eu ON THE HYPER-ZAGREB INDEX OF SOME OPERATIONS ON GRAPHS

In general, a topological index, also known as a graph-theoretic index, is a numerical invariant of a chemical graph (Plavs̆ić et.al. 1993) [20]. There are two major types of topological indices, namley the distance based and the degree based. The very first paper studied in this aspect was the Wiener index of a graph by using distance between pair of vertices in the graph by the famous chemist Herold Wiener while he was trying to find the boiling point of paraffin [22] by considering the structural property of molecular graphs. Another topo-


Introduction
In general, a topological index, also known as a graph-theoretic index, is a numerical invariant of a chemical graph (Plavsić et.al. 1993) [20].There are two major types of topological indices, namley the distance based and the degree based.The very first paper studied in this aspect was the Wiener index of a graph by using distance between pair of vertices in the graph by the famous chemist Herold Wiener while he was trying to find the boiling point of paraffin [22] by considering the structural property of molecular graphs.Another topo-logical index is the Zagreb indices, which dates back to 30 years, see [9,19].The first and second Zagreb indices of a graph are the most studied degree based topological indices introduced by Gutman and Trinajstic(1972) used to analyze the structure-dependency of the total π-electron energy(ε), see [10].For more historical background of Zagreb indices we refer to [2,3,5,7,[14][15][16][17][18]23].
G.H Shirdel and H.Rezapour introduced the hyper-Zagreb index of a graph in 2013 and studied for some graph operations.Recently B.Basavanagoud and S.Patil published a paper on hyper-Zagreb index [2] and rectified some errors of [22].Wei gao et.al.studied hyper-Zagreb for cartesian product of some special graphs and applied these results in some chemical graphs [21].Eliasi and Taeri [7] introduced four new operations of F-sums and obtained the expression for the Wiener index Recently in [11], first and second Zagreb indices for the F-sums of graphs were presented.In this paper we obtain the explicit formulae for hyper-Zagreb index of a graph for four operations in F-sums.For the computational techniques we refer [8,11,12,13].

Definition and Preliminaries
Let G be a simple connected undirected graph with a vertex set V (G) and edge set E(G).A topological representation of a molecule is known as molecular graph.A molecular graph is a collection of vertices representing atoms of molecule and edges representing the covalent bonds between the atoms.Topological indices can be used directly as the numerical values in biological parameters of molecules in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) [6].For any vertex vǫV (G), d G (v) denotes the degree of a vertex v in G.The first and second Zagreb indices of a graph G are denoted by M 1 (G) and M 2 (G) and they are defined as, The hyper-Zagreb index of a graph G is denoted by HM (G) and defined as Cartesian product of two connected graphs G 1 and G 2 , denoted by G 1 ×G 2 , is a graph such that the set of vertices is

F-Sums
For any simple finite connected graph G, Suppose that G 1 and G 2 are two connected graphs.Based on these above operations, four new operations on these graphs are as defined below: The Figure 1 and Figure 2 depicts the graphs of the four operations

Main Results
In this section we obtain the results of hyper-Zagreb index of graphs for four operations of F-sums.
Theorem 3.1.Let G 1 and G 2 be two connected graphs.Then Proof.We have Using ( 1) and ( 2), we receive Theorem 3.2.Let G 1 and G 2 be two connected graphs.Then Proof.We receive Since here we consider edges of G only, then Hence, ( 1) and ( 2) gives Theorem 3.3.Let G 1 and G 2 be connected graphs.Then Proof.We have X i and X j are vertices of L(G 1 ), so we have Therefore, (1) and (2) gives Theorem 3.4.Let G 1 and G 2 be two connected graphs.Then Proof.
By use of Theorem 3.2, we receive