# New Results on Zagreb Energy of Graphs

Authors: Seyed Mahmoud Sheikholeslami; Akbar Jahanbani; Rana Khoeilar
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

LetG be a graph with vertex set VG=v1,…,vn, and let di be the degree of vi. The Zagreb matrix of G is the square matrix of order n whose i,j-entry is equal to di+dj if the vertices vi and vj are adjacent, and zero otherwise. The Zagreb energy ZEG of G is the sum of the absolute values of the eigenvalues of the Zagreb matrix. In this paper, we determine some classes of Zagreb hyperenergetic, Zagreb borderenergetic, and Zagreb equienergetic graphs.


---

## Body

## 1. Introduction

In this paper,G is a simple undirected graph, with vertex set V=VG and edge set E=EG. The integers n=nG=VG and m=mG=EG are the order and the size of the graph G, respectively. For a vertex v∈V, the open neighborhood of v is the set Nv=u∈V|uv∈E and the degree of v is dv=Nv. We write Pn, Cn, and Kn for the path, cycle, and complete graph of order n, respectively. A bipartite graph is a graph such that its vertex set can be partitioned into two sets X and Y (called the partite sets) such that every edge meets both X and Y. A complete bipartite graph is a bipartite graph such that any vertex of a partite set is adjacent to all vertices of the other partite set. A complete bipartite graph with partite set of cardinalities p and q is denoted by Kp,q. The complement G¯ of G is the simple graph whose vertex set is V and whose edges are the pairs of nonadjacent vertices of G. The line graph of a graph G, written LG, is the graph whose vertices are the edges of G, with ef∈ELG when e=uv and f=vw in G. The line graph LG of a r-regular graph G with n vertices is 2r−2-regular with nr/2 vertices.For each vertexv of a graph G, take a new vertex v′ and join v′ to all vertices of G adjacent to v. The graph S′G thus obtained is called the splitting graph of G. The cocktail party graphCPa (for a≥3) is a graph obtained from the complete graph K2a by deleting a perfect matching.Any graph onn vertices, with n≥2, has at least two vertices with the same degree. The graphs with at most two vertices with the same degree are called antiregular; for more information, see [1, 17]. For any positive integer n, there exists only one connected antiregular graph on n vertices, denoted by An (see Figure 1).Figure 1
Two antiregular graph with verticesn=6 and n=7.
(a)(b)Theadjacency matrixAG of G is defined by its entries as aij=1 if vivj∈EG and 0 otherwise. Let λ1⩾λ2⩾…⩾λn denote the eigenvalues of AG. The energy of the graph G is defined as(1)E=EG=∑i=1nλi,where λi, i=1,2,…,n, are the eigenvalues of graph G.This concept was introduced by Gutman and is intensively studied inchemistry, since it can be used to approximate the total π-electron energy of a molecule (see, e.g., [8, 9]). Since then, numerous other bounds for E were found (see, e.g., [11–14]).The Zagreb indices are widely studied degree-based topological indices and were introduced by Gutman and Trinajstić [7] in 1972. The Zagreb matrix of a graph G is a square matrix AzG=mij of order n, defined in [10], as follows:(2)mij=di+dj,if the verticesviandvjofGare adjacent,0,otherwise.The eigenvalues ofAzG labeled as z1⩾z2⩾…⩾zn are said to be the Zagreb eigenvalues or Az-eigenvalues of G and their collection is called Zagreb spectrum or Az-spectrum of G.Ifz1,z2,…,zs are the distinct Zagreb eigenvalues of G having the multiplicities m1,m2,…,ms, then the Zagreb spectrum of G is denoted as(3)SpecAz=z1z2…zsm1m2…ms,where m1+m2+⋯+ms=n.The sum of all absolute Zagreb eigenvalues is the Zagreb energy denoted byZEG and defined in [10] as follows:(4)ZE=ZEG=∑i=1nzi.Now, we prove the next lemma that will be needed to obtain our results.Lemma 1.
For a complete graphKn, the Zagreb eigenvalues are −2n−1 and 2n−12 with multiplicities n−1 and 1, respectively, and ZEKn=4n−12.Proof.
LetG be a graph with vertices v1,v2,v3,…,vn. Then, the Zagreb matrix is as follows:(5)AzG=v1v2v3…vpv1v2v3⋮vp0d1+d2d1+d3…d1+dnd2+d10d2+d3…d2+dnd3+d1d3+d20…d3+dn⋮⋮⋮⋱⋮dn+d1dn+d2dn+d3…0.
Since,Kn is a regular graph of degree n−1, we have(6)AzKn=02n−22n−2…2n−22n−202n−2…2n−22n−22n−20…2n−2⋮⋮⋮⋱⋮2n−22n−22n−2…0.
It can be easily seen that the Zagreb spectrum ofKn is as follows:(7)SpecAzKn=2n−12−2n−11n−1.
Therefore, by the definition of the Zagreb energy, we have(8)ZEKn=4n−12.
Gutman [5] introduced energy in 1978 and conjectured that the complete graph Kn possesses the maximum energy among all graphs with n vertices. Gutman [6] also proved this to be false leading to the new concept of hyperenergetic graphs.
A graph ishyperenergetic [6] if EG>2n−2, non-hyperenergetic if EG<2n−2, and broderenergetic [4] (other than Kn) if EG=2n−2. If EG=EH, then graphs G and H are equienergetic [2].
Following the above ideas, a graphG of order n is said to be Zagreb hyperenergetic if ZEG>4n−12, Zagreb non-hyperenergetic if ZEG<4n−12, and Zagreb broderenergetic (other than Kn) if ZEG=4n−12. If ZEG=ZEH, then two graphs G and H are called Zagreb equienergetic.
In [10], the authors obtained some lower and upper bounds for Zagreb energy, Das [3] presented some new bounds for Zagreb energy, Rakshith [16] discussed the new bounds for Zagreb energy, and Jahanbani et al. [15] obtained new bounds for Zagreb energy.
In this paper, we study the Zagreb energy of line graphs, Zagreb energy of complement graphs, and Zagreb hyperenergetic, Zagreb borderenergetic, and Zagreb equienergetic graphs.


