Nordhaus–Gaddum-Type Relations for Arithmetic-Geometric Spectral Radius and Energy

Spectral graph theory plays an important role in engineering. LetG be a simple graph of order nwith vertex setV � v1, v2, . . . , vn 􏼈 􏼉. For vi ∈ V, the degree of the vertex vi, denoted by di, is the number of the vertices adjacent to vi. )e arithmetic-geometric adjacencymatrixAag(G) ofG is defined as the n × n matrix whose (i, j) entry is equal to ((di + dj)/2 ���� didj 􏽱 ) if the vertices vi and vj are adjacent and 0 otherwise. )e arithmetic-geometric spectral radius and arithmetic-geometric energy of G are the spectral radius and energy of its arithmetic-geometric adjacencymatrix, respectively. In this paper, some new upper bounds on arithmeticgeometric 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.


Introduction
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. e 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.
e combinatorial forms of spectral of weighted graphs, based on topological indices, are closely related to the molecular orbital energy levels of π-electrons in conjugated hydrocarbons, as well as the spectrum properties. e 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 arithmeticgeometric spectral radius and arithmetic-geometric energy.
Let G � (V, E) be a simple graph with vertex set V � v 1 , v 2 , . . . , v n and edge set E, and let |E| � m. An edge e ∈ E with end vertices v i and v j is denoted by v i v j . e degree of the vertex v i ∈ V of G for i � 1, 2, . . . , n is denoted by d i . Let Δ and δ represent the maximum and minimum degrees of G, respectively. e graph G is called the complement to the graph G, if its vertex set is the same as V, and v i and v j (i, j � 1, 2, . . . , n, i ≠ j) are adjacent in G if and only if they are not adjacent in G.
e adjacency matrix of a graph G is the matrix (a ij ) n×n , denoted by A(G), where a ij � 1 if v i and v j are adjacent; if not, a ij � 0. All eigenvalues of A(G) are denoted by the nonincreasing sequence λ 1 , λ 2 , . . . , λ n . e spectral radius of G is the greatest eigenvalue λ 1 . e energy of G 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 [1] proposed the arithmetic-geometric index of a graph G and defined the arithmetic-geometric adjacency matrix (AG matrix) of G, denoted by A ag (G). An element of AG matrix is defined in the following manner: e AG eigenvalues of G are the eigenvalues of its corresponding AG matrix. Note that AG matrix is real and symmetric so that all eigenvalues of A ag (G) are real, which can be recorded as η 1 ≥ η 2 ≥ · · · ≥ η n . Similarly, the greatest AG eigenvalue η 1 is called the arithmetic-geometric spectral radius (AG spectral radius) of G. In addition, the arithmeticgeometric energy (AG energy) of G is defined in an analogue way as e first Zagreb index M 1 (G) is a kind of important topological indices, which is defined in [2] as A graph G is said to be strongly regular with parameters (n, r, μ, ]) if it is r− regular, every pair of adjacent vertices has μ ≥ 0 common neighbors, and every pair of distinct nonadjacent vertices has ] ≥ 0 common neighbors [3]. If ] � 0, then G is a disjoint union of complete graphs, whereas if ] ≥ 1 and G is noncomplete, then the eigenvalues of G are r (the trivial eigenvalue) and the roots x 1 and x 2 (the nontrivial eigenvalues) of the quadratic equation: It is straightforward to show that the complement of a strong regular graph with parameters (n, r, μ, ]) is still a strong regular graph with parameters (n, n − 1 − r, n − 2 − 2r + ], n − 2r + μ). 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 [6], 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][16][17], on the energy of graphs with given parameters.
In the paper of Nordhaus and Gaddum [18], the lower and upper bounds on χ(G) + χ(G) and χ(G) · χ(G) were given, where χ(G) and χ(G) were the chromatic number of a graph G and its complement G, separately. Since then, in terms of an invariant ψ(G) of graph G, any bound on ψ(G) + ψ(G) or ψ(G) · ψ(G) is referred to as a Nordhaus-Gaddum-type inequality or relation. And Nordhaus-Gaddum-type relations have received wide attention (e.g., [19]).
We write G 1 � G 2 if the graphs G 1 and G 2 are isomorphic. As usual, let K n denote the complete graph of order n and K(p 1 , p 2 , . . . , p n ) the complete multipartite graph, having p i vertices in the ith partite set for each i � 1, 2, . . . , n.
us, K n � K(p 1 , p 2 , . . . , p n ), where p 1 � p 2 � · · · � p n � 1. For other undefined notions and terminologies from graph theory, the readers are referred to [20,21]. e 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 G, respectively.

