Let G be a finite and undirected simple graph, with vertex set V ( G ) and edge set E ( G ) . The number of vertices of G is n, and its vertices are labeled by v 1 , v 2 , ⋯ , v n . 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 v i and v j are adjacent and is equal to zero otherwise. The characteristic polynomial of the adjacency matrix, i.e., d e t ( x I n − A ( G ) ) , where I n is the unit matrix of order n, is said to be the characteristic polynomial of the graph G and will be denoted by ϕ ( G , x ) . 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 ϕ ( G , x ) = 0 . since A ( G ) is a real symmetric matrix, so its eigenvalues are all real. Denoting them by λ 1 , λ 2 , ⋯ , λ n and as a whole, they are called the spectrum of G. Spectral pro- perties 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

ε ( G ) = ∑ i = 1 n | λ i | (1)

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 G 1 × G 2 is equal to the product of the energy of graphs G 1 and G 2 , 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 D m G , the line graph L ( G ) , the sub- division graph S ( G ) , and the total graph T ( G ) of a regular graph G, re- spectively. In particularly, we obtained a lot of graphs pair of equienergetic.

Two nonisomorphic graphs are said to be equienergetic if they have the same energy. Let G and H be two vertex disjoint graphs, G ∪ H denotes the union graph of G and H. m G denoted the union graph of m copies of G. K n denotes the complete graph with n vertices. For more notation and terminology, we refer the readers to standard textbooks [10] .

Let G 1 and G 2 be two graphs with vertex set V ( G 1 ) and V ( G 2 ) respec- tively. the product G 1 × G 2 is the graph with vertex set V ( G 1 ) × V ( G 2 ) , in which two vertices, say ( x 1 , y 1 ) and ( x 2 , y 2 ) , are adjacent if and only if x 1 is adjacent to x 2 in G 1 and y 1 is adjacent to y 2 in G 2 . Let A = ( a i j ) m × n , B = ( b i j ) p × q be two matrices, the Kronecker product A ⊗ B of A and B is the m p × n q matrix obtained from A by replacing each element a i j with the block a i j B .

Lemma 2.1. [1] Let A ( G 1 ) , A ( G 2 ) be adjacency matrices of graphs G 1 , G 2 , respectively. Then the product graph G 1 × G 2 has as adjacency matrix A ( G 1 ) ⊗ A ( G 2 ) .

Lemma 2.2. [11] Let A, B, C, D be matrices and the products AC, BD exist. Then

( A ⊗ B ) ( C ⊗ D ) = ( A C ) ⊗ ( B D ) . (2)

Theorem 2.1. Let G, H be two graphs. Then

ε ( G × H ) = ε ( G ) × ε ( H ) . (3)

Proof. Let λ 1 , λ 2 , ⋯ , λ n and μ 1 , μ 2 , ⋯ , μ m be the eigenvalues of G and H, respectively, suppose x i ( i = 1 , 2 , ⋯ , n ) are linearly independent eigenvectors of A ( G ) corresponding to λ 1 , λ 2 , ⋯ , λ n respectively, and y i ( i = 1 , 2 , ⋯ , m ) are linearly independent eigenvectors of A ( H ) corresponding to μ 1 , μ 2 , ⋯ , μ m respectively, Consider the vector z i j = x i ⊗ y j ( i = 1 , 2 , ⋯ , n , j = 1 , 2 , ⋯ , m ) . Using Lemma 2.1, we see that

( A ( G ) ⊗ A ( H ) ) z i j = ( A ( G ) x i ) ⊗ ( A ( H ) y j ) = λ i μ j x i ⊗ y j = λ i μ j z i j .

In this way ,we find m n linearly independent eigenvectors, and hence λ i μ j ( i = 1 , 2 , ⋯ , n , j = 1 , 2 , ⋯ , m ) are the eigenvalues of G × H .

And so

ε ( G × H ) = ∑ i = 1 n ∑ j = 1 m | λ i μ j | = ∑ i = 1 n | λ i | ∑ j = 1 m | μ j | = ε ( G ) ε ( H ) .

,

Corollary 2.1. Let G 1 , G 2 , ⋯ , G k be k graphs. Then

ε ( G 1 × G 2 × ⋯ × G k ) = ε ( G 1 ) ε ( G 2 ) ⋯ ε ( G k ) . (4)

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 + m n 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 H ( i = 1 , 2 , ⋯ , n ) .

Lemma 2.3. [1] 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

ϕ ( G ∘ H , x ) = ϕ ( G , x − m x − r ) ( ϕ ( H , x ) ) n . (5)

Theorem 2.2. Let G be a graph with n vertices, and let H be an r-regular graph with m vertices. If λ 1 , λ 2 , ⋯ , λ n and r , μ 2 , ⋯ , μ m be the eigenvalues of G and H, respectively. then

ε ( G ∘ H ) = 1 2 ∑ i = 1 n ( | r + λ i + ( r − λ i ) 2 + 4 m | + | r + λ i − ( r − λ i ) 2 + 4 m | ) + n ( ε ( H ) − r ) . (6)

Proof. By Lemma 2.3, we have

ϕ ( G ∘ H , x ) = ( x − r ) n ( x − μ 2 ) n ⋯ ( x − μ m ) n ∏ i = 1 n ( x − m x − r − λ i ) = ( x − μ 2 ) n ⋯ ( x − μ m ) n ∏ i = 1 n ( x 2 − ( r + λ i ) x + r λ i − m ) .

