Bounds of the Spectral Radius and the Nordhaus-Gaddum Type of the Graphs

Abstract

The Laplacian spectra are the eigenvalues of Laplacian matrix 𝐿(𝐺)=𝐷(𝐺)−𝐴(𝐺), where 𝐷(𝐺) and 𝐴(𝐺) are the diagonal matrix of vertex degrees and the adjacency matrix of a graph 𝐺, respectively, and the spectral radius of a graph 𝐺 is the largest eigenvalue of 𝐴(𝐺). The spectra of the graph and corresponding eigenvalues are closely linked to the molecular stability and related chemical properties. In quantum chemistry, spectral radius of a graph is the maximum energy level of molecules. Therefore, good upper bounds for the spectral radius are conducive to evaluate the energy of molecules. In this paper, we first give several sharp upper bounds on the adjacency spectral radius in terms of some invariants of graphs, such as the vertex degree, the average 2-degree, and the number of the triangles. Then, we give some numerical examples which indicate that the results are better than the mentioned upper bounds in some sense. Finally, an upper bound of the Nordhaus-Gaddum type is obtained for the sum of Laplacian spectral radius of a connected graph and its complement. Moreover, some examples are applied to illustrate that our result is valuable.

1. Introduction

The graphs in this paper are simple and undirected. Let 𝐺 be a simple graph with 𝑛 vertices and 𝑚 edges. For 𝑣∈𝑉, denote by 𝑑_𝑣, 𝑚_𝑣, and 𝑁_𝑣 the degree of 𝑣, the average 2-degree of 𝑣, and the set of neighbors of 𝑣, respectively. Then 𝑑_𝑣𝑚_𝑣 is the 2-degree of 𝑣. Let Δ, Δ^′, 𝛿, and 𝛿^′ denote the maximum degree, second largest degree, minimum degree, and second smallest degree of vertices of 𝐺, respectively. Obviously, we have Δ^′<Δ and 𝛿^′>𝛿. A graph is 𝑑-regular if Δ=𝛿=𝑑.

The complement graph 𝐺^𝑐 of 𝐺 is the graph with the same set of vertices as G, where two distinct vertices are adjacent if and only if they are independent in 𝐺. The line graph 𝐿_𝐺 of 𝐺 is defined by 𝑉(𝐿_𝐺)=𝐸(𝐺), where any two vertices in 𝐿_𝐺 are adjacent if and only if they are adjacent as edges of G.

Let 𝑋 be a nonnegative square matrix. The spectral radius 𝜌(𝑋) of 𝑋 is the maximum eigenvalue of 𝑋. Denote by 𝐵 the adjacency matrix of 𝐿_𝐺, then 𝜌(𝐵) is the spectral radius of 𝐵. Let 𝐷(𝐺) and 𝐴(𝐺) denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Then the matrix 𝐿(𝐺)=𝐷(𝐺)−𝐴(𝐺) is called the Laplacian matrix of a graph 𝐺. Obviously, it is symmetric and positive semidefinite. Similarly, the quasi-Laplacian matrix is defined as 𝑄(𝐺)=𝐷(𝐺)+𝐴(𝐺), which is a nonnegative irreducible matrix. The largest eigenvalue of the Laplacian matrix, denoted by 𝜇(𝐺), is called the Laplacian spectral radius. The Laplacian eigenvalues of a graph are important in graph theory, because they have close relations to many graph invariants, including connectivity, isoperimetric number, diameter, and maximum cut. Particularly, good upper bounds for 𝜇(𝐺) are applied in many fields. For instance, it is used in theoretical chemistry, within the Heilbronner model, to determine the first ionization potential of alkanes, in combinatorial optimization to provide an upper bound on the size of the maximum cut in graph, in communication networks to provide a lower bound on the edge-forwarding index, and so forth. To learn more information on the applications of Laplacian spectral radius and other Laplacian eigenvalues of a graph, see references [1–4].

In the recent thirty years, the researchers obtained many good upper bounds for 𝜇(𝐺) [5–8]. These upper bounds improved the previous results constantly. In this paper, we focus on the bounds for the spectral radius of a graph, and the bound of Nordhaus-Gaddum type is also considered, which is the sum of Laplacian spectral radius of a connected graph 𝐺 and its complement 𝐺^𝑐.

At the end of this section, we introduce some lemmas which will be used later on.

