The Energy and Operations of Graphs

Let G be a finite and undirected simple graph on n vertices, ( ) A G is the adjacency matrix of G, 1 2 , , , n λ λ λ  are eigenvalues of ( ) A G , then the energy of G is ( ) 1 n i i ε λ = = ∑  . In this paper, we determine the energy of graphs obtained from a graph by other unary operations, or graphs obtained from two graphs by other binary operations. In terms of binary operation, we prove that the energy of product graphs 1 2 G G × is equal to the product of the energy of graphs 1 G and 2 G , and give the computational formulas of the energy of Corona graph G H  , join graph G H ∇ of two regular graphs G and H, respectively. In terms of unary operation, we give the computational formulas of the energy of the duplication graph m D G , the line graph ( ) L G , the subdivision graph ( ) S G , and the total graph ( ) T G of a regular graph G, respectively. In particularly, we obtained a lot of graphs pair of equienergetic.



are eigenvalues of ( ) A G , then the energy of G is ( ) In this paper, we determine the energy of graphs obtained from a graph by other unary operations, or graphs obtained from two graphs by other binary operations.In terms of binary operation, we prove that the energy of product graphs 1 2 G G × is equal to the product of

Introduction
Let G be a finite and undirected simple graph, with vertex set ( ) The number of vertices of G is n, and its vertices are labeled by 1 2 , , , n v v v  .The adjacency matrix ( ) A G of the graph G is a square matrix of order n, whose ( ) , i j -entry is equal to 1 if the vertices i v and j v are adjacent and is equal to zero otherwise.The characteristic polynomial of the adjacency matrix, i.e., ( ) ( ) , where n I is the unit matrix of order n, is said to be the characteristic polynomial of the graph G and will be denoted by ( ) The eigenvalues of a graph G are defined as the eigenvalues of its adjacency matrix ( ) A G , and so they are just the roots of the equation ( ) A G is a real symmetric matrix, so its eigenvalues are all real.Denoting them by 1 2 , , , n λ λ λ  and as a whole, they are called the spectrum of G. Spectral properties of graphs, including properties of the characteristic polynomial, have been extensively studied, for details, we refer to [1].In the 1970s, I. Gutman in [2] introduced the concept of the energy of G by ( ) In the Hückel molecular orbital (HMO) theory, the energy approximates the the molecular orbital energy levels of π-electrons in conjugated hydrocarbons (see [3] [4] [5] [6]).Up to now, the energy of G has been extensively studied, for details, we refer to [7] [8] [9].In this paper, we determine the energy of graphs obtained from a graph by other unary operations, or graphs obtained from two graphs by other binary operations.In terms of binary operation, we prove that the energy of product graphs

The Binary Operations of Graphs
Let 1 G and 2 G be two graphs with vertex set ( ) which two vertices, say ( ) , x y and ( ) Lemma 2.2.[11] Let A, B, C, D be matrices and the products AC, BD exist. Then Theorem 2.1.Let G, H be two graphs.Then Proof.Let .
In this way ,we find mn linearly independent eigenvectors, and hence ( ) And so . .
Let G be a graph with n vertices, and let H be a graph with m vertices.The Corona G H  is the graph with n mn + vertices obtained from G and n copies of H by joining the i-th vertex of G to each vertex in i-th copy of ( ) Let G be a graph with n vertices, and let H be an r-regular graph with m vertices.Then the characteristic polynomial of the corona G H  is given by Proof.By Lemma 2.3, we have .
. where the relations vertices and is regular of degree r n s = − , the charac- teristic polynomial of the join G is given by . where the relations

The Unary Operations of Graphs
Let G be a graph with vertex set ( ) { }  the duplication graph m D G is the graph with mn vertices obtained from mG by joining i v to each neighbors of i v in the j-th copy of ( ) Proof.If ( ) A G is the adjacency matrix of graph G, then, it is obviously that the adjacency matrix of the duplication graph m D G is ( ) , where m J is all 1 matrix of order m. the spectrum of m J is Let G be a graph, the line graph ( ) L G of graph G is the graph whose vertices are the edges of G, with two vertices in ( ) L G adjacent whenever the corresponding edge in G have exactly one vertex in common.Lemma 3.1 [1] If G is a regular graph of degree r, with n vertices and m n Corollary 3.2.Let G be a regular graph of degree r, with n vertices and , , , n r Lemma 3.2.[1] If G is a regular graph of degree r, with n vertices and , , , n r , , .

Conclusion
In this paper, we prove that ( ) ( ) ( ), For regular graphs G and H, we give the computational formulas of ( )

Corollary 3 . 1 .
Let G and H be two equienergetic graph, then m D G and m D H are equienergetic.
a graph, the subdivision graph ( ) S G of graph G is the graph obtained by inserting a new vertex into every edge of G.The graph ( ) R G of graph G is the graph obtained from G by adding, for each edge uv, a new vertex whose neighbours are u and v.The graph ( ) Q G of graph G is the graph obtained from G by inserting a new vertex into every edge of G, and joining by edges those pairs of new vertices which lie on adjacent edges of G.The total graph ( ) T G of graph G is the graph whose vertices are the vertices and edges of G, with two vertices of ( ) T G adjacent if and only if the corresponding element of G are adjacent or incident.