## 2. Main Results

In this section, we provide Zagreb energy of complementG¯ and Zagreb energy of line graph LG of a graph G, and furthermore, we develop results to determine the nature of graphs like complement G¯, line graph LG, and splitting graph S′G to be Zagreb hyperenergetic and Zagreb borderenergetic.We start with the following proposition that helps us to obtain our results.Proposition 1.
LetG be an r-regular graph (r≥3) of order n with Zagreb eigenvalues z1⩾z2⩾…⩾zn. The Zagreb eigenvalues of AzG¯ are 2n−r−12 with multiplicity one and 2n−r−1−zi/2r−1, for i=2,3,…,n.Theorem 1.
LetG be an r-regular graph (r≥3) of order n with Zagreb eigenvalues z1⩾z2⩾…⩾zn. The Zagreb energy of complement G¯ is(9)ZEG¯=2n−r−1n−r−1+∑i=2n−zi2r−1.Proof.
SinceG is r-regular, the complement G¯ is n−r−1-regular. By Equality (4) and Proposition 1, we obtain(10)ZEG¯=2n−r−1n−r−1+∑i=2n2n−r−1−zi2r−1=2n−r−1n−r−1+∑i=2n−zi2r−1.Theorem 2.
For anr-regular graph G of order n, the complement G¯ is Zagreb non-hyperenergetic if r≥3.Proof.
From Equality (10), we have(11)ZEG¯=2n−r−1n−r−1+∑i=2n−zi2r−1.
It is easy to verify that(12)2n−r−1n−r−1+∑i=2n−zi2r−1<4n−12.
Hence, the complementG¯ is a Zagreb non-hyperenergetic graph.Proposition 2.
LetG be an r-regular graph (r≥3) of order n with Zagreb eigenvalues z1⩾z2⩾…⩾zn. The Zagreb eigenvalues of AzLG are 8−8r with multiplicity nr−2/2 and 4r−1zi/2r+r−2 for i=1,2,…,n.Theorem 3.
LetG be an r-regular graph (r≥3) of order n with Zagreb eigenvalues z1⩾z2⩾…⩾zn. The Zagreb energy of line graph LG is(13)ZELG=4r−1∑i=1nzi2r+r−2+8−8rnr−22.Proof.
The line graphLG of a r-regular graph G is a 2r−2-regular graph of order nr/2. By definition of Zagreb energy and Proposition 2, we have(14)ZELG=∑i=1n4r−1zi2r+r−2+8−8rnr−22=4r−1∑i=1nzi2r+r−2+8−8rnr−22.Theorem 4.
LetG be an r-regular graph (r≥3) of order n different from K2andK3. Then, LG is Zagreb non-hyperenergetic.Proof.
Applying Theorem3, we have(15)ZELG=4r−1∑i=1nzi2r+r−2+8−8rnr−22.
It is not hard to see that(16)4r−1∑i=1nzi2r+r−2+8−8rnr−22<4nr2−12.
Thus,LG is a Zagreb non-hyperenergetic graph.Remark 1.
Note that the graphsK2 or K3 are Zagreb borderenergetic.Example 1.
The antiregular graphsA6 and A7 illustrated in Figure 1 are non-hyperenergetic.
LetA6 be a graph with vertices v1,v2,v3,v4,v5,andv6. The Zagreb matrix of A6 is(17)AzA6=v1v2v3v4v5v6v1v2v3v4v5v6060000607988070600096077080706080760.
Therefore, the Zagreb spectrum ofA6 is as follows:(18)SpecAzA6=25.022.3391.087−6−13.149−9.297111111.