Lemma 1 (see [9]). Let 𝑀=(𝑚_𝑖𝑗)_𝑛×𝑛 be an irreducible nonnegative matrix with spectral radius 𝜌(𝑀), and let 𝑅_𝑖(𝑀) be the 𝑖th row sum of M; that is, 𝑅_𝑖(𝑀)=∑_𝑗𝑚_𝑖𝑗. Then min_{1≤𝑖≤𝑛}𝑅_𝑖(𝑀)≤𝜌(𝑀)≤max_{1≤𝑖≤𝑛}𝑅_𝑖(𝑀). (1)
Moreover, if the row sums of 𝑀 are not all equal, then both inequalities are strict.

Lemma 2 (see [10]). Let 𝐺=[𝑉,𝐸] be a connected graph with 𝑛 vertices; then 𝜌(𝐺)≤12𝜌𝐿_𝐺+1. (2)
The equality holds if and only if 𝐺 is a regular graph.

This lemma gives a relation between the spectral radius of a graph and its line graph. Therefore, we can estimate the spectral radius of the adjacency matrix of graph by estimating that of its line graph.

Lemma 3 (see [11]). Let 𝐵 be a real symmetric 𝑛×𝑛 matrix, and let 𝜌(𝐵) be the largest eigenvalue of 𝐵. If 𝑃(𝜆) is a polynomial on λ, then min_𝑣∈𝑉𝑅_𝑣(𝑃(𝐵))≤𝑃𝜌(𝐵)≤max_𝑣∈𝑉𝑅_𝑣(𝑃(𝐵)). (3)
Here 𝑅_𝑣(𝑃(𝐵)) is the 𝑣th row sum of matrix 𝑃(𝐵). Moreover, if the row sums of 𝑃(𝐵) are not all equal, then both inequalities are strict.

Lemma 4 (see [11]). Let 𝐺 be a simple connected graph with 𝑛 vertices and let 𝜌(𝑄) be the largest eigenvalue of the quasi-Laplacian matrix of graph 𝐺. Then 𝜇(𝐺)≤𝜌(𝑄), (4) with equality holds if and only if 𝐺 is a bipartite graph.

By these lemmas, we will give some improved upper bounds for the spectral radius and determine the corresponding extremal graphs.

This paper is organized as follows. In Section 2, we will give several sharp upper and lower bounds for the spectral radius of graphs and determine the extremal graphs which achieve these bounds. In Section 3, some bounds of Nordhaus-Gaddum type will be given. Furthermore, in Sections 2 and 3, we present some examples to illustrate that our results are better than all of the mentioned upper bounds in this paper, in some sense.

2. Bounds on the Spectral Radius

2.1. Previous Results

The eigenvalues of adjacency matrix of the graph have wide applications in many fields. For instance, it can be used to present the energy level of specific electrons. Specially, the spectral radius of a graph is the maximum energy level of molecules. Hence, good upper bound for the spectral radius helps to estimate the energy level of molecules [12–15]. Recently, there are some classic upper bounds for the spectral radius of graphs.

In the early time Cao [16] gave a bound as follows: 𝜌(𝐺)≤2𝑚−𝛿(𝑛−1)+Δ(𝛿−1). (5)

The equality holds if and only if 𝐺 is regular graph or a star plus of 𝐾₂, or a complete graph plus a regular graph with smaller degree of vertices.

Hu [17] obtained an upper bound with simple form as follows: 𝜌(𝐺)≤√2𝑚−𝑛−𝛿+2. (6)

The equality holds if and only if 𝐺 is 𝑛−2 regular graph.

In 2005, Xu [18] proved that 𝜌(𝐺)≤2𝑚−𝑛+1−(𝛿−1)(𝑛−1−Δ). (7)

The equality holds if and only if 𝐺 is regular graph or a star graph.

Using the average 2-degree of the vertices, the rese-archers got more upper bounds.

Cao’s [16] another upper bound: 𝜌(𝐺)≤max_{(𝐺)𝑢∈𝑉}𝑑_𝑢𝑚_𝑢. (8)

The equality holds if and only if 𝐺 is a regular graph or a semiregular bipartite graph.

Similarly, Abrham and Zhang [19] proved that 𝜌(𝐺)≤max_{(𝐺)𝑢𝑣∈𝐸}𝑑_𝑢𝑑_𝑣. (9)

The equality holds if and only if 𝐺 is a regular graph or a semiregular bipartite graph.

