New Results on Zagreb Energy of Graphs

Let G be a graph with vertex set V(G) � v1, . . . , vn 􏼈 􏼉, and let di be the degree of vi. (e 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. (e Zagreb energy ZE(G) 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.


Introduction
In this paper, G is a simple undirected graph, with vertex set V � V(G) and edge set E � E(G). e integers n � n(G) � |V(G)| and m � m(G) � |E(G)| are the order and the size of the graph G, respectively. For a vertex v ∈ V, the open neighborhood of v is the set N(v) � u ∈ V|uv ∈ E { } and the degree of v is d(v) � |N(v)|. We write P n , C n , and K n 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 K p,q . e complement G of G is the simple graph whose vertex set is V and whose edges are the pairs of nonadjacent vertices of G. e line graph of a graph G, written L(G), is the graph whose vertices are the edges of G, with ef ∈ E(L(G)) when e � uv and f � vw in G. e line graph L(G) of a r-regular graph G with n vertices is (2r − 2)-regular with nr/2 vertices.
For each vertex v of a graph G, take a new vertex v ′ and join v ′ to all vertices of G adjacent to v. e graph S ′ (G) thus obtained is called the splitting graph of G. e cocktail party graph CP(a) (for a ≥ 3) is a graph obtained from the complete graph K 2a by deleting a perfect matching.
Any graph on n vertices, with n ≥ 2, has at least two vertices with the same degree. e 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 A n (see Figure 1). e adjacency matrix A(G) of G is defined by its entries as a ij � 1 if v i v j ∈ E(G) and 0 otherwise. Let λ 1 ⩾λ 2 ⩾ . . . ⩾λ n denote the eigenvalues of A(G). e energy of the graph G is defined as where λ i , i � 1, 2, . . . , n, are the eigenvalues of graph G. is concept was introduced by Gutman and is intensively studied in chemistry, 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][12][13][14]).
e Zagreb indices are widely studied degree-based topological indices and were introduced by Gutman and Trinajstić [7] in 1972. e Zagreb matrix of a graph G is a square matrix A z (G) � [m ij ] of order n, defined in [10], as follows: e eigenvalues of A z (G) labeled as z 1 ⩾z 2 ⩾ . . . ⩾z n are said to be the Zagreb eigenvalues or A z -eigenvalues of G and their collection is called Zagreb spectrum or A z -spectrum of G.
If z 1 , z 2 , . . . , z s are the distinct Zagreb eigenvalues of G having the multiplicities m 1 , m 2 , . . . , m s , then the Zagreb spectrum of G is denoted as where m 1 + m 2 + · · · + m s � n. e sum of all absolute Zagreb eigenvalues is the Zagreb energy denoted by ZE(G) and defined in [10] as follows: Now, we prove the next lemma that will be needed to obtain our results. Lemma 1. For a complete graph K n , the Zagreb eigenvalues are −2(n − 1) and 2(n − 1) 2 with multiplicities (n − 1) and 1, respectively, and ZE(K n ) � 4(n − 1) 2 .
Proof. Let G be a graph with vertices v 1 , v 2 , v 3 , . . . , v n . en, the Zagreb matrix is as follows: Since, K n is a regular graph of degree n − 1, we have It can be easily seen that the Zagreb spectrum of K n is as follows: erefore, by the definition of the Zagreb energy, we have Gutman [5] introduced energy in 1978 and conjectured that the complete graph K n 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 is hyperenergetic [6] if E(G) > 2n − 2, nonhyperenergetic if E(G) < 2n − 2, and broderenergetic [4] (other than K n ) if E(G) � 2n − 2. If E(G) � E(H), then graphs G and H are equienergetic [2].
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.

Main Results
In this section, we provide Zagreb energy of complement G and Zagreb energy of line graph L(G) of a graph G, and furthermore, we develop results to determine the nature of graphs like complement G, line graph L(G), 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.
Proof. Since G is r-regular, the complement G is (n− r − 1)-regular. By Equality (4) and Proposition 1, we obtain □ Theorem 2. For an r-regular graph G of order n, the complement G is Zagreb non-hyperenergetic if r ≥ 3.
Proof. From Equality (10), we have It is easy to verify that Hence, the complement G is a Zagreb non-hyperenergetic graph.
Proof. e line graph L(G) 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 □ Theorem 4. Let G be an r-regular graph (r ≥ 3) of order n different from K 2 and K 3 . en, L(G) is Zagreb nonhyperenergetic.
Proof. Applying eorem 3, we have Mathematical Problems in Engineering 3 It is not hard to see that us, L(G) is a Zagreb non-hyperenergetic graph. □ Remark 1. Note that the graphs K 2 or K 3 are Zagreb borderenergetic.
Example 1. e antiregular graphs A 6 and A 7 illustrated in Figure 1 are non-hyperenergetic.
Let A 6 be a graph with vertices v 1 , v 2 , v 3 , v 4 , v 5 , and v 6 . e Zagreb matrix of A 6 is erefore, the Zagreb spectrum of A 6 is as follows: Analogously, we can see that erefore, the Zagreb spectrum of A 7 is as follows: SpecA z A 7 � 34.501 −17.57 3.189 −11.737 1.094 −9.477 0 Hence, by the definition of the Zagreb energy, we have By definition and Equalities (19) and (22), we deduce that A 6 and A 7 are non-hyperenergetic.

Some Classes of Zagreb Hyperenergetic and Zagreb Equienergetic Graphs.
is section contributes some results towards Zagreb hyperenergetic and Zagreb equienergetic graphs.
Theorem 5. For a regular graph G, the splitting graph S ′ (G) is a Zagreb hyperenergetic graph.
Proof. Let G be a graph with vertices v 1 , v 2 , v 3 , . . . , v p . en, the Zagreb matrix is as follows: Let v 1 ′ , v 2 ′ , v 3 ′ , . . . , v p ′ be the vertices added in G corresponding to v 1 , v 2 , v 3 , . . . , v p to obtain S ′ (G) such that N(v i ) � N(v i ′ ). Note that the degree of v i ′ is d i . en, the