product.


## 2. Some Properties of Sigma Coindex

All operations examined in this section are binary. Therefore, we will consider graphs ofG and H as two finite and simple graphs. Let us examine the sigma coindices of some special graphs before moving on to the basic results.Proposition 1.
σ¯G=σG¯.Proof.
From definition of the sigma coindex, we have(9)σ¯G=∑g1g2∉EGdGg1−dGg22=∑g1g2∈EG¯nG−1−dG¯g1−nG−1−dG¯g22=∑g1g2∈EG¯dG¯g1−dG¯g22=σG¯,as required.Proposition 2.
σ¯G¯=σG.Proof.
From definition of the sigma coindex, we have as required(10)σ¯G¯=∑g1g2∉EG¯dG¯g1−dG¯g22=∑g1g2∈EGnG−1−dGg1−nG−1−dGg22=∑g1g2∈EGdGg1−dGg22=σG.Proposition 3.
σ¯G=nGM1G−σG−4mG2.Proof.
The proof follows by the expressionσG+σ¯G=nGM1G−4mG2 from Lemma 4 of [20].By the following proposition, we can give the sigma coindices of the complete graphs, star graphs, cycles, and path graphs.Proposition 4.
(11)σ¯Kn=0,σ¯Sn=0,σ¯Cn=0,σ¯Pn=2n−3.


