Nordhaus–Gaddum-Type Relations for Arithmetic-Geometric Spectral Radius and Energy
Spectral graph theory plays an important role in engineering. Let be a simple graph of order with vertex set . For , the degree of the vertex , denoted by , is the number of the vertices adjacent to . The arithmetic-geometric adjacency matrix of is defined as the matrix whose entry is equal to if the vertices and are adjacent and 0 otherwise. The arithmetic-geometric spectral radius and arithmetic-geometric energy of are the spectral radius and energy of its arithmetic-geometric adjacency matrix, respectively. In this paper, some new upper bounds on arithmetic-geometric energy are obtained. In addition, we present the Nordhaus–Gaddum-type relations for arithmetic-geometric spectral radius and arithmetic-geometric energy and characterize corresponding extremal graphs.
Spectral properties of graphs have been widely applied in the field of engineering technology. In systems engineering, complex networks are often used as abstract models to study and reveal the relationship among system elements. The mathematical study of complex network depends on graph theory. In network dynamics, the analysis of threshold in the epidemic model and synchronization condition of coupling oscillators is essentially the study of spectral of the corresponding graph. Naturally, the development of systems engineering depends on the results and conclusions of basic studies in graph spectral theory. Similarly, in the chemical engineering, a chemical molecule can also be abstracted as a graph. The combinatorial forms of spectral of weighted graphs, based on topological indices, are closely related to the molecular orbital energy levels of -electrons in conjugated hydrocarbons, as well as the spectrum properties. The combinatorial form of the spectrum, significantly correlated with the stability of conjugated alternant molecules, is defined as the sum of the absolute values of the eigenvalues of graphs. In this paper, we further study the arithmetic-geometric spectral radius and arithmetic-geometric energy.
Let be a simple graph with vertex set and edge set , and let . An edge with end vertices and is denoted by . The degree of the vertex of for is denoted by . Let and represent the maximum and minimum degrees of , respectively. The graph is called the complement to the graph G, if its vertex set is the same as , and and are adjacent in if and only if they are not adjacent in .
The adjacency matrix of a graph is the matrix , denoted by , where if and are adjacent; if not, . All eigenvalues of are denoted by the nonincreasing sequence . The spectral radius of is the greatest eigenvalue . The energy of is defined as
As a graph invariant, topological indices are used to understand physicochemical properties of chemical compounds since they capture some properties of molecules. In 2015, the paper  proposed the arithmetic-geometric index of a graph and defined the arithmetic-geometric adjacency matrix (AG matrix) of , denoted by . An element of AG matrix is defined in the following manner:
The AG eigenvalues of are the eigenvalues of its corresponding AG matrix. Note that AG matrix is real and symmetric so that all eigenvalues of are real, which can be recorded as . Similarly, the greatest AG eigenvalue is called the arithmetic-geometric spectral radius (AG spectral radius) of . In addition, the arithmetic-geometric energy (AG energy) of is defined in an analogue way as
The first Zagreb index is a kind of important topological indices, which is defined in  as
A graph is said to be strongly regular with parameters if it is regular, every pair of adjacent vertices has common neighbors, and every pair of distinct nonadjacent vertices has common neighbors . If , then is a disjoint union of complete graphs, whereas if and is noncomplete, then the eigenvalues of are (the trivial eigenvalue) and the roots and (the nontrivial eigenvalues) of the quadratic equation:
It is straightforward to show that the complement of a strong regular graph with parameters is still a strong regular graph with parameters . Recently, some bounds on AG spectral radius and AG energy were obtained in a couple of papers [4, 5]. Until now, hundreds of such “energy” have been introduced, such as inverse sum index energy , distance energy [7, 8], ABC energy [9, 10], matching energy [11, 12], and Randić energy [13, 14]. Recently, there are many studies, such as [15–17], on the energy of graphs with given parameters.