Preliminaries
Lemma 1 (see [22]). If B is an n × n real symmetric matrix with eigenvalues λ 1 ≥ λ 2 ≥ · · · ≥ λ n , then for any 0 ≠ x ∈ R n , Equality holds if and only if x is an eigenvector of B corresponding to λ 1 .
Lemma 2 (see [23]). Let G be a graph of order n and size m with maximum degree Δ and minimum degree δ ≥ 1. en, with equality holding if and only if G is regular, a star plus copies of K 2 , or a complete graph plus a regular graph with a smaller degree of vertices.
Lemma 3 (see [5]). Let G be a graph of order n with the maximum degree Δ and minimum degree δ ≥ 1. en, with equality holding if and only if G is regular.
Lemma 4 (see [5]). Let G be a graph of order n. en, with equality holding if and only if G is regular.
Lemma 5 (see [24]). A connected graph G of order n has only one positive eigenvalue in its adjacency spectrum if and only if G is a complete multipartite graph.

On AG Energy of a Graph
Lemma 6. Let G be a graph of order n and size m, then equality holds if and only if G is a regular graph.
Proof. Let us take any unit vector x � (x 1 , x 2 , . . . , x n ) T in R n . By Lemma 1, we have 2 Mathematical Problems in Engineering (10), we have then (9) holds. For the equality in (9) to hold, all the inequalities in the above argument must be equalities. From (10), we have en, the equality holds in (9). □ Theorem 1. Let G be a graph of order n and size m, then with equality holding if and only if G is isomorphic to a regular complete multipartite graph or G � K n .
Proof. Applying Lemma 6, we have e first equality attained if and only if G has at most one positive AG eigenvalue, and the second equality holds if and only if G is regular. If G has no positive eigenvalue, then G � K n . If G has only one positive eigenvalue, from Lemma 5, then G is isomorphic to a complete multipartite graph. And, if G is a regular graph, then its AG matrix is identical to its adjacency matrix. If so, we know that the equality in eorem 1 holds if and only if G is isomorphic to a regular complete multipartite graph, or G � K n . □ Theorem 2. Let G be a graph of order n and size m, and δ > 0. en, where is bound is achieved if G is either (n/2)K 2 , K n , or a noncomplete connected strongly regular graph with two nontrivial eigenvalues both with absolute value: Proof. By Lemma 6, we have Combining the above inequality with the Cauchy-Schwartz inequality, we can see that the following inequality is obvious: us, Since the function reaches the maximum for x � ( From this fact and inequality (19), it immediately follows that inequality (14) holds when It is obvious that 2m ≥ n and we know that if G is either (n/2)K 2 , K n , or a noncomplete connected strongly regular graph, the inequality (2m/n) ≥ ( As the graph (n/2)K 2 is regular, its AG matrix is the same as its adjacency matrix. at is to say, the AG eigenvalues for (n/2)K 2 are ± 1 (both with multiplicity (n/2)). Similarly, the AG eigenvalues for K n are n − 1 (multiplicity 1) and − 1 (multiplicity n − 1) so that equality must hold in (14), if G is isomorphic to (n/2)K 2 or K n .
□ Mathematical Problems in Engineering Remark 1. e paper [4] proved that if G is a simple graph of order n and size m having no isolated vertices, then the following bound must hold: e reason is following: F(x) defined in the proof of eorem 2 reaches the maximum for x � ( . Hence, inequality (14) is an improvement on the bound in [4].

where Γ is the graph obtained from G by deleting all isolated vertices. Equality holds if and only if Γ � (n/2)K 2 .
Proof. It is easy to know that G has at least n − 2m isolated vertices if 2m ≤ n. We can obtain a graph Γ with no isolated vertices by removing all isolated vertices. It follows that Γ has at most 2m vertices. Hence, by applying inequality (22) to Γ, we can immediately see that Moreover, equality holds if and only if 2m � n; that is, G is the disjoint union of edges. en, holds. If G is a strongly regular graph with parameters (n, ((n + � n √ )/2), ((n + 2 � n √ )/4), ((n + 2 � n √ )/4)), the bound is achieved.
Proof. It is obvious that 2m ≥ n. If (2m/n) ≥ ( (14) is as follows: Using routine calculus, it is seen that the left-hand side of inequality (26)-considered as a function of m-is maximized when holds. Inequality (25) now follows by substituting this value of m into (26).
, then by inequality (22) and 2m ≥ n, we know that Since

