The Energy and Operations of Graphs

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1. Introduction

Let G be a finite and undirected simple graph, with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$ . The number of vertices of G is n, and its vertices are labeled by ${v}_{1}\mathrm{,}{v}_{2}\mathrm{,}\cdots ,{v}_{n}$ . The adjacency matrix $A\left(G\right)$ of the graph G is a square matrix of order n, whose $\left(i\mathrm{,}j\right)$ -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., $det\left(x{I}_{n}-A\left(G\right)\right)$ , 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 $\varphi \left(G\mathrm{,}x\right)$ . The eigenvalues of a graph G are defined as the eigenvalues of its adjacency matrix $A\left(G\right)$ , and so they are just the roots of the equation $\varphi \left(G,x\right)=0$ . since $A\left(G\right)$ is a real symmetric matrix, so its eigenvalues are all real. Denoting them by ${\lambda}_{1}\mathrm{,}{\lambda}_{2}\mathrm{,}\cdots \mathrm{,}{\lambda}_{n}$ 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

$\epsilon \left(G\right)={\displaystyle \underset{i=1}{\overset{n}{\sum}}}\left|{\lambda}_{i}\right|$ (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 ${G}_{1}\times {G}_{2}$ 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 $G\circ H$ , join graph $G\nabla H$ 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 $L\left(G\right)$ , the sub- division graph $S\left(G\right)$ , and the total graph $T\left(G\right)$ 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, $G\cup H$ denotes the union graph of G and H. $mG$ 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] .

2. The Binary Operations of Graphs

Let ${G}_{1}$ and ${G}_{2}$ be two graphs with vertex set $V\left({G}_{1}\right)$ and $V\left({G}_{2}\right)$ respec- tively. the product ${G}_{1}\times {G}_{2}$ is the graph with vertex set $V\left({G}_{1}\right)\times V\left({G}_{2}\right)$ , in which two vertices, say $\left({x}_{1}\mathrm{,}{y}_{1}\right)$ and $\left({x}_{2}\mathrm{,}{y}_{2}\right)$ , 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 $A={\left({a}_{ij}\right)}_{m\times n}$ , $B={\left({b}_{ij}\right)}_{p\times q}$ be two matrices, the Kronecker product $A\otimes B$ of A and B is the $mp\times nq$ matrix obtained from A by replacing each element ${a}_{ij}$ with the block ${a}_{ij}B$ .

Lemma 2.1. [1] Let $A\left({G}_{1}\right)$ , $A\left({G}_{2}\right)$ be adjacency matrices of graphs ${G}_{1}$ , ${G}_{2}$ , respectively. Then the product graph ${G}_{1}\times {G}_{2}$ has as adjacency matrix $A\left({G}_{1}\right)\otimes A\left({G}_{2}\right)$ .

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

$\left(A\otimes B\right)\left(C\otimes D\right)=\left(AC\right)\otimes \left(BD\right).$ (2)

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

$\epsilon \left(G\times H\right)=\epsilon \left(G\right)\times \epsilon \left(H\right).$ (3)

Proof. Let ${\lambda}_{1}\mathrm{,}{\lambda}_{2}\mathrm{,}\cdots \mathrm{,}{\lambda}_{n}$ and ${\mu}_{1}\mathrm{,}{\mu}_{2}\mathrm{,}\cdots \mathrm{,}{\mu}_{m}$ be the eigenvalues of G and H, respectively, suppose ${x}_{i}\left(i=1,2,\cdots ,n\right)$ are linearly independent eigenvectors of $A\left(G\right)$ corresponding to ${\lambda}_{1}\mathrm{,}{\lambda}_{2}\mathrm{,}\cdots \mathrm{,}{\lambda}_{n}$ respectively, and ${y}_{i}\left(i=1,2,\cdots ,m\right)$ are linearly independent eigenvectors of $A\left(H\right)$ corresponding to ${\mu}_{1}\mathrm{,}{\mu}_{2}\mathrm{,}\cdots \mathrm{,}{\mu}_{m}$ respectively, Consider the vector ${z}_{ij}={x}_{i}\otimes {y}_{j}\left(i=1,2,\cdots ,n,j=1,2,\cdots ,m\right).$ Using Lemma 2.1, we see that