In recent years, Feng et al. [10] give some upper bounds for the spectral radius as follows: 𝜌(𝐺)≤max_{(𝐺)𝑢∈𝑉}𝑑²_𝑢+𝑑_𝑢𝑚_𝑢2. (10)

The equality holds if and only if 𝐺 is regular graph. 𝜌(𝐺)≤max_{(𝐺)𝑢𝑣∈𝐸}𝑑_𝑢𝑑_𝑢+𝑚_𝑢+𝑑_𝑣𝑑_𝑣+𝑚_𝑣2. (11)

The equality holds if and only if 𝐺 is regular graph. 𝜌(𝐺)≤max_{(𝐺)𝑢∈𝑉}𝑑_𝑢+𝑑_𝑢𝑚_𝑢2. (12)

The equality holds if and only if 𝐺 is regular graph. 𝜌(𝐺)≤max_{(𝐺)𝑢𝑣∈𝐸}𝑑_𝑢+𝑑_𝑣+𝑑_𝑢−𝑑_𝑣²+4𝑚_𝑢𝑚_𝑣4. (13)

The equality holds if and only if 𝐺 is regular graph.

2.2. Main Results

All of these upper bounds mentioned in Section 2.1 are characterized by the degree and the average 2-degree of the vertices. Actually, we can also use other invariants of the graph to estimate the spectral radius. In the following, such an invariant will be introduced.

In a graph, a circle with length 3 is called a triangle. If 𝑢 is a triangle’s vertex in a graph, then 𝑢 is incident with this triangle. Denote by 𝑇_𝑢 the number of the triangles associated with the vertex 𝑢. For example, in Figure 1, we have 𝑇_𝑢 = 3 and 𝑇_𝑣 = 𝑇_𝑤= 0.

Figure 1 

Graph with triangles.

Let 𝑁_𝑢∩𝑁_𝑣 be the set of the common adjacent points of vertex 𝑢 and v; then |𝑁_𝑢∩𝑁_𝑣| present the cardinality of 𝑁_𝑢∩𝑁_𝑣.

Now, some new and sharp upper and lower bounds for the spectral radius will be given.

Theorem 5. Let 𝐺 be a simple connected graph with 𝑛 vertices. Then 𝜌(𝐺)≤max_{(𝐺)𝑢𝑣∈𝐸}𝑑²_𝑢𝑚_𝑢+𝑑²_𝑣𝑚_𝑣−2𝑇_𝑢+𝑇_𝑣2𝑑_𝑢𝑑_𝑣−𝑁_𝑢∩𝑁_𝑣; (14) the equality holds if and only if 𝐺 is a regular graph.

Proof. Let 𝐾=diag(𝑑_𝑢𝑑_𝑣−|𝑁_𝑢∩𝑁_𝑣|∶𝑢𝑣∈𝐸(𝐺)) is a diagonal matrix and 𝐵 is the adjacency matrix of the line graph. Denote 𝑁=𝐾⁻¹𝐵𝐾, then 𝑁 and 𝐵 have the same eigenvalues. Since 𝐺 is a simple connected graph, it is easy to obtain that 𝑁 is nonnegative and irreducible matrix. The (𝑢𝑣,𝑝𝑞)th entry of 𝑁 is equal to 𝑑_𝑝𝑑_𝑞−𝑁_𝑝∩𝑁_𝑞𝑑_𝑢𝑑_𝑣−𝑁_𝑢∩𝑁_𝑣,𝑝𝑞∼𝑢𝑣,0,else, (15) here 𝑝𝑞∼𝑢𝑣 implies that 𝑝𝑞 and 𝑢𝑣 are adjacent in graph. Hence, the 𝑢𝑣th row sum 𝑅_𝑢𝑣(𝑁) of 𝑁 is _{𝑝𝑞∼𝑢𝑣}𝑑_𝑝𝑑_𝑞−𝑁_𝑝∩𝑁_𝑞𝑑_𝑢𝑑_𝑣−𝑁_𝑢∩𝑁_𝑣=∑_𝑞∼𝑢𝑑_𝑢𝑑_𝑞+∑_𝑝∼𝑣𝑑_𝑝𝑑_𝑣−2𝑑_𝑢𝑑_𝑣𝑑_𝑢𝑑_𝑣−𝑁_𝑢∩𝑁_𝑣−∑_𝑞∼𝑢𝑁_𝑢∩𝑁_𝑞+∑_𝑝∼𝑣𝑁_𝑝∩𝑁_𝑣−2𝑁_𝑢∩𝑁_𝑣𝑑_𝑢𝑑_𝑣−𝑁_𝑢∩𝑁_𝑣=𝑑²_𝑢𝑚_𝑢+𝑑²_𝑣𝑚_𝑣−2𝑑_𝑢𝑑_𝑣−2𝑇_𝑢+𝑇_𝑣+2𝑁_𝑢∩𝑁_𝑣𝑑_𝑢𝑑_𝑣−𝑁_𝑢∩𝑁_𝑣=𝑑²_𝑢𝑚_𝑢+𝑑²_𝑣𝑚_𝑣−2𝑇_𝑢+𝑇_𝑣𝑑_𝑢𝑑_𝑣−𝑁_𝑢∩𝑁_𝑣−2. (16)
From Lemmas 1 and 2, we have 𝜌(𝐺)≤12𝜌(𝐵)1+1≤max2𝑅_𝑢𝑣(𝑁)+1∶𝑢𝑣∈𝑉(𝐻). (17)
It means that (14) holds and the equality in (14) holds if and only if 𝐺 is a regular graph.