## 3. Sigma Coindex under Graph Operations

In this section, we give some formulae for the sigma coindices of some graph operations as union, join, corona product, tensor product, Cartesian product, lexicographic product, and strong product.The tensor product of graphsG and H, denoted as G⊗H, is the graph with VG⊗H=VG×VH. The vertices g1,h1 and g2,h2 are adjacent if and only if g1 is adjacent to g2 in G and h1 is adjacent to h2 in H. Also we know that dG⊗Hg1,h1=dGg1dHh1 and mG⊗H=2mGmH.Let us first formulate the sigma index of the tensor products of any two graphs as shown in Theorem1.Theorem 1.
LetΓ=G⊗H. Then(12)σΓ=FGFH−4M2GM2H.The proof follows from the expressionsM2G⊗H=2M2GM2H in Theorem 2.1 of [21], FG⊗H=FGFH in Theorem 7 of [22], and (7).Now we can express the sigma coindex under the tensor product of any two graphs using Theorem1.Theorem 2.
LetΓ=G⊗H. Then,(13)σ¯Γ=nGnHM1GM1H+4M2GM2H−FGFH−16mG2mH2.Proof.
From the expressionσΓ+σ¯Γ=nΓM1Γ−4mΓ2 in Lemma 4 of [20], we get(14)σ¯Γ=nG⊗HM1G⊗H−σG⊗H−4mG⊗H2.
The proof follows from the expressionsM1G1⊗G2=M1G1M1G2 in Theorem 2.1 of [21] and Theorem 1.Example 1.
Using Theorem2, we have(1)
σ¯Pn⊗Pm=8n+mnm+2−16n+m2+4nm+44,(2)
σ¯Pn⊗Cm=8mnm−2m−2,(3)
σ¯Pn⊗Km=mm−122nm−6m+2.The Cartesian product ofG and H; denoted by G×H; is the graph with the vertex set VG×VH. The vertices g1,h1 and g2,h2 are adjacent if either g1=g2 and h1 is adjacent to h2 in H; or h1=h2 and g1 is adjacent to g2 in G. Also we know that mG×H=nGmH+mGnH and dG×Hg1,h1=dGg1+dHh1, respectively.Theorem 3.
LetΓ=G×H be the Cartesian product of two graphs G and H. Then(15)σ¯Γ=nG2σtH+nH2σtG−nGσH−nHσG.Proof.
In Theorem 1 of [23] and Theorem 1 of [9], the following formulae are given, respectively:(16)M1G×H=nHM1G+nGM1H+8mGmH,(17)σG×H=nHσG+nGσH,respectively. So the proof is completed by applying (8), (16), and (17) in Proposition 3.Pn×Pm and Pn×Cm are known as the rectangular grid and the C4 nanotube (see [24]), respectively.Example 2.
By using Theorem3, we have(1)
σ¯Pn×Pm=2n2m+2nm2−4n2−4m2−2n−2m,(2)
σ¯Pn×Cm=2nm2−4m2−2m.The lexicographic productΓ=GH of graphs G and H is a graph with the vertex set VG×VH. Any two vertices g1,h1 and g2,h2 are adjacent in GH if and only if either g1 is adjacent to g2 in G or g1=g2 and h1 is adjacent to h2 in H. Also we know that mGH=nGmH+nH2mG and dGHg1,h1=nHdGg1+dHh1.We need sigma index to calculate the sigma coindex of the lexicographic product of two graphs. But Theorem 2 in [9] is not true. The correct statement is shown in Theorem 4.Theorem 4.
LetΓ=GH be the lexicographic product of two graphs G and H. Then(18)σΓ=nH4σG+nGσH+2mGnHM1H−8mGmH2.In Theorem5, we present the sigma coindex of the lexicographic product of the two graphs depending on some topological indices of these graphs.Theorem 5.
LetΓ=GH be the lexicographic product of two graphs G and H. Then(19)σ¯Γ=nH4σ¯G+nG2−2mGσtH−nGσH.Proof.
From Proposition3, we have(20)σ¯Γ=nΓM1Γ−σΓ−4mΓ2.
In Theorem 3 of [23], the following formula is given as(21)M1Γ=nH3M1G+nGM1H+8nHmHmG.
By applying (18) and (21) in (20), we get(22)σ¯Γ=nH4nGM1G−σG+nG2nH−2mGnHM1H−nGσH−4nG2mH2−2mGmH2+nH4mG2.
Also we have(23)σtG=σG+σ¯G=nGM1G−4mG2.
So the proof is completed by applying (23) in (22).Example 3.
The sigma coindices of the fence graphPnPm and the closed fence graph CnPm are given as follows:(1)
σ¯PnPm=2nm4+2n2m−6m4−4nm+4m−4n2+6n−8,(2)
σ¯CnPm=2n2m−4nm+6n−4n2.The disjunction product ofG and H; denoted by G∨H ; is a graph with the vertex set VG×VH. The vertices g1,h1 and g2,h2 are adjacent iff either g1 is adjacent to g2 in G; or h1 is adjacent to h2 in H. Also we know that mG∨H=nG2mH+mGnH2−2mGmH and dG∨Hg1,h1=nHdGg1+nGdHh1−dGg1dHh1.Theorem 6.
LetΓ=G∨H be the disjunctive product of the graphs G and H. Then(24)σ¯Γ=nH3−4nHmH+M1H+2nH2mH¯−2nHM1¯H+12HM¯Hσ¯G+nG3−4nGmG+M1G+2nG2mG¯−2nGM1¯G+12HM¯Gσ¯H.Proof.
From definition of the disjunctive product and the sigma coindex, we have(25)σ¯Γ=∑xy∉EΓdΓx−dΓy2=∑g1∈VG∑h1h2∉EHdG∨Hg1,h1−dG∨Hg1,h22∑h1∈VH∑g1g2∉EGdG∨Hg1,h1−dG∨Hg2,h12+∑g1g2∉EG∑h1h2∉EHdG∨Hg1,h1−dG∨Hg2,h22.
In other world, we have(26)σ¯Γ=∑g1∈VG∑h1h2∉EHnG−dGg12dHh1−dHh22+∑h1∈VH∑g1g2∉EGnH−dHh12dGg1−dGg22+∑g1g2∉EG∑h1h2∉EHnH−12dHh1+dHh2dGg1−dGg2+nG−12dGg1+dGg2dHh1−dHh22.
