#### 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.

#### 1. Introduction

Let be a simple graph with a vertex set and edge set , where and . The degree of a vertex in , denoted by , is defined as the number of incident edges to it. The complement of a graph, denoted by , is the graph with the same vertex set. Here, any two vertices and are adjacent if and only if they are not adjacent in . The number of edges of the graph is denoted by , where . For other undefined notations and terminology from graph theory, the readers are referred to .

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 . These indices are defined as follows:

The first and second Zagreb coindices are defined as respectively.

The forgotten topological index was introduced by Furtula and Gutman , F(G), as the sum of cubes of vertex degrees:

The forgotten coindex of a graph is introduced as follows :

The hyper-Zagreb index was first introduced by . This index is defined as follows:

The hyper-Zagreb coindex was introduced by Veylaki et al. , as are defined as follows:

The sigma index of a graph is defined as 

In , the authors studied sigma index and its properties, especially the inverse problem for it. After that Jahanbani and Ediz  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,

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 , the algebraic properties of tensor, lexicographic, and Cartesian products of monogenic semigroup graphs were presented. Azari  put forward some results on the eccentric connectivity coindex of several graph operations. In , the upper bounds on the multiplicative Zagreb indices of graph operations were given. Nacaroglu et al.  gave some bounds on the multiplicative Zagreb coindices of graph operations. Ascioglu et al.  presented formulae for omega invariant of some graph operations. In , F index of different corona products of two given graphs was calculated. Das et al.  examined the Harary index of graph operations. We refer the reader to  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 of and 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. .

Proof. From definition of the sigma coindex, we haveas required.

Proposition 2. .

Proof. From definition of the sigma coindex, we have as required

Proposition 3. .

Proof. The proof follows by the expression from Lemma 4 of .

By the following proposition, we can give the sigma coindices of the complete graphs, star graphs, cycles, and path graphs.

Proposition 4.

#### 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 graphs and , denoted as , is the graph with . The vertices and are adjacent if and only if is adjacent to in and is adjacent to in . Also we know that and .

Let us first formulate the sigma index of the tensor products of any two graphs as shown in Theorem 1.

Theorem 1. Let . Then

The proof follows from the expressions in Theorem 2.1 of , in Theorem 7 of , and (7).

Now we can express the sigma coindex under the tensor product of any two graphs using Theorem 1.

Theorem 2. Let . Then,

Proof. From the expression in Lemma 4 of , we getThe proof follows from the expressions in Theorem 2.1 of  and Theorem 1.

Example 1. Using Theorem 2, we have(1),(2),(3).

The Cartesian product of and ; denoted by ; is the graph with the vertex set . The vertices and are adjacent if either and is adjacent to in ; or and is adjacent to in . Also we know that and , respectively.

Theorem 3. Let be the Cartesian product of two graphs and . Then

Proof. In Theorem 1 of  and Theorem 1 of , the following formulae are given, respectively:respectively. So the proof is completed by applying (8), (16), and (17) in Proposition 3.

and are known as the rectangular grid and the nanotube (see ), respectively.

Example 2. By using Theorem 3, we have(1),(2).

The lexicographic product of graphs and is a graph with the vertex set . Any two vertices and are adjacent in if and only if either is adjacent to in or and is adjacent to in . Also we know that and .

We need sigma index to calculate the sigma coindex of the lexicographic product of two graphs. But Theorem 2 in  is not true. The correct statement is shown in Theorem 4.

Theorem 4. Let be the lexicographic product of two graphs and . Then

In Theorem 5, we present the sigma coindex of the lexicographic product of the two graphs depending on some topological indices of these graphs.

Theorem 5. Let be the lexicographic product of two graphs and . Then

Proof. From Proposition 3, we haveIn Theorem 3 of , the following formula is given asBy applying (18) and (21) in (20), we getAlso we haveSo the proof is completed by applying (23) in (22).

Example 3. The sigma coindices of the fence graph and the closed fence graph are given as follows:(1),(2).

The disjunction product of and ; denoted by ; is a graph with the vertex set . The vertices and are adjacent iff either is adjacent to in ; or is adjacent to in . Also we know that and .

Theorem 6. Let be the disjunctive product of the graphs and . Then

Proof. From definition of the disjunctive product and the sigma coindex, we haveIn other world, we haveThe proof is completed by applying (1), (2), (6), and (8) in (26).

Let and be two graphs. The strong product of and is a graph with the vertex set of , denoted by . The vertices and are adjacent iff either and ; or and ; or and . Also we know that and .

Theorem 7. Let be the strong product of two graphs and . Then

Proof. From definition of the strong product and the sigma coindex, we haveThe sum can be split into five parts:which we denote by , and , respectively. We haveSimilarly, we getOn the contrary, we haveFinally, similar to (32), we getThe proof is completed by using (30)–(34) and the relations in Theorem 2.1 of , in Theorem 1 of .

As an application of Theorem 7, we give below the sigma coindices of and .

Example 4.

Let and be vertex-disjoint graphs. The union of graphs and , denoted , is the graph with vertex set and edge set .

Theorem 8. Let . Then

Proof. From the definition of the sigma coindex of a graph, we have

Example 5. .

Let and be vertex-disjoint graphs. Then the join, , of and , is the supergraph of in which each vertex of is adjacent to every vertex of . The join of two graphs is also known as their sum. Thus, for example, the complete bipartite graph is . The degree of a vertex of is defined by

Theorem 10. Let . Then

Proof. From the definition of the sigma coindex of a graph, we have

Let . Thus . Also we have

From Theorem 10, we have the following result.

Corollary 1. Let be vertex-disjoint graphs. If , then

The graph is called suspension of (see ). By using Theorem 10, we get Example 6.

Example 6. .

The corona product of graphs and , denoted , is the graph obtained by taking one copy of and copies of and then joining the vertex of to every vertex in the copy of for . Corona product operation is closed with identity (see ).

Theorem 11. Let . Thenwhere

Proof. From definition of the sigma coindex, we have

Let be nonnegative integers. The thorn graph of the graph , denoted by , is a graph obtained by attaching new vertices of degree one to the vertex of the graph , (see ). If , then , where is the complement of a complete graph .

Corollary 2. .

#### 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.

#### Data Availability

No data were used to support the study.

#### Conflicts of Interest

The author declares no conflicts of interest.