$\left(A\left(G\right)\otimes A\left(H\right)\right){z}_{ij}=\left(A\left(G\right){x}_{i}\right)\otimes \left(A\left(H\right){y}_{j}\right)={\lambda}_{i}{\mu}_{j}{x}_{i}\otimes {y}_{j}={\lambda}_{i}{\mu}_{j}{z}_{ij}.$

In this way ,we find $mn$ linearly independent eigenvectors, and hence ${\lambda}_{i}{\mu}_{j}\left(i=1,2,\cdots ,n,j=1,2,\cdots ,m\right)$ are the eigenvalues of $G\times H$ .

And so

$\epsilon \left(G\times H\right)={\displaystyle \underset{i=1}{\overset{n}{\sum}}}{\displaystyle \underset{j=1}{\overset{m}{\sum}}}\left|{\lambda}_{i}{\mu}_{j}\right|={\displaystyle \underset{i=1}{\overset{n}{\sum}}}\left|{\lambda}_{i}\right|{\displaystyle \underset{j=1}{\overset{m}{\sum}}}\left|{\mu}_{j}\right|=\epsilon \left(G\right)\epsilon \left(H\right).$

,

Corollary 2.1. Let ${G}_{1}\mathrm{,}{G}_{2}\mathrm{,}\cdots \mathrm{,}{G}_{k}$ be k graphs. Then

$\epsilon \left({G}_{1}\times {G}_{2}\times \cdots \times {G}_{k}\right)=\epsilon \left({G}_{1}\right)\epsilon \left({G}_{2}\right)\cdots \epsilon \left({G}_{k}\right).$ (4)

Let G be a graph with n vertices, and let H be a graph with m vertices. The Corona $G\circ H$ 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 $H\left(i=1,2,\cdots ,n\right).$

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 $G\circ H$ is given by

$\varphi \left(G\circ H\mathrm{,}x\right)=\varphi \left(G\mathrm{,}x-\frac{m}{x-r}\right){\left(\varphi \left(H\mathrm{,}x\right)\right)}^{n}\mathrm{.}$ (5)

Theorem 2.2. Let G be a graph with n vertices, and let H be an r-regular graph with m vertices. If ${\lambda}_{1}\mathrm{,}{\lambda}_{2}\mathrm{,}\cdots \mathrm{,}{\lambda}_{n}$ and $r\mathrm{,}{\mu}_{2}\mathrm{,}\cdots \mathrm{,}{\mu}_{m}$ be the eigenvalues of G and H, respectively. then

$\epsilon \left(G\circ H\right)=\frac{1}{2}{\displaystyle \underset{i=1}{\overset{n}{\sum}}}\left(\left|r+{\lambda}_{i}+\sqrt{{\left(r-{\lambda}_{i}\right)}^{2}+4m}\right|+\left|r+{\lambda}_{i}-\sqrt{{\left(r-{\lambda}_{i}\right)}^{2}+4m}\right|\right)+n\left(\epsilon \left(H\right)-r\right).$ (6)

Proof. By Lemma 2.3, we have

$\begin{array}{c}\varphi \left(G\circ H,x\right)={\left(x-r\right)}^{n}{\left(x-{\mu}_{2}\right)}^{n}\cdots {\left(x-{\mu}_{m}\right)}^{n}{\displaystyle \underset{i=1}{\overset{n}{\prod}}}\left(x-\frac{m}{x-r}-{\lambda}_{i}\right)\\ ={\left(x-{\mu}_{2}\right)}^{n}\cdots {\left(x-{\mu}_{m}\right)}^{n}{\displaystyle \underset{i=1}{\overset{n}{\prod}}}\left({x}^{2}-\left(r+{\lambda}_{i}\right)x+r{\lambda}_{i}-m\right).\end{array}$

