Abstract
Let be a graph with vertex set , and let be the degree of . The Zagreb matrix of is the square matrix of order whose -entry is equal to if the vertices and are adjacent, and zero otherwise. The Zagreb energy of 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.
1. Introduction
In this paper, is a simple undirected graph, with vertex set and edge set . The integers and are the order and the size of the graph , respectively. For a vertex , the open neighborhood of is the set and the degree of is . We write , , and for the path, cycle, and complete graph of order , respectively. A bipartite graph is a graph such that its vertex set can be partitioned into two sets and (called the partite sets) such that every edge meets both and . 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 and is denoted by . The complement of is the simple graph whose vertex set is and whose edges are the pairs of nonadjacent vertices of . The line graph of a graph , written , is the graph whose vertices are the edges of , with when and in . The line graph of a -regular graph with vertices is -regular with vertices.
For each vertex of a graph G, take a new vertex and join to all vertices of adjacent to . The graph thus obtained is called the splitting graph of . The cocktail party graph (for ) is a graph obtained from the complete graph by deleting a perfect matching.
Any graph on vertices, with , 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 , there exists only one connected antiregular graph on vertices, denoted by (see Figure 1).
(a)
(b)
The of is defined by its entries as if and 0 otherwise. Let denote the of . The of the graph is defined aswhere , , are the of graph .
This concept was introduced by Gutman and is intensively studied in , since it can be used to approximate the total - energy of a (see, e.g., [8, 9]). Since then, numerous other bounds for 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 is a square matrix of order , defined in [10], as follows:
The eigenvalues of labeled as are said to be the Zagreb eigenvalues or -eigenvalues of and their collection is called Zagreb spectrum or -spectrum of .
If are the distinct Zagreb eigenvalues of having the multiplicities , then the Zagreb spectrum of is denoted aswhere .
The sum of all absolute Zagreb eigenvalues is the Zagreb energy denoted by 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 , the Zagreb eigenvalues are and with multiplicities and 1, respectively, and .
Proof. Let be a graph with vertices . Then, the Zagreb matrix is as follows:Since, is a regular graph of degree , we haveIt can be easily seen that the Zagreb spectrum of is as follows:Therefore, by the definition of the Zagreb energy, we haveGutman [5] introduced energy in 1978 and conjectured that the complete graph 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 , non-hyperenergetic if , and broderenergetic [4] (other than ) if . If , then graphs and are equienergetic [2].
Following the above ideas, a graph of order is said to be Zagreb hyperenergetic if , Zagreb non-hyperenergetic if , and Zagreb broderenergetic (other than ) if . If , then two graphs and 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 complement and Zagreb energy of line graph of a graph , and furthermore, we develop results to determine the nature of graphs like complement , line graph , and splitting graph to be Zagreb hyperenergetic and Zagreb borderenergetic.
We start with the following proposition that helps us to obtain our results.
Proposition 1. Let be an -regular graph () of order with Zagreb eigenvalues . The Zagreb eigenvalues of are with multiplicity one and , for .
Theorem 1. Let be an -regular graph () of order with Zagreb eigenvalues . The Zagreb energy of complement is
Proof. Since is -regular, the complement is -regular. By Equality (4) and Proposition 1, we obtain
Theorem 2. For an -regular graph of order , the complement is Zagreb non-hyperenergetic if .
Proof. From Equality (10), we haveIt is easy to verify thatHence, the complement is a Zagreb non-hyperenergetic graph.
Proposition 2. Let be an -regular graph () of order with Zagreb eigenvalues . The Zagreb eigenvalues of are with multiplicity and for .
Theorem 3. Let be an -regular graph () of order with Zagreb eigenvalues . The Zagreb energy of line graph is
Proof. The line graph of a -regular graph is a -regular graph of order . By definition of Zagreb energy and Proposition 2, we have
Theorem 4. Let be an -regular graph () of order different from . Then, is Zagreb non-hyperenergetic.
Proof. Applying Theorem 3, we haveIt is not hard to see thatThus, is a Zagreb non-hyperenergetic graph.
Remark 1. Note that the graphs or are Zagreb borderenergetic.
Example 1. The antiregular graphs and illustrated in Figure 1 are non-hyperenergetic.
Let be a graph with vertices . The Zagreb matrix of isTherefore, the Zagreb spectrum of is as follows:By the definition of the Zagreb energy, we haveAnalogously, we can see thatTherefore, the Zagreb spectrum of is as follows:Hence, by the definition of the Zagreb energy, we haveBy definition and Equalities (19) and (22), we deduce that and 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 graph is a Zagreb hyperenergetic graph.
Proof. Let be a graph with vertices . Then, the Zagreb matrix is as follows:Let be the vertices added in corresponding to to obtain such that . Note that the degree of is . Then, the Zagreb matrix of can be written as a block matrix as follows:orTherefore, the Zagreb spectrum of is as followswhere for are the eigenvalues of and are the eigenvalues of . Therefore, by the definition of the Zagreb energy, we can writeHence, we haveEquality (28) gives the desired result.
Theorem 6. For , .
Proof. Consider the complete graph and the complete bipartite graph for . The Zagreb spectrum of isTherefore, by the definition of Zagreb energy, we haveOn the other hand, the Zagreb spectrum of isBy the definition of Zagreb energy, we can writeThus, from Equalities (30) and (32), the required result follows.
Theorem 7. For , .
Proof. The Zagreb spectrum of the complete graph isAlso, the Zagreb spectrum of the cocktail party graph isTherefore,andNow, Equalities (35) and (36) lead to the result.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.