Computing F-I ndex of D ifferent C orona P roducts of G raphs

. F-index of a graph is equal to the sum of cubes of degree of all the vertices of a given graph. Among different products of graphs, as corona product of two graphs is one of most important, in this study, the explicit expressions for F-index of different types of corona product of are obtained.


Introduction
A topological index is defined as a real valued function, which maps each molecular graph to a real number and is necessarily invariant under automorphism of graphs.There are various topological indices having strong correlation with the physicochemical characteristics and have been found to be useful in isomer discrimination, quantitative structure-activity relationship (QSAR) and structure-property relationship (QSPR).
In this article, as a molecular graph, we consider only finite, connected and undirected graphs without any self-loops or multiple edges.Let G be such a graph with vertex set V (G) and edge set E(G) so that the order and size of G is equal to n and m respectively.Let the edge connecting the vertices u and v is denoted by uv.Let, the degree of the vertex v in G is denoted by d G (v), which is the number of edges incident to v, that is, the number of first neighbors of v.
Among various degree-based topological indices, the first (M 1 (G)) and the second (M 2 (G)) Zagreb index of a G are one of the oldest and most studied topological indices introduced in [13] by Gutman and Trinajstić and defined as These indices have extensively studied both with respect to mathematical and chemical point of view.
Other than first and second Zagreb indices another topological indices was introduced in [13] and is defined as Furtula and Gutman in 2015 [12] studied this index again and named as "forgotten topological index" or "F-index".In that paper they showed that the predictive ability of this index is almost similar to that of first Zagreb index and for the entropy and acetic factor, both of them yield correlation coefficients greater than 0.95.There are various recent mathematical as well as chemical study of F-index (for details see [4,5,6,7,1,8]).Throughout this paper, as usual, C n and P n denote the cycle and path graphs on n vertices.
Among the most well known products of graphs, the corona product of graphs is one of the most important graph operations as different important class of graphs can be obtained by corona product of some general and particular graphs.For example, Corona product of graphs appears in chemical literature as plerographs of the hydrogen suppressed molecular graphs known as kenographs.Also, by specializing the components of corona product of graphs different interesting classes of graphs such as t-thorny graph, sunlet graph, bottleneck graph, suspension of graphs and some classes of bridge graphs can be obtained (for details see [3,9,10,2,17,11].In this paper, we derive some explicit expressions of different type of corona product of graphs such as subdivision-vertex corona, subdivision-edge corona, subdivision-vertex neighborhood corona, subdivision-edge neighborhood corona and vertexedge corona of two graphs.

Main Results
Let G 1 and G 2 be two simple connected graphs with n i number of vertices and m i number of edges respectively, for i ∈ {1, 2}.Different topological indices under the corona product of graphs have already been studied by some researchers.The corona product of G 1 • G 2 of these two graphs is obtained by taking one copy of G 1 and n 1 copies of G 2 ; and by joining each vertex of the i-th copy of G 2 to the i-th vertex of G 1 , where 1 ≤ i ≤ n 1 .The corona product of G 1 and G 2 has total number of (n 1 n 2 +n 1 ) vertices and (m 1 +n 1 m 2 +n 1 n 2 ) edges.Note that corona product operation of two graphs is not commutative and in case of corona product of graphs we can obtain connected graphs having pendent vertices.Inspired by this original corona product of graphs, several authors defined other different versions of corona product of graphs such as subdivision-vertex corona, subdivision-edge corona, subdivision-vertex neighborhood corona, subdivision-edge neighborhood corona and vertexedge corona etc. and investigated them (see [14,15,16]).In this section, we proceed to introduce different type of corona product of graphs and hence find the explicit expressions of F-index of that corona product of two graphs.Recall that, the subdivision graph S = S(G) is the graph obtained from G by replacing each of its edges by a path of length two, or equivalently, by inserting an additional vertex into each edge of G.

Subdivision-vertex corona
Definition 1. [14] Let G 1 and G 2 be two vertex disjoint graphs.The subdivision-vertex corona of G 1 and G 2 is denoted by G 1 ⊙ G 2 and obtained from S(G 1 ) and n 1 copies of G 2 , all vertex-disjoint, by joining the i-th vertex of V (G 1 ) to every vertex in the i-th copy of G 2 .