By the definition of the Zagreb energy, we have(19)ZEA6=56.892.
Analogously, we can see that(20)AzA7=v1v2v3v4v5v6v7v1v2v3v4v5v6v707000007091191990907000011708980908070010097070908070.
Therefore, the Zagreb spectrum ofA7 is as follows:(21)SpecAzA7=34.501−17.573.189−11.7371.094−9.47701111111.
Hence, by the definition of the Zagreb energy, we have(22)ZEA7=77.568.
By definition and Equalities (19) and (22), we deduce that A6 and A7 are non-hyperenergetic.


### 2.1. Some Classes of Zagreb Hyperenergetic and Zagreb Equienergetic Graphs

This section contributes some results towards Zagreb hyperenergetic and Zagreb equienergetic graphs.Theorem 5.
For a regular graph G, the splitting graphS′G is a Zagreb hyperenergetic graph.Proof.
LetG be a graph with vertices v1,v2,v3,…,vp. Then, the Zagreb matrix is as follows:(23)AzG=v1v2v3…vpv1v2v3⋮vp0d1+d2d1+d3…d1+dpd2+d10d2+d3…d2+dpd3+d1d3+d20…d3+dp⋮⋮⋮⋱⋮dp+d1dp+d2dp+d3…0.
Letv1′,v2′,v3′,…,vp′ be the vertices added in G corresponding to v1,v2,v3,…,vp to obtain S′G such that Nvi=Nvi′. Note that the degree of vi′ is di. Then, the Zagreb matrix of S′G can be written as a block matrix as follows:(24)AzS′G=2AzG32AzG32AzG0or(25)AzS′G=232320⊗AzG.
Therefore, the Zagreb spectrum ofS′G is as follows(26)SpecAzS′G=2−132zi2+132zipp,where zi for i=1,2,3,…,p are the eigenvalues of AzG and 2±13/2 are the eigenvalues of 23/23/20. Therefore, by the definition of the Zagreb energy, we can write(27)ZES′G=∑i=1p2−132zi=∑i=1pzi2−132+2+132=2ZEG.
Hence, we have(28)ZES′G=2ZEG.
Equality (28) gives the desired result.Theorem 6.
Forn≥3, ZEKn=ZEKn−1,n−1.Proof.
Consider the complete graphKn and the complete bipartite graph Kn−1,n−1 for n≥3. The Zagreb spectrum of Kn is(29)SpecAzKn=2n−12−2n−11n−1.
Therefore, by the definition of Zagreb energy, we have(30)ZEKn=2n−12+−2n−1n−1=4n−12.
On the other hand, the Zagreb spectrum ofKn−1,n−1 is(31)SpecAzKn−1,n−1=02n−122n−11−n2n−411.
By the definition of Zagreb energy, we can write(32)ZEKn−1,n−1=2n−12+21−nn−1=4n−12.
Thus, from Equalities (30) and (32), the required result follows.Theorem 7.
Fora≥3, ZEK2a−1=ZECPa.Proof.
The Zagreb spectrum of the complete graphK2a−1 is(33)SpecAzK2a−1=8a−12−4a−112a−1.Also, the Zagreb spectrum of the cocktail party graph CPa is(34)SpecAzCPa=−8a−18a−12a−11.
Therefore,(35)ZEK2a−1=8a−12+81−aa−1=16a−12,and(36)ZECPa=8a−12+−8a−1a−1=16a−12.
Now, Equalities (35) and (36) lead to the result.