And so

ε ( G ∘ H ) = 1 2 ∑ i = 1 n ( | r + λ i + ( r − λ i ) 2 + 4 m | + | r + λ i − ( r − λ i ) 2 + 4 m | ) + n ( ∑ j = 2 m | μ j | ) = 1 2 ∑ i = 1 n ( | r + λ i + ( r − λ i ) 2 + 4 m | + | r + λ i − ( r − λ i ) 2 + 4 m | ) + n ( ε ( H ) − r ) .

,

Corollary 2.2. Let H 1 and H 2 be two equienergetic r-regular graph with m vertices, and let G be a graph with n vertices. Then G ∘ H 1 and G ∘ H 2 are equienergetic.

Corollary 2.3. Let m ≥ 2, n ≥ 3 . Then ε ( K n ∘ K m ) = m n + m − 2 + ( n − 1 ) m 2 + 4 m .

Proof. K m has spectrum n − 1, − 1 ( n − 1 times). Since

( m − 1 ) − 1 − ( m − 1 + 1 ) 2 + 4 m ≤ 0 , and m ≥ 2, n ≥ 3 means

( m − 1 ) + ( n − 1 ) − ( m − n ) 2 + 4 m ≥ 0 . Hence

ε ( K n ∘ K m ) = 1 2 ∑ i = 1 n ( | ( m − 1 ) + λ i + ( m − 1 − λ i ) 2 + 4 m |     + | m − 1 + λ i − ( m − 1 − λ i ) 2 + 4 m | ) + n ( ε ( H ) − ( m − 1 ) ) = m + n − 2 + ( n − 1 ) m 2 + 4 m + n ( m − 1 ) = m n + m − 2 + ( n − 1 ) m 2 + 4 m .

Let G and H be two graphs, The join G ∇ H of (disjoint) grapgs G and H is the graph obtained from G ∪ H by joining each vertex of G to each vertex of H.

Lemma 2.4. [1] If G 1 is r 1 -regular with n 1 vertices, and G 2 is r 2 - regular with n 2 vertices, then the characteristic polynomial of the join G 1 ∇ G 2 is given by

ϕ ( G 1 ∇ G 2 , x ) = ϕ ( G 1 , x ) ϕ ( G 2 , x ) ( x − r 1 ) ( x − r 2 ) ( ( x − r 1 ) ( x − r 2 ) − n 1 n 2 ) . (7)

Corollary 2.4. Let G i be r i -regular graph with n i vertices, i = 1 , 2. Then

ε ( G 1 ∇ G 2 ) = ε ( G 1 ) + ε ( G 2 ) − ( r 1 + r 2 ) + ( r 1 + r 2 ) 2 + 4 ( n 1 n 2 − r 1 r 2 ) . (8)

Corollary 2.5. Let G 1 and H 1 be two equienergetic r 1 -regular graph with n 1 vertices, and let G 2 and H 2 be two equienergetic r 2 -regular graph with n 2 vertices, then G 1 ∇ G 2 and H 1 ∇ H 2 are equienergetic.

Lemma 2.5. [1] Let G 1 , G 2 , ⋯ , G k be regular graphs, let G i have degree r i and n i vertices ( i = 1 , 2 , ⋯ , k ) . where the relations

n 1 − r 1 = n 2 − r 2 = ⋯ = n k − r k = s hold. Then the graph G = G 1 ∇ G 2 ∇ ⋯ ∇ G k has n = n 1 + n 2 + ⋯ + n k vertices and is regular of degree r = n − s , the charac- teristic polynomial of the join G is given by

ϕ ( G , x ) = ( x − r ) ( x + n − r ) k − 1 ∏ i = 1 k ϕ ( G i , x ) x − r i . (9)

By Lemma 2.5, we have following Corollary.

Corollary 2.6. Let G 1 , G 2 , ⋯ , G k be regular graphs, let G i have degree r i and n i vertices ( i = 1 , 2 , ⋯ , k ) . where the relations n 1 − r 1 = n 2 − r 2 = ⋯ = n k − r k = s hold. Then

ε ( G 1 ∇ G 2 ∇ ⋯ ∇ G k ) = 2 ( k − 1 ) s + ∑ i = 1 k     ε ( G i ) . (10)

Let G be a graph with vertex set V ( G ) = { v 1 , v 2 , ⋯ , v n } , the duplication graph D m G is the graph with m n vertices obtained from m G by joining v i to each neighbors of v i in the j-th copy of G ( j = 1 , 2 , ⋯ , m , i = 1 , 2 , ⋯ , n ) .