And so

$\begin{array}{c}\epsilon \left(G\circ H\right)=\frac{1}{2}{\displaystyle \underset{i=1}{\overset{n}{\sum}}}\left(\left|r+{\lambda}_{i}+\sqrt{{\left(r-{\lambda}_{i}\right)}^{2}+4m}\right|+\left|r+{\lambda}_{i}-\sqrt{{\left(r-{\lambda}_{i}\right)}^{2}+4m}\right|\right)+n\left({\displaystyle \underset{j=2}{\overset{m}{\sum}}}\left|{\mu}_{j}\right|\right)\\ =\frac{1}{2}{\displaystyle \underset{i=1}{\overset{n}{\sum}}}\left(\left|r+{\lambda}_{i}+\sqrt{{\left(r-{\lambda}_{i}\right)}^{2}+4m}\right|+\left|r+{\lambda}_{i}-\sqrt{{\left(r-{\lambda}_{i}\right)}^{2}+4m}\right|\right)+n\left(\epsilon \left(H\right)-r\right).\end{array}$

,

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 $G\circ {H}_{1}$ and $G\circ {H}_{2}$ are equienergetic.

Corollary 2.3. Let $m\ge \mathrm{2,}n\ge 3$ . Then $\epsilon \left({K}_{n}\circ {K}_{m}\right)=mn+m-2+\left(n-1\right)\sqrt{{m}^{2}+4m}.$

Proof. ${K}_{m}$ has spectrum $n-\mathrm{1,}-1$ ( $n-1$ times). Since

$\left(m-1\right)-1-\sqrt{{\left(m-1+1\right)}^{2}+4m}\le 0$ , and $m\ge \mathrm{2,}n\ge 3$ means

$\left(m-1\right)+\left(n-1\right)-\sqrt{{\left(m-n\right)}^{2}+4m}\ge 0$ . Hence

$\begin{array}{c}\epsilon \left({K}_{n}\circ {K}_{m}\right)=\frac{1}{2}{\displaystyle {\sum}_{i=1}^{n}}(\left|\left(m-1\right)+{\lambda}_{i}+\sqrt{{\left(m-1-{\lambda}_{i}\right)}^{2}+4m}\right|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left|m-1+{\lambda}_{i}-\sqrt{{\left(m-1-{\lambda}_{i}\right)}^{2}+4m}\right|)+n\left(\epsilon \left(H\right)-\left(m-1\right)\right)\\ =m+n-2+\left(n-1\right)\sqrt{{m}^{2}+4m}+n\left(m-1\right)\\ =mn+m-2+\left(n-1\right)\sqrt{{m}^{2}+4m}.\end{array}$

Let G and H be two graphs, The join $G\nabla H$ of (disjoint) grapgs G and H is the graph obtained from $G\cup H$ 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 ${G}_{1}\nabla {G}_{2}$ is given by

$\varphi \left({G}_{1}\nabla {G}_{2},x\right)=\frac{\varphi \left({G}_{1},x\right)\varphi \left({G}_{2},x\right)}{\left(x-{r}_{1}\right)\left(x-{r}_{2}\right)}\left(\left(x-{r}_{1}\right)\left(x-{r}_{2}\right)-{n}_{1}{n}_{2}\right).$ (7)

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

$\epsilon \left({G}_{1}\nabla {G}_{2}\right)=\epsilon \left({G}_{1}\right)+\epsilon \left({G}_{2}\right)-\left({r}_{1}+{r}_{2}\right)+\sqrt{{\left({r}_{1}+{r}_{2}\right)}^{2}+4\left({n}_{1}{n}_{2}-{r}_{1}{r}_{2}\right)}.$ (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 ${G}_{1}\nabla {G}_{2}$ and ${H}_{1}\nabla {H}_{2}$ are equienergetic.

Lemma 2.5. [1] Let ${G}_{1}\mathrm{,}{G}_{2}\mathrm{,}\cdots \mathrm{,}{G}_{k}$ be regular graphs, let ${G}_{i}$ have degree ${r}_{i}$ and ${n}_{i}$ vertices $\left(i=1,2,\cdots ,k\right)$ . where the relations

${n}_{1}-{r}_{1}={n}_{2}-{r}_{2}=\cdots ={n}_{k}-{r}_{k}=s$ hold. Then the graph $G={G}_{1}\nabla {G}_{2}\nabla \cdots \nabla {G}_{k}$ has $n={n}_{1}+{n}_{2}+\cdots +{n}_{k}$ vertices and is regular of degree $r=n-s$ , the charac- teristic polynomial of the join G is given by

$\varphi \left(G,x\right)=\left(x-r\right){\left(x+n-r\right)}^{k-1}{\displaystyle \underset{i=1}{\overset{k}{\prod}}}\frac{\varphi \left({G}_{i},x\right)}{x-{r}_{i}}.$ (9)

By Lemma 2.5, we have following Corollary.