## 2.1. Some Classes of Zagreb Hyperenergetic and Zagreb Equienergetic Graphs

This section contributes some results towards Zagreb hyperenergetic and Zagreb equienergetic graphs.Theorem 5.
For a regular graph G, the splitting graphS′G is a Zagreb hyperenergetic graph.Proof.
LetG be a graph with vertices v1,v2,v3,…,vp. Then, the Zagreb matrix is as follows:(23)AzG=v1v2v3…vpv1v2v3⋮vp0d1+d2d1+d3…d1+dpd2+d10d2+d3…d2+dpd3+d1d3+d20…d3+dp⋮⋮⋮⋱⋮dp+d1dp+d2dp+d3…0.
Letv1′,v2′,v3′,…,vp′ be the vertices added in G corresponding to v1,v2,v3,…,vp to obtain S′G such that Nvi=Nvi′. Note that the degree of vi′ is di. Then, the Zagreb matrix of S′G can be written as a block matrix as follows:(24)AzS′G=2AzG32AzG32AzG0or(25)AzS′G=232320⊗AzG.
Therefore, the Zagreb spectrum ofS′G is as follows(26)SpecAzS′G=2−132zi2+132zipp,where zi for i=1,2,3,…,p are the eigenvalues of AzG and 2±13/2 are the eigenvalues of 23/23/20. Therefore, by the definition of the Zagreb energy, we can write(27)ZES′G=∑i=1p2−132zi=∑i=1pzi2−132+2+132=2ZEG.
Hence, we have(28)ZES′G=2ZEG.
Equality (28) gives the desired result.Theorem 6.
Forn≥3, ZEKn=ZEKn−1,n−1.Proof.
Consider the complete graphKn and the complete bipartite graph Kn−1,n−1 for n≥3. The Zagreb spectrum of Kn is(29)SpecAzKn=2n−12−2n−11n−1.
Therefore, by the definition of Zagreb energy, we have(30)ZEKn=2n−12+−2n−1n−1=4n−12.
On the other hand, the Zagreb spectrum ofKn−1,n−1 is(31)SpecAzKn−1,n−1=02n−122n−11−n2n−411.
By the definition of Zagreb energy, we can write(32)ZEKn−1,n−1=2n−12+21−nn−1=4n−12.
Thus, from Equalities (30) and (32), the required result follows.Theorem 7.
Fora≥3, ZEK2a−1=ZECPa.Proof.
The Zagreb spectrum of the complete graphK2a−1 is(33)SpecAzK2a−1=8a−12−4a−112a−1.Also, the Zagreb spectrum of the cocktail party graph CPa is(34)SpecAzCPa=−8a−18a−12a−11.
Therefore,(35)ZEK2a−1=8a−12+81−aa−1=16a−12,and(36)ZECPa=8a−12+−8a−1a−1=16a−12.
Now, Equalities (35) and (36) lead to the result.



---

*Source: 9969845-2021-05-24.xml*