In the paper of Nordhaus and Gaddum , the lower and upper bounds on and were given, where and were the chromatic number of a graph and its complement , separately. Since then, in terms of an invariant of graph , any bound on or is referred to as a Nordhaus–Gaddum-type inequality or relation. And Nordhaus–Gaddum-type relations have received wide attention (e.g., ).
We write if the graphs and are isomorphic. As usual, let denote the complete graph of order and the complete multipartite graph, having vertices in the th partite set for each . Thus, , where . For other undefined notions and terminologies from graph theory, the readers are referred to [20, 21]. The rest of the paper is structured as follows: in Section 2, we give some useful lemmas; in Section 3, we get some new bounds on the AG energy; and in Sections 4 and 5, we obtain the Nordhaus–Gaddum-type relations for the AG spectral radius and AG energy of graph , respectively.
Lemma 1. (see ). If is an real symmetric matrix with eigenvalues , then for any , . Equality holds if and only if is an eigenvector of corresponding to .
Lemma 2. (see ). Let be a graph of order and size with maximum degree and minimum degree . Then,with equality holding if and only if is regular, a star plus copies of , or a complete graph plus a regular graph with a smaller degree of vertices.
Lemma 3. (see ). Let be a graph of order with the maximum degree and minimum degree . Then,with equality holding if and only if is regular.
Lemma 4. (see ). Let be a graph of order . Then,with equality holding if and only if is regular.
Lemma 5. (see ). A connected graph of order has only one positive eigenvalue in its adjacency spectrum if and only if is a complete multipartite graph.
3. On AG Energy of a Graph
Lemma 6. Let be a graph of order and size , thenequality holds if and only if is a regular graph.
Proof. Let us take any unit vector in . By Lemma 1, we haveTaking into (10), we havethen (9) holds.
For the equality in (9) to hold, all the inequalities in the above argument must be equalities. From (10), we have . Then, is a regular graph.
Conversely, if is regular, then . So is an eigenvector of corresponding to the eigenvalue . Then, the equality holds in (9).
Theorem 1. Let be a graph of order and size , thenwith equality holding if and only if is isomorphic to a regular complete multipartite graph or .
Proof. Applying Lemma 6, we haveThe first equality attained if and only if has at most one positive AG eigenvalue, and the second equality holds if and only if is regular. If has no positive eigenvalue, then . If has only one positive eigenvalue, from Lemma 5, then is isomorphic to a complete multipartite graph. And, if is a regular graph, then its AG matrix is identical to its adjacency matrix. If so, we know that the equality in Theorem 1 holds if and only if is isomorphic to a regular complete multipartite graph, or .
Theorem 2. Let be a graph of order and size , and . Then,where . This bound is achieved if is either , , or a noncomplete connected strongly regular graph with two nontrivial eigenvalues both with absolute value:
Proof. By Lemma 6, we have . Moreover, sincemust hold, we haveCombining the above inequality with the Cauchy–Schwartz inequality, we can see that the following inequality is obvious:Thus,Since the functionreaches the maximum for , we see that must hold as well. In addition, if , then . From this fact and inequality (19), it immediately follows that inequality (14) holds when .
It is obvious that and we know that if is either , , or a noncomplete connected strongly regular graph, the inequality holds. As the graph is regular, its AG matrix is the same as its adjacency matrix. That is to say, the AG eigenvalues for are (both with multiplicity ). Similarly, the AG eigenvalues for are (multiplicity 1) and (multiplicity ) so that equality must hold in (14), if is isomorphic to or .
Moreover, if is a noncomplete connected strongly regular graph with the trivial eigenvalue and other two nontrivial eigenvalues both with absolute value , the equality holds in (14) as well.
Remark 1. The paper  proved that if is a simple graph of order and size having no isolated vertices, then the following bound must hold:The reason is following: defined in the proof of Theorem 2 reaches the maximum for , and the maximum is . It follows thatmust hold. Moreover, . Hence, inequality (14) is an improvement on the bound in .
Theorem 3. If and is a graph on vertices with edges, thenwhere is the graph obtained from by deleting all isolated vertices. Equality holds if and only if .
