The Extended Adjacency Indices for Several Types of Graph Operations

: Let G be a simple graph without multiple edges and any loops . At first, the extended adjacency matrix of a graph was first proposed by Yang et al in 1994

As a class of degree-based topological molecular descriptor, the extended adjacency index of a connected graph was proposed by Yang [21] in 1994, and defined as where d u and d v are the degrees of vertices u and v, respectively.
Recently, some new applications [10,11,21] and properties of the EA indices [4,20] are found.In particular, some extremal graphs and equienergetic graphs are characterized [1,4].In this work, we show the extended adjacency indices for several types of graph operations such as tensor product, disjunction and strong product.
The definitions below will be used.
Definition 1 [21] The extended adjacency index of a connected graph G is defined as where is the degree of vertex in G. Definition 2 [6] The first Zagreb index defined as, ∑ ∈ ∑ , ∈ .
Definition 3 [18] The tensor product of two graphs and is the graph ⊗ with the vertex set , and the two vertices For other undefined notations and terminologies, refer to [3].Examples of graph operations are as follows, see Figures 1-3.

Main Results
In this section, we calculate the EA indices for the graph operations above., where ∆ ( @ ) and ∆ @ are the maximum (minimum) degrees of and , respectively.
Proof Let where, .

Conclusion
In this work, we compute the EA indices for three graph operations, and use the results to calculate some special graphs.Except that, more other graph operations for EA index and combinations of general graphs can be considered in further researches.Furthermore, it is a meaningful and challenging problem to generalize the EA index of ordinary graph to hypergraph.
!, " and #, 2 Let D B and D E be two cycles with orders n and m, respectively.Then ⊗ 2,9.Corollary 2.3 Let 4 B and 4 E be two complete graphs with orders n and m, respectively.Then ⊗ BE BF EF .
Corollary 2.10 Let D B and D E be two cycles with orders n and m, respectively.Then ⊠ ) = 49,.Corollary 2.11 Let 4 B and 4 E be two complete graphs with orders n and m, respectively.Then