The Sigma Coindex of Graph Operations

&e 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.


Introduction
Let G be a simple graph with a vertex set V(G) and edge set E(G), where |V(G)| � n G and |E(G)| � m G . e degree of a vertex g in G, denoted by deg G (g), is defined as the number of incident edges to it. e complement of a G graph, denoted by G, is the graph with the same vertex set. Here, any two vertices g 1 and g 2 are adjacent if and only if they are not adjacent in G. e number of edges of the graph G is denoted by m G , where m G � n G 2 − m G . 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. e 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. ese 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. e oldest degree-based topological indices are the first and second Zagreb indices being considered in [2]. ese indices are defined as follows: (1) e first and second Zagreb coindices are defined as [3] M 1 (G) � respectively. e forgotten topological index was introduced by Furtula and Gutman [4], F(G), as the sum of cubes of vertex degrees: e forgotten coindex of a graph G is introduced as follows [5]: e hyper-Zagreb coindex was introduced by Veylaki et al. [7], as are defined as follows: e sigma index of a graph is defined as [8] σ(G) � 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, 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][11][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.

Some Properties of Sigma Coindex
All operations examined in this section are binary. erefore, we will consider graphs of G 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.
Proof. From definition of the sigma coindex, we have as required.
Proof. From definition of the sigma coindex, we have as required Proof. e proof follows by the expression σ(G) □ By the following proposition, we can give the sigma coindices of the complete graphs, star graphs, cycles, and path graphs.

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. e tensor product of graphs G and H, denoted as G ⊗ H, is the graph with V(G ⊗ H) � V(G) × V(H). e vertices (g 1 , h 1 ) and (g 2 , h 2 ) are adjacent if and only if g 1 is adjacent to g 2 in G and h 1 is adjacent to h 2 in H. Also we know that and Let us first formulate the sigma index of the tensor products of any two graphs as shown in eorem 1.
e proof follows from the expressions [22], and (7). Now we can express the sigma coindex under the tensor product of any two graphs using eorem 1.
e proof follows from the expressions e Cartesian product of G and H; denoted by G × H; is the graph with the vertex set V(G) × V(H). e vertices (g 1 , h 1 ) and (g 2 , h 2 ) are adjacent if either g 1 � g 2 and h 1 is adjacent to h 2 in H; or h 1 � h 2 and g 1 is adjacent to g 2 in G. Also we know that m G×H � n G m H + m G n H and Proof. In eorem 1 of [23] and eorem 1 of [9], the following formulae are given, respectively: respectively. So the proof is completed by applying (8), (16), and (17) in Proposition 3.
□ P n × P m and P n × C m are known as the rectangular grid and the C 4 nanotube (see [24]), respectively. Example 2. By using eorem 3, we have e lexicographic product Γ � G[H] of graphs G and H is a graph with the vertex set V(G) × V(H). Any two vertices (g 1 , h 1 ) and (g 2 , h 2 ) are adjacent in G [H] if and only if either g 1 is adjacent to g 2 in G or g 1 � g 2 and h 1 is adjacent We need sigma index to calculate the sigma coindex of the lexicographic product of two graphs. But eorem 2 in [9] is not true. e correct statement is shown in eorem 4.
In eorem 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 Γ � G[H] be the lexicographic product of two graphs G and H. en
Proof. From Proposition 3, we have In eorem 3 of [23], the following formula is given as By applying (18) and (21) in (20), we get Also we have So the proof is completed by applying (23) in (22). □ Example 3. e sigma coindices of the fence graph P n [P m ] and the closed fence graph C n [P m ] are given as follows: e vertices (g 1 , h 1 ) and (g 2 , h 2 ) are adjacent iff either g 1 is adjacent to Journal of Mathematics g 2 in G; or h 1 is adjacent to h 2 in H. Also we know that

Theorem 6. Let Γ � G∨H be the disjunctive product of the graphs G and H. en
Proof. From definition of the disjunctive product and the sigma coindex, we have In other world, we have e proof is completed by applying (1), (2), (6), and (8) in (26).

□
Let G and H be two graphs. e strong product of G and H is a graph with the vertex set of V(G) × V(H), denoted by G⊠H. e vertices (g 1 , h 1 ) and (g 2 , h 2 ) are adjacent iff either g 1 � g 2 and h 1 h 2 ∈ E(H); or h 1 � h 2 and g 1 g 2 ∈ E(G); or

H). Also we know that m(G⊠H) � n(G)m(H) + m(G)n(H) + 2m(G)m(H) and
Proof. From definition of the strong product and the sigma coindex, we have e sum can be split into five parts: which we denote by S 1 , S 2 , S 3 , S 4 , and S 5 , respectively. We have Similarly, we get On the contrary, we have Finally, similar to (32), we get As an application of eorem 7, we give below the sigma coindices of P n ⊠P m and P n ⊠C m .

Example 4
Let G and H be vertex-disjoint graphs. e union of graphs G and H, denoted G ∪ H, is the graph with vertex set V(G) ∪ V(H) and edge set E(G) ∪ E(H).
Proof. From the definition of the sigma coindex of a graph, we have Let G and H be vertex-disjoint graphs. en 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. e join of two graphs is also known as their sum. us, for example, the complete bipartite graph is K n + K m � K n,m . e degree of a vertex x of G + H is defined by Proof. From the definition of the sigma coindex of a graph, we have From eorem 10, we have the following result.
e graph G + K 1 is called suspension of G(see [27]). By using eorem 10, we get Example 6.
e corona product of graphs G and H, denoted G ∘ H, is the graph obtained by taking one copy of G and n H 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 ≤ n G . Corona product operation is closed with identity (see [28]).
□ Let k 1 , k 2 , . . . , k n be nonnegative integers. e thorn graph of the graph G, denoted by G * , is a graph obtained by attaching k i new vertices of degree one to the vertex g i of the graph G, i � 1, 2, . . . , n (see [29]). If k 1 � k 2 � · · · � k n � k, then G * ,k � G ∘ K k , where K k is the complement of a complete graph K x .

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. is remains as an open problem.

Data Availability
No data were used to support the study.

Conflicts of Interest
e author declares no conflicts of interest.