Proof. It is easy to know that has at least isolated vertices if . We can obtain a graph with no isolated vertices by removing all isolated vertices. It follows that has at most vertices. Hence, by applying inequality (22) to , we can immediately see thatMoreover, equality holds if and only if ; that is, is the disjoint union of edges.
Theorem 4. Let be a graph of order and size and . Then,holds. If is a strongly regular graph with parameters , the bound is achieved.
Proof. It is obvious that . If , then the inequality (14) is as follows:Using routine calculus, it is seen that the left-hand side of inequality (26)—considered as a function of —is maximized whenholds. Inequality (25) now follows by substituting this value of into (26).
If , then by inequality (22) and , we know thatSince , then it is clear that (25) follows. Moreover, it follows by Theorem 2 that if is a strongly regular graph with parameters , equality holds in (25), which means that the upper bound is achieved. The theorem follows.
4. Nordhaus–Gaddum-Type Relation for AG Spectral Radius
Let , , and be the maximum degree, minimum degree, and spectral radius of the complement graph of , respectively. Here, we are presenting the lower and upper bounds on .
Theorem 5. Let be a graph of order . Then,and the equality holds if and only if is a regular graph.
Theorem 6. Let be a graph of order . Then,with equality holding if and only if is a regular graph.
Theorem 7. Let be a connected graph of order and size . And we write the number of vertices with by and the second largest degree of graph by :(1)If or , then(2)If and , then
Moreover, the bounds are sharp.
Proof. (1) Without loss of generality, let . Since , by Lemmas 2 and 3, we haveIt clearly follows that has isolated vertices. Let be the graph obtained from by removing the isolated vertices. It is clear that , , and , soThe proof of the inequality (34) follows.
(2) Let and . It follows that and . From Lemmas 2 and 3, we haveHence, the inequality (35) holds.
To show the sharpness of upper bounds in Theorem 7, we consider the following examples.
Example 1. Let be a star of order . That is to say, and . Then, is the union of an isolated vertex and a complete graph . We have
Example 2. Let be a regular graph. It is clear that is still a regular graph. We have
5. Nordhaus–Gaddum-Type Relation for AG Energy
In the following theorems, some lower and upper bounds for are obtained.
Theorem 8. Let be a connected graph of order and size and be all connected components of . Then,where and are the number of vertices and edges in , respectively. Moreover, the equality holds if and only if is isomorphic to a regular complete multipartite graph, or .
Proof. Applying Theorem 1, we haveHence, we can obtainFrom Theorem 1, we know that equality is possible if and only if is isomorphic to regular complete multipartite graph, or . The complement of regular complete multipartite graph is the disjoint union of complete graphs of order , and is the complement graph of . If so, holds. Hence, the equality holds if and only if is isomorphic to a regular complete multipartite graph, or .
Theorem 9. Let be a graph of order and size , thenwith equality holding if and only if either or .
Proof. Note thatSimilar to the analysis of equality in Theorem 8, we consider the following two cases:(1), which implies .(2)Both and are isomorphic to regular complete multipartite graphs. This is impossible because complete multipartite graphs are connected, whereas their complements are disconnected.
Theorem 10. Let be a graph of order and is the complement graph of . If both and have no isolated vertices, then
If is a strongly regular graph with parameters , the equality holds.
Proof. From Theorem 4, we haveIf is a strongly regular graph, we know that is also a strongly regular graph. Moreover, all inequalities in the above argument must be equalities. The first equality holds if is isomorphic to a strongly regular graph with parameters . And, the last equality holds if is regular. The theorem now follows immediately.
In this paper, we obtain an upper bound on AG spectral radius, and furthermore, the Nordhaus–Gaddum-type relation of AG spectral radius is derived. Some new upper and lower bounds on arithmetic-geometric energy are obtained. Moreover, we illustrate one of the upper bounds is better than the result in paper . In the end, we state the Nordhaus–Gaddum-type relation of AG energy.
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was funded by the Shanxi Scholarship Council of China (no. 201901D211227).
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