From definition it is clear that the subdivision-vertex corona
The subdivision-vertex corona of P 4 and P 2 is given in Figure 1.In the following theorem we calculate the F-index of the subdivision-vertex corona

Bulletin of Mathematical Sciences and Applications Vol. 19 Theorem 2. The F-index of the subdivision-vertex corona
from where the desired result follows.⊓ ⊔ Example 1.Using the last theorem, we have (i) Subdivision-edge corona Definition 3. [14] Let G 1 and G 2 be two vertex disjoint graphs.The subdivision-edge corona of G 1 and G 2 is denoted by G 1 ΘG 2 and obtained from S(G 1 ) and n 1 copies of G 2 , all vertex-disjoint, by joining the i-th vertex of V (G 1 ) to every vertex in the i-th copy of G 2 .
From definition, we have the subdivision-edge corona G 1 ΘG 2 has m 1 (1 + n 2 ) + n 1 vertices and m 1 (n 2 + m 2 + 2) edges.Also, the degree of the vertices of G 1 ΘG 2 are given by .., m 1 and j = 1, 2, ..., n 2 .The subdivision-edge corona of P 4 and P 2 is also depicted in Figure 1.In the following theorem we determine the F-index of subdivision-edge corona of two graphs G 1 and G 2 .

Theorem 4. The F-index of the subdivision-edge corona
Proof.From the definition of the subdivision-edge corona G 1 ΘG 2 , we have from where the desired result follows.⊓ ⊔

BMSA Volume 19
Example 2. The following results are direct consequence of the previous theorem.
Subdivision-vertex neighborhood corona Definition 5. [15] For two vertex disjoint graphs G 1 and G 2 , the subdivision-vertex neighborhood corona of G 1 and G 2 is denoted by G 1 G 2 and obtained from S(G 1 ) and n 1 copies of G 2 , all vertexdisjoint, by joining the neighbors of the i-th vertex of V (G 1 ) to every vertex in the i-th copy of G 2 . Let .., n 1 and j = 1, 2, ..., n 2 .The subdivision-vertex neighborhood corona of P 4 and P 2 is given in Figure 2. In the following theorem we determine the F-index of subdivision-vertex neighborhood corona of two graphs.
Proof.From definition of G 1 G 2 , we have from where the desired result follows.
⊓ ⊔ Example 3. The following results are follows from the last theorem.(i) Subdivision-edge neighborhood corona Definition 7. [15] For two vertex disjoint graphs G 1 and G 2 , the subdivision-edge neighborhood corona of G 1 and G 2 is denoted by G 1 ⋄ G 2 and obtained from S(G 1 ) and n 1 copies of G 2 , all vertexdisjoint, by joining the neighbors of the i-th vertex of V (G 1 ) to every vertex in the i-th copy of G 2 .
.., n 1 and j = 1, 2, ..., n 2 .The subdivision-edge neighborhood corona of P 4 and P 2 is given in Figure 2. In the following theorem we determine the F-index of subdivision-edge neighborhood corona of two graphs G 1 and G 2 .

Bulletin of Mathematical Sciences and Vol. 19
Proof.From definition of subdivision-edge neighborhood corona product of two graphs we have Hence we get the desired result.⊓ ⊔ Example 4. From the last theorem we have the following results.(i) The vertex-edge corona Definition 9. [16] The vertex-edge corona of two graphs G 1 and G 2 is denoted by G 1 ⊗ G 2 , is the graph obtained by taking one copy of G 1 , n 1 copies of G 2 and also m 1 copies of G 2 , then joining the i-th vertex of G 1 to every in the i-th vertex copy of G 2 and also joining the end vertices of j-th edge of G 1 to every vertex in the j-th edge copy of G 2 , where Let the vertex set of the j-th edge copy of G 2 is denoted by V je (G 2 ) = {u j1 , u j2 , ..., u jn 2 } and the vertex set of the i-th vertex copy of G 2 is denoted by V iv (G 2 ) = {w i1 , w i2 , ..., w in 2 } .Also let us denote the edge set of the j-th edge and i-th vertex copy of G 2 by E je (G 2 ) and E iv (G 2 ) respectively.From definition we have the vertex-edge corona In the following theorem we determine the F-index of vertex-edge corona of two graphs G 1 and G 2 .

BMSA Volume 19
Proof.From definition of vertex-edge corona of graphs, we have Now to calculate the contribution of A 1 , we have Also, we have the contribution of A 2 as Similarly, we get the contribution of A 3 as follows.
Adding A 1 , A 2 and A 3 , we get the desired result.

Figure 1 :
Figure 1: Subdivision-vertex and subdivision-edge corona products of P 4 and P 2 .

Figure 2 :
Figure 2: Subdivision-vertex and subdivision-edge neighborhood corona products of P 4 and P 2 .