Corollary 2.6. Let ${G}_{1}\mathrm{,}{G}_{2}\mathrm{,}\cdots \mathrm{,}{G}_{k}$ be regular graphs, let ${G}_{i}$ have degree ${r}_{i}$ and ${n}_{i}$ vertices $\left(i=1,2,\cdots ,k\right)$ . where the relations ${n}_{1}-{r}_{1}={n}_{2}-{r}_{2}=\cdots ={n}_{k}-{r}_{k}=s$ hold. Then

$\epsilon \left({G}_{1}\nabla {G}_{2}\nabla \cdots \nabla {G}_{k}\right)=2\left(k-1\right)s+{\displaystyle \underset{i=1}{\overset{k}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\epsilon \left({G}_{i}\right).$ (10)

3. The Unary Operations of Graphs

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

Theorem 3.1. Let G be a graph. Then

$\epsilon \left({D}_{m}G\right)=m\epsilon \left(G\right).$ (11)

Proof. If $A\left(G\right)$ is the adjacency matrix of graph G, then, it is obviously that the adjacency matrix of the duplication graph ${D}_{m}G$ is ${J}_{m}\otimes A\left(G\right)$ , where ${J}_{m}$ is all 1 matrix of order m. the spectrum of ${J}_{m}$ is $m\mathrm{,0}\left(m-1\right)$ times, similar to the proof of Theorem 2.1, we have $\epsilon \left({D}_{m}G\right)=m\epsilon \left(G\right).$

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 $L\left(G\right)$ of graph G is the graph whose vertices are the edges of G, with two vertices in $L\left(G\right)$ 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

$m\left(=\frac{1}{2}nr\right)$ edges, then

$\varphi \left(L\left(G\right)\mathrm{,}x\right)={\left(x+2\right)}^{m-n}\varphi \left(G\mathrm{,}x-r+2\right)\mathrm{.}$ (12)

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

$m\left(=\frac{1}{2}nr\right)$ edges, If ${\lambda}_{1}\left(=r\right),{\lambda}_{2},\cdots ,{\lambda}_{n}$ is the eigenvalues of G, then

$\epsilon \left(L\left(G\right)\right)=2\left(m-n\right)+{\displaystyle \underset{i=1}{\overset{n}{\sum}}}\left|r+{\lambda}_{i}-2\right|.$ (13)

Corollary 3.3.

$\epsilon \left(L\left({K}_{n}\right)\right)=\{\begin{array}{ll}2{n}^{2}-6n\hfill & 4\le n,\text{\hspace{0.05em}}\hfill \\ 4\left(n-2\right)\hfill & 2\le n\le 3.\hfill \end{array}$ (14)

Let G be a graph, the subdivision graph $S\left(G\right)$ of graph G is the graph obtained by inserting a new vertex into every edge of G. The graph $R\left(G\right)$ 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 $Q\left(G\right)$ 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 $T\left(G\right)$ of graph G is the graph whose vertices are the vertices and edges of G, with two vertices of $T\left(G\right)$ 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

$m\left(=\frac{1}{2}nr\right)$ edges, then

1) $\varphi \left(S\left(G\right),x\right)={x}^{m-n}\varphi \left(G,{x}^{2}-r\right),$

2) $\varphi \left(R\left(G\right),x\right)={x}^{m-n}{\left(x+1\right)}^{n}\varphi \left(G,\frac{{x}^{2}-r}{x+1}\right),$

3) $\varphi \left(Q\left(G\right),x\right)={\left(x+2\right)}^{m-n}{\left(x+1\right)}^{n}\varphi \left(G,\frac{{x}^{2}-\left(r-2\right)x-r}{x+1}\right).$

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

$\frac{1}{2}\left(2{\lambda}_{i}+r-2\pm \sqrt{4{\lambda}_{i}+{r}^{2}+4}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(i=1,2,\cdots ,n\right).$