Nordhaus-Gaddum-Type Relation for AG Spectral Radius
Let Δ, δ, and η 1 be the maximum degree, minimum degree, and spectral radius of the complement graph G of G, respectively. Here, we are presenting the lower and upper bounds on η 1 + η 1 .

Theorem 5.
Let G be a graph of order n. en, and the equality holds if and only if G is a regular graph.
Proof. e proof of inequality in (29) follows directly from Lemma 4.
If the equality in (29) holds, that is, η 1 � �������� M 1 (G)/n and η 1 � �������� M 1 (G)/n, then by Lemma 4, G is regular. If G is a regular graph, then is finishes the proof. □ Theorem 6. Let G be a graph of order n. en, with equality holding if and only if G is a regular graph.
Proof. Lemma 6 gives that where |E(G)| denotes the number of edges in G. en, we have From Lemma 6, equality holds if and only if G is isomorphic to a regular graph. □ Theorem 7. Let G be a connected graph of order n and size m. And we write the number of vertices with d i � Δ by t and the second largest degree of graph G by τ: (2) If Δ ≤ n − 2 and Δ ≤ n − 2, then Moreover, the bounds are sharp.
Proof. (1) Without loss of generality, let Δ � n − 1. Since δ ≥ 1, by Lemmas 2 and 3, we have It clearly follows that G has t isolated vertices. Let Γ 1 be the graph obtained from G by removing the isolated vertices. It is clear that e proof of the inequality (34) follows. (2) Let Δ ≤ n − 2 and Δ ≤ n − 2. It follows that δ ≥ 1 and δ ≥ 1. From Lemmas 2 and 3, we have Hence, the inequality (35) holds.
To show the sharpness of upper bounds in eorem 7, we consider the following examples. □ Example 1. Let G be a star of order n. at is to say, t � 1 and τ � 1.
en, G is the union of an isolated vertex and a complete graph K n− 1 . We have Example 2. Let G be a regular graph. It is clear that G is still a regular graph. We have Mathematical Problems in Engineering

Nordhaus-Gaddum-Type Relation for AG Energy
In the following theorems, some lower and upper bounds for E ag (G) + E ag (G) are obtained.

Theorem 8.
Let G be a connected graph of order n and size m and Γ 1 , Γ 2 , . . . , Γ r be all connected components of G. en, where |V(Γ i )| and |E(Γ i )| are the number of vertices and edges in Γ i (i � 1, . . . , r), respectively. Moreover, the equality holds if and only if G is isomorphic to a regular complete multipartite graph, or G � K n .
Proof. Applying eorem 1, we have Hence, we can obtain From eorem 1, we know that equality is possible if and only if G is isomorphic to regular complete multipartite graph, or G � K n . e complement of regular complete multipartite graph is the disjoint union of k complete graphs of order (n/k), and K n is the complement graph of K n . If so, E ag (Γ i ) � (4|E(Γ i )|/|V(Γ i )|) holds. Hence, the equality holds if and only if G is isomorphic to a regular complete multipartite graph, or G � K n . □ Theorem 9. Let G be a graph of order n and size m, then with equality holding if and only if either G � K n or G � K n .
Proof. Note that Similar to the analysis of equality in eorem 8, we consider the following two cases: (1) G � K n , which implies G � K n .
Proof. From eorem 4, we have If G is a strongly regular graph, we know that G is also a strongly regular graph. Moreover, all inequalities in the above argument must be equalities. e first equality holds if G is isomorphic to a strongly regular graph with parameters (n, ((n + � n √ )/2), ((n + 2 � n √ )/4), ((n + 2 � n √ )/4)). And, the last equality holds if G is regular. e theorem now follows immediately.

Conclusions
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 [4]. In the end, we state the Nordhaus-Gaddum-type relation of AG energy.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.