Three Topological Indices of Two New Variants of Graph Products

. Graph operations play an important role to constructing complex network structures from simple graphs, and these complex networks play vital roles in diﬀerent ﬁelds such as computer science, chemistry, and social sciences. Computation of topological indices of these complex network structures via graph operation is an important task. In this study, we deﬁned two new variants of graph products, namely, corona join and subdivision vertex join products and investigated exact expressions of the ﬁrst and second Zagreb indices and ﬁrst reformulated Zagreb index for these new products.


Introduction
In mathematics, the graph theory is the study of graphs which are mathematical structures used to model pairwise connection between objects.e graph theory is applied in the various fields such as computer science, biology, chemistry, social sciences, and operation research [1,2].
Let G � (V(G), E(G)) be a simple, connected graph with vertex set V(G) and edge set E(G). e number of vertices and number of edges are called the order n and size m, respectively, of the graph G.A graph of order n and size m will be denoted by G(n, m).For any vertex v ∈ V(G), the degree of vertex v is the number of edges incident on the vertex v, and it is written as d G (v) or simply d(v).For a simple graph G, the subdivision of the graph G is denoted by S(G) and obtained by inserting a new vertex on every edge of G.
Topological index is a numeric value which is associated with a chemical structure of a certain chemical compound.
is numeric value can help to predict certain properties of that chemical compound.Hundreds of topological indices have been introduced, but few of them gain attention of the scientific community.Zagreb indices are among the oldest and useful topological indices.For a graph G, the first and second Zagreb indices are defined as In 1972, these topological indices were applied for the first time to find the total electron π-energy of molecular graphs [3].Later, the Zagreb indices developed important applications in QSPR/QSAR studies, and a lot of research studies have been published on these [4][5][6][7][8][9][10].
Milićević et al. in 2004 reformulated the Zagreb indices in terms of edge degree which is defined as where d(e) shows the degree of the edge e in G, which is defined as d(e) � d(u) + d(v) − 2 with edge e � uv and e ∼ f shows that the edge e and f are adjacent [11].
Graph operations, especially graph products, play a significant role not only in pure and applied mathematics but also in computer science, chemistry, electrical engineering, and pharmaceutics.For instance, the Cartesian product provides a significant model for connecting computers [12].
Let G 1 (n 1 , m 1 ) and G 2 (n 2 , m 2 ) be two connected simple graphs.Corona product of graphs G 1 and G 2 , denoted by G 1 °G2 , is obtained by taking one copy of G 1 and n 1 copies of G 2 , and joining each vertex of i th copy of G 2 to i th vertex of G 1 [8].In [13], authors introduced two variants of the corona product and discussed their spectral properties.e subdivision vertex variant of corona of G 1 and G 2 is attained from S(G 1 ) and n 1 copies of G 2 by joining the i th vertex of V(G 1 ) to every vertex in the i th copy of G 2 .Similarly, the subdivision-edge neighborhood corona is obtained by attaching the neighbors of the i th vertex of V(G 1 ) to every vertex in the i th copy of G 2 .
e join graph of G 1 and G 2 is obtained by joining each vertex of G 1 to each vertex of G 2 , and it is denoted by G 1 + G 2 [14].
Khalifeh et al. [15] computed the first and second Zagreb indices of Cartesian product, composition, join, disjunction, and symmetric difference of graphs and applied the results on C 4 tube, torus, and multiwalled polyhex nanotorus.Authors in [16] investigated the upper bounds on the multiplicative Zagreb indices of some product of graphs.Azari and Iranmanesh [17] discussed the rooted product of graphs and found the exact expression of first and second Zagreb indices for this product.Jamil and Tomescu [18] found the exact formulas of the first reformulated Zagreb index for Cartesian product, composition, join, corona product, splice, link, and chain of graphs.Some graph operations and their topological indices are presented in [13,[15][16][17][18][19][20][21][22][23][24][25][26][27][28].Now, we define variants of these graphs' product.
and G 2 (n 2 , m 2 ) be simple connected graphs, and the corona join graph of G 1 and G 2 is obtained by taking one copy of G 1 , n 1 copies of G 2 , and joining each vertex of the i th copy of G 2 with all vertices of G 1 .e corona join product of G 1 and G 2 is denoted by G 1 ⊕ G 2 and shown in Figure 1.
Definition 2. For G 1 (n 1 , m 1 ) and G 2 (n 2 , m 2 ), the subdivision vertex join is denoted by G 1 ∔G 2 and obtained by joining the each new vertex of S(G 1 ) to all vertices of G 2 .Figure 2 shows the illustration of subdivision vertex join for P 3 and P 2 .

Main Results
In this section, we present the main results.e following lemmas are useful to obtain the exact expressions of topological indices of new variants of graph products.e proofs of the following two lemmas are directly from the definitions of corona join product G 1 ⊕ G 2 and subdivision vertex join G 1 ∔G 2 .

be two graphs; then, the degree behavior of vertices in the graph
Lemma 2. Let we have three simple connected graphs G 1 � (n 1 , m 1 ) and S(G 1 ) � (n 1 ′ , m 1 ′ ); then, the degree behavior of vertices in the graph e following result gives the formula of the first Zagreb index for G 1 ⊕ G 2 .
Theorem 1.Let G 1 (n 1 , m 1 ) and G 2 (n 2 , m 2 ) be two simple graphs; then, the first Zagreb index of corona join product G 1 ⊕ G 2 is given as Proof.From the definition of the first Zagreb index, we have 2 Mathematical Problems in Engineering Now, we apply Lemma 1: which is our required result.e next theorem is about the exact expression of the second Zagreb index for G 1 ⊕ G 2 .

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Theorem 2. For simple graphs G 1 (n 1 , m 1 ) and G 2 (n 2 , m 2 ), the second Zagreb index of corona join product G 1 ⊕ G 2 is given as Proof.From the definition of the second Zagreb index, we have 9) Now, we apply the Lemma 1:

Mathematical Problems in Engineering
which is our required result.

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Theorem 3. Let G 1 � (n 1 , m 1 ) and G 2 � (n 2 , m 2 ) be two simple graphs; then, the first reformulated Zagreb index of corona join product Proof.From the definition of the first reformulated Zagreb index and by Lemma 1, we have 4 Mathematical Problems in Engineering Mathematical Problems in Engineering Mathematical Problems in Engineering which is our required result.

Conclusion
In this study, we proposed two new variants of special graph products and found their exact expressions for the first Zagreb index, second Zagreb index, and first reformulated Zagreb index.ese new graph invariants can be used to construct come cellular networks, for example, the factor cellular network or some hybrid cellular networks.In future, other degree and distance-based topological indices of these graph operations can be found.e obtained results may help to construct and investigate the topological indices of complex networks structures.