THE SECOND HYPER-ZAGREB INDEX OF GRAPH OPERATIONS

. A graph can be recognized by numeric number, polynomial or matrix which represent the whole graph. Topological index is a numerical descriptor of a molecule, based on a certain topological feature of the corresponding molecular graph, it is found that there is a strong correlation between the properties of chemical compounds and their molecular structure. Zagreb indices are numeric numbers related to graphs. In this study, the second Hyper-Zagreb index for some special graphs, and graph operations has been computed, that have been applied to compute the second Hyper-Zagreb index for Nano-tube and Nano-torus.


INTRODUCTION
Topological indices are real numbers related to graphs, They have many applications as tools for modeling chemical and other properties of molecules.In practical applications, Zagreb Indices are among the best topological indices applications to recognize the physical properties, chemical reactions and biological activities.[4,15].Throughout this paper, we consider a finite connected graph G that has no loops or multiple edges.The vertex and the edge sets of a graph G are denoted by V (G) and E(G), respectively.The degree of the vertex a is the number of edges joined with this vertex denoted by δ (a).
The first Zagreb index M 1 (G), and the second Zagreb index M 2 (G) were firstly considered by Gutman and Trinajstic in 1972 [1,4].They are defined as: In 2005, Li and Zheng [17] introduced the first general Zagreb index as: In 2013, Shirdel et al [14].introduced degree-based of Zagreb indices named Hyper-Zagreb index as: In 2013, Ranjini et al [13,18].re-defined the Zagreb indices, the third Zagreb indices for a graph G as: Furtula and Gutman in 2015 introduced forgotten index (F-index) [7,9] which defined as: In In this paper, we present some exact formulaes of the second Hyper-Zagreb index for some special graphs and some graph binary operations such as tensor product of graphs.We apply some results to compute the second Hyper-Zagreb index for some important classes of nano-structures such as nano-tube and nano-torus.
In order to calculate the second Hyper-Zagreb index of graph operations, we need one modern versions of Zagreb index is given by [7].
We named this index "Y-index", which studied of some graph operations [2].

PRELIMINARIES
In this part, we give the second Hyper-Zagreb index of some special graphs as: complete graph K n , cycle C n , path P n , complete bipartite graph K m,n and conical graph C m,n (cf.Fig. 1) [5].

MAIN RESULTS
In the following section, we study the second Hyper-Zagreb index of some graph operations.
Theorem 3.1: The second Hyper-Zagreb index of (G 1 ⊗ G 2 ) is given by.
Proof.By Definition 1.1 and Lemma 1.3, we have Theorem 3.2: The second Hyper-Zagreb index of (G 1 × G 2 ) is given by.
It is easy to see that the summation of step 1, step 2 complete the proof.
Theorem 3.4: The second Hyper-Zagreb index of (G 1 * G 2 ) is given by.
Proof.By Definition 1.1 and Lemma 1.3, we have Proof.By Definition 1.1, we have Therefore, Step1. ∑

Definition 1 . 1 :
2016, computed exact formulas for the Zagreb and Hyper-Zagreb indices of Carbon Nanocones CNC k [n] by Gao et al.They defined a new degree-based of Zagreb indices named second Hyper-Zagreb index [16].THE SECOND HYPER-ZAGREB INDEX OF GRAPH OPERATIONS 1457 The second Hyper-Zagreb index of a graph G defined as:

Definition 1 . 2 :
Product binary operations create a new graph G from two initial graphs G 1 , G 2 , the resulting graph has the same set of vertices V (G 1 ),V (G 2 ) and |V (G)| = |V (G 1 )||V (G 2 )| but its set of edges depends of the considered operation, i.e., if