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

$m\left(=\frac{1}{2}nr\right)$ edges, If ${\lambda}_{1}\left(=r\right),{\lambda}_{2},\cdots ,{\lambda}_{n}$ is the eigenvalues of G, then

1) $\epsilon \left(S\left(G\right)\right)=2{\displaystyle {\sum}_{i=1}^{n}}\sqrt{r+{\lambda}_{i}},$

2) $\epsilon \left(R\left(G\right)\right)={\displaystyle {\sum}_{i=1}^{n}}\sqrt{{\lambda}_{i}^{2}+4\left(r+{\lambda}_{i}\right)},$

3) $\epsilon \left(Q\left(G\right)\right)=2\left(m-n\right)+{\displaystyle {\sum}_{i=1}^{n}}\sqrt{{\left(r+{\lambda}_{i}\right)}^{2}+4},$

4) $\begin{array}{c}\epsilon \left(T\left(G\right)\right)=2\left(m-n\right)+\frac{1}{2}{\displaystyle {\sum}_{i=1}^{n}}(\left|2{\lambda}_{i}+r-2+\sqrt{4{\lambda}_{i}+{r}^{2}+4}\right|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left|2{\lambda}_{i}+r-2-\sqrt{4{\lambda}_{i}+{r}^{2}+4}\right|).\end{array}$

Proof. (1) By Lemma 3.2 (1), we know that the spectrum of $S\left(G\right)$ is $\left\{0\left(m-n\text{\hspace{0.17em}}\text{times}\right),\pm \sqrt{r+{\lambda}_{i}}\left(i=1,2,\cdots ,n\right)\right\}$ . So $\epsilon \left(S\left(G\right)\right)=2{\displaystyle {\sum}_{i=1}^{n}}\sqrt{r+{\lambda}_{i}}.$

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

$\left\{0\left(m-n\text{\hspace{0.17em}}\text{times}\right),\frac{{\lambda}_{i}\pm \sqrt{{\lambda}_{i}^{2}+4\left(r+{\lambda}_{i}\right)}}{2}\left(i=1,2,\cdots ,n\right)\right\}$ . So $\epsilon \left(R\left(G\right)\right)={\displaystyle {\sum}_{i=1}^{n}}\sqrt{{\lambda}_{i}^{2}+4\left(r+{\lambda}_{i}\right)}.$

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

Corollary 3.4. 1) If $n\ge \mathrm{2,}$ then $\epsilon \left(S\left({K}_{n}\right)\right)=2\left(\sqrt{2n-2}+\left(n-1\right)\sqrt{n-2}\right).$

2) If $n\ge \mathrm{2,}$ then $\epsilon \left(R\left({K}_{n}\right)\right)=\sqrt{{n}^{2}+6n-7}+\left(n-1\right)\sqrt{4n-7}.$

3) $\epsilon \left(Q\left({K}_{n}\right)\right)={n}^{2}-3n+2\sqrt{{n}^{2}-2n+2}+\left(n-1\right)\sqrt{{n}^{2}-4n+8}.$

4) If $n\ge \mathrm{2,}$ then $\epsilon \left(T\left({K}_{n}\right)\right)=\{\begin{array}{ll}2{n}^{2}-2n-4\hfill & n\ge 3,\hfill \\ 4\hfill & n=2.\hfill \end{array}$

4. Conclusion

In this paper, we prove that $\epsilon \left(G\times H\right)=\epsilon \left(G\right)\times \epsilon \left(H\right),$ $\epsilon \left({D}_{m}G\right)=m\epsilon \left(G\right).$ For regular graphs G and H, we give the computational formulas of $\epsilon \left(G\nabla H\right)$ , $\epsilon \left(G\circ H\right)$ , $\epsilon \left(L\left(G\right)\right)$ , $\epsilon \left(S\left(G\right)\right)$ , $\epsilon \left(R\left(G\right)\right)$ , $\epsilon \left(Q\left(G\right)\right)$ , and $\epsilon \left(T\left(G\right)\right)$ re- spectively. In particularly, we obtained a lot of graphs pair of equienergetic.

Acknowledgements

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).

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