Theorem 3.1. Let G be a graph. Then

ε ( D m G ) = m ε ( G ) . (11)

Proof. If A ( G ) is the adjacency matrix of graph G, then, it is obviously that the adjacency matrix of the duplication graph D m G is J m ⊗ A ( G ) , where J m is all 1 matrix of order m. the spectrum of J m is m ,0 ( m − 1 ) times, similar to the proof of Theorem 2.1, we have ε ( D m G ) = m ε ( G ) .

Corollary 3.1. Let G and H be two equienergetic graph, then D m G and D m H are equienergetic.

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 corre- sponding 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 ( = 1 2 n r ) edges, then

ϕ ( L ( G ) , x ) = ( x + 2 ) m − n ϕ ( G , x − r + 2 ) . (12)

Corollary 3.2. Let G be a regular graph of degree r, with n vertices and

m ( = 1 2 n r ) edges, If λ 1 ( = r ) , λ 2 , ⋯ , λ n is the eigenvalues of G, then

ε ( L ( G ) ) = 2 ( m − n ) + ∑ i = 1 n | r + λ i − 2 | . (13)

Corollary 3.3.

ε ( L ( K n ) ) = { 2 n 2 − 6 n 4 ≤ n ,   4 ( n − 2 ) 2 ≤ n ≤ 3. (14)

Let G be 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.

Lemma 3.2. [1] If G is a regular graph of degree r, with n vertices and

m ( = 1 2 n r ) edges, then

1) ϕ ( S ( G ) , x ) = x m − n ϕ ( G , x 2 − r ) ,

2) ϕ ( R ( G ) , x ) = x m − n ( x + 1 ) n ϕ ( G , x 2 − r x + 1 ) ,

3) ϕ ( Q ( G ) , x ) = ( x + 2 ) m − n ( x + 1 ) n ϕ ( G , x 2 − ( r − 2 ) x − r x + 1 ) .

4) The total graph T ( G ) has m − n eigenvalues equal to −2 and the following 2n eigenvalues:

1 2 ( 2 λ i + r − 2 ± 4 λ i + r 2 + 4 ) ,     ( i = 1 , 2 , ⋯ , n ) .

Theorem 3.2. Let G be a regular graph of degree r, with n vertices and

m ( = 1 2 n r ) edges, If λ 1 ( = r ) , λ 2 , ⋯ , λ n is the eigenvalues of G, then

1) ε ( S ( G ) ) = 2 ∑ i = 1 n r + λ i ,

2) ε ( R ( G ) ) = ∑ i = 1 n λ i 2 + 4 ( r + λ i ) ,

3) ε ( Q ( G ) ) = 2 ( m − n ) + ∑ i = 1 n ( r + λ i ) 2 + 4 ,

4) ε ( T ( G ) ) = 2 ( m − n ) + 1 2 ∑ i = 1 n ( | 2 λ i + r − 2 + 4 λ i + r 2 + 4 |     + | 2 λ i + r − 2 − 4 λ i + r 2 + 4 | ) .

Proof. (1) By Lemma 3.2 (1), we know that the spectrum of S ( G ) is { 0 ( m − n   times ) , ± r + λ i ( i = 1 , 2 , ⋯ , n ) } . So ε ( S ( G ) ) = 2 ∑ i = 1 n r + λ i .

(2) By Lemma 3.2 (2), we know that the spectrum of R ( G ) is

{ 0 ( m − n   times ) , λ i ± λ i 2 + 4 ( r + λ i ) 2 ( i = 1 , 2 , ⋯ , n ) } . So ε ( R ( G ) ) = ∑ i = 1 n λ i 2 + 4 ( r + λ i ) .

(3), (4) Proof is similar to (1).

Corollary 3.4. 1) If n ≥ 2, then ε ( S ( K n ) ) = 2 ( 2 n − 2 + ( n − 1 ) n − 2 ) .

2) If n ≥ 2, then ε ( R ( K n ) ) = n 2 + 6 n − 7 + ( n − 1 ) 4 n − 7 .

3) ε ( Q ( K n ) ) = n 2 − 3 n + 2 n 2 − 2 n + 2 + ( n − 1 ) n 2 − 4 n + 8 .

4) If n ≥ 2, then ε ( T ( K n ) ) = { 2 n 2 − 2 n − 4 n ≥ 3 , 4 n = 2.

In this paper, we prove that ε ( G × H ) = ε ( G ) × ε ( H ) , ε ( D m G ) = m ε ( G ) . For regular graphs G and H, we give the computational formulas of ε ( G ∇ H ) , ε ( G ∘ H ) , ε ( L ( G ) ) , ε ( S ( G ) ) , ε ( R ( G ) ) , ε ( Q ( G ) ) , and ε ( T ( G ) ) re- spectively. In particularly, we obtained a lot of graphs pair of equienergetic.

This work is supported by National Natural Science Foundation of China (11561056, 11661066), National Natural Science Foundation of Qinghai Provence (2016-ZJ-914), and Scientific Research Fund of Qinghai University for Nationalities (2015G02).

Ma, H.C. and Liu, X.H. (2017) The Energy and Operations of Graphs. Advances in Pure Mathematics, 7, 345-351. https://doi.org/10.4236/apm.2017.76021