The proof is completed by applying (1), (2), (6), and (8) in (26).LetG and H be two graphs. The strong product of G and H is a graph with the vertex set of VG×VH, denoted by G⊠H. The vertices g1,h1 and g2,h2 are adjacent iff either g1=g2 and h1h2∈EH; or h1=h2 and g1g2∈EG; or g1g2∈EG and h1h2∈EH. Also we know that mG⊠H=nGmH+mGnH+2mGmH and dG⊠Hg1,h1=dGg1+dHh1+dGg1dHg2.Theorem 7.
LetΓ=G⊠H be the strong product of two graphs G and H. Then(27)σ¯Γ=2mH¯+2M1¯H+12HM¯HσG+2mG¯+2M1¯G+12HM¯GσH+12nHM1H+4mHnH+2mH2+nH2σ¯G+12nGM1G+4mGnG+2mG2+nG2σ¯H.Proof.
From definition of the strong product and the sigma coindex, we have(28)σ¯G⊠H=∑g1,h1g2,h2∉EG⊠HdG⊠Hg1,h1−dG⊠Hg1,h22.
The sum can be split into five parts:(29)∑g1=g2∈VG∑h1h2∉EH,∑h1=h2∈VH∑h1h2∉EH,∑g1g2∈EG∑h1h2∉EH,∑g1g2∉EG∑h1h2∈EH,∑g1g2∉EG∑h1h2∉EH,which we denote by S1,S2,S3,S4, and S5, respectively. We have(30)S1=∑g1∈VG∑h1h2∉EHdG⊠Hg1,h1−dG⊠Hg1,h22=∑g1∈VG∑h1h2∉EHdGg1+12dHh1−dHh22=M1G+4mG+nGσ¯H.
Similarly, we get(31)S2=∑h1∈VH∑g1g2∉EGdG⊠Hg1,h1−dG⊠Hg2,h12=M1H+4mH+nHσ¯G.
On the contrary, we have(32)S3=∑g1g2∈EG∑h1h2∉EHdG⊠Hg1,h1−dG⊠Hg2,h22=∑g1g2∈EG∑h1h2∉EH1+12dHh1+dHh2dGg1−dGg2+1+12dGg1+dGg2dHh1−dHh22=2∑g1g2∈EG∑h1h2∉EH1+dHh1+dHh2+14dHh1+dHh22dGg1−dGg22+1+dGg1+dGg2+14dGg1+dGg22dHh1−dHh22=2mH¯+2M1¯H+12HM¯HσG+2mG+2M1G+12HMGσ¯H.
Finally, similar to (32), we get(33)S4=∑g1g2∉EG∑h1h2∈EHdG⊠Hg1,h1−dG⊠Hg2,h22=2mG¯+2M1¯G+12HM¯GσH+2mH+2M1H+12HMHσ¯G,(34)S5=∑g1g2∉EG∑h1h2∉EHdG⊠Hg1,h1−dG⊠Hg2,h22=2mH¯+2M1¯H+12HM¯Hσ¯G+2mG¯+2M1¯G+12HM¯Gσ¯H.
The proof is completed by using (30)–(34) and the relations HMG+HM¯G=nG−2M1G+4mG2 in Theorem 2.1 of [25], M1G+M1¯G=2mGnG−1 in Theorem 1 of [26].As an application of Theorem7, we give below the sigma coindices of Pn⊠Pm and Pn⊠Cm.Example 4.
(35)Pn⊠Pm=6n+m3nm−6n−6m−1+28nm+140,Pn⊠Cm=18m2+18nm2−108m.LetG and H be vertex-disjoint graphs. The union of graphs G and H, denoted G∪H, is the graph with vertex set VG∪VH and edge set EG∪EH.Theorem 8.
LetΓ=G∪H. Then(36)σ¯Γ=σ¯G+σ¯H+nHM1G+nGM1H−8mGmH.Proof.
From the definition of the sigma coindex of a graph, we have(37)σ¯Γ=∑xy∉EGdΓx−dΓy2=∑g1g2∉EGdGg1−dGg22+∑h1h2∉EHdHh1−dHh22+∑g∈VG∑h∈VHdGg−dHh2=σ¯G+σ¯H+∑g∈VG∑h∈VHdG2g+dH2h−2dGgdHh=σ¯G+σ¯H+nHM1G+nGM1H−8mGmH.Example 5.
σ¯Kn1∪Kn2=n1n2n1−n22.LetG and H be vertex-disjoint graphs. Then the join, G+H, of G and H, is the supergraph of G∪H in which each vertex of G is adjacent to every vertex of H. The join of two graphs is also known as their sum. Thus, for example, the complete bipartite graph is Kn+Km=Kn,m. The degree of a vertex x of G+H is defined by(38)dG+Hx=dGx+nH,ifx∈VG,dHx+nG,ifx∈VH.Theorem 10.
LetΓ=G+H. Then(39)σ¯Γ=σ¯G+σ¯H.Proof.
From the definition of the sigma coindex of a graph, we have(40)σ¯Γ=∑xy∉EΓdΓx−dΓy2=∑g1g2∉EGdGg1+nH−dGg2+nH2+∑h1h2∉EHdHh1+nG−dHh2+nG2=σ¯G+σ¯H.LetG=G1+G2+⋯+Gp. Thus nGi¯=n−nGi1≤i≤p. Also we have(41)dGu=dGiu+nGi,u∈VGi.From Theorem10, we have the following result.Corollary 1.
LetG1,G2,…,Gp be vertex-disjoint graphs. If G=G1+G2+⋯+Gp, then(42)σ¯G=∑i=1pσ¯Gi.The graphG+K1 is called suspension of G(see [27]). By using Theorem 10, we get Example 6.Example 6.
σ¯G+K1=σ¯G.The corona product of graphsG and H, denoted G∘H, is the graph obtained by taking one copy of G and nH copies of H and then joining the i−th vertex of G to every vertex in the i−th copy of H for 1≤i≤nG. Corona product operation is closed with identity (see [28]).Theorem 11.
LetΓ=G∘H. Then(43)σ¯Γ=σ¯G+nGσ¯H+2nG2σtH+nG−1K,where(44)K=nHM1G+nGM1H+4mGnHnH−1−4mHnGnH−1−8mGmH+nGnHnH−12.Proof.
From definition of the sigma coindex, we have(45)σ¯Γ=∑xy∉EΓdΓx−dΓy2=∑g1g2∉EGdGg1+nH−dGg2+nH2+nG∑h1h2∉EHdHh1+1−dHh2+12+nG−1∑g1∈VG∑h1∈VHdGg1+nH−dHh1+12+nG2∑h1,h2∈VHdHh1+1−dHh2−12=σ¯G+nGσ¯H+2nG2σtH+nG−1K.Letk1,k2,…,kn be nonnegative integers. The thorn graph of the graph G, denoted by G∗, is a graph obtained by attaching ki new vertices of degree one to the vertex gi of the graph G, i=1,2,…,n (see [29]). If k1=k2=⋯=kn=k, then G∗,k≅G∘Kk¯, where Kk¯ is the complement of a complete graph Kx.Corollary 2.
σ¯G∗,k=σ¯G+kn−1M1G+kk−1n−14m+nk−1.


## 4. Conclusions

In this paper, we have presented the exact formulae for the sigma coindices of graphs under some graph operations. We have also applied these results to some special graph types. However, there are also graph products that are not presented here. This remains as an open problem.



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