In a graph, let 𝛼 and 𝛽 represent the number of vertices with the maximum degree and minimum degree, respectively. Then, we get the following results.

Theorem 6. Let 𝐺 be a simple connected graph with 𝑛 vertices. If Δ≤min{𝑛−1−𝛽,𝑛−1−𝛼}, then 𝜌(𝐺)≤2𝑚+Δ𝛿^′−1−𝛽𝛿−𝑛−1−𝛽𝛿^′,𝜌(𝐺)≥2𝑚+Δ^′−1𝛿−𝛼Δ−(𝑛−1−𝛼)Δ^′; (18)(19) the equality holds if and only if 𝐺 is a regular graph.

Proof. Since 𝑅_𝑣(𝐴²) is exactly the number of walks of length 2 in 𝐺 with a starting point 𝑣, thus 𝑅_𝑣𝐴²=_𝑢∼𝑣𝑑_𝑢=2𝑚−𝑑_𝑣−_{𝑢̸∼𝑣}𝑑_𝑢. (20)
Therefore, from Lemmas 1 and 3, if Δ≤𝑛−1−𝛽, we have 𝑑_𝑣≤𝑛−1−𝛽 for any 𝑣∈𝑉(𝐺). Then 𝜌𝐴²≤max_{𝑣∈𝑉(𝐺)}2𝑚−𝑑_𝑣−_{𝑢̸∼𝑣}𝑑_𝑢≤max_{𝑣∈𝑉(𝐺)}2𝑚−𝑑_𝑣−𝛽𝛿+𝑛−𝑑_𝑣−1−𝛽𝛿^′=max_{𝑣∈𝑉(𝐺)}2𝑚+𝛿^′−1𝑑_𝑣−𝛽𝛿−𝑛−1−𝛽𝛿^′≤2𝑚+Δ𝛿^′−1−𝛽𝛿−𝑛−1−𝛽𝛿^′. (21)
Hence, it is easy to obtain that (18) holds.
If equality in (18) holds, then all equalities in the above argument must hold. Thus, for all 𝑣∈𝑉(𝐺)_{𝑢̸∼𝑣}𝑑_𝑢=𝛽𝛿+𝑛−𝑑_𝑣−1−𝛽𝛿^′. (22)
It means that 𝑑_𝑣=𝑛−1 and 𝛿^′=𝛿, or 𝑑_𝑢=𝛿=𝛿^′; this shows that the graph 𝐺 is regular. Conversely, if 𝐺 is k-regular, it is not difficult to check that 𝜌(𝐺) attains the upper bound by direct calculation.
Similarly for the lower bound, if Δ≤𝑛−1−𝛼, we have 𝜌𝐴²≥min_{𝑣∈𝑉(𝐺)}2𝑚−𝑑_𝑣−_{𝑢̸∼𝑣}𝑑_𝑢≥min_{𝑣∈𝑉(𝐺)}2𝑚−𝑑_𝑣−𝛼Δ+𝑛−𝑑_𝑣−1−𝛼Δ^′=min_{𝑣∈𝑉(𝐺)}2𝑚+Δ^′−1𝑑_𝑣−𝛼Δ−(𝑛−1−𝛼)Δ^′≥2𝑚+Δ^′−1𝛿−𝛼Δ−(𝑛−1−𝛼)Δ^′. (23)
It means that (19) holds and the equality in (19) holds if and only if 𝐺 is a regular graph by similar discussion.

Theorem 7. Let 𝐺 be a simple connected graph with 𝑛 vertices. If Δ≤𝑛−1−𝛽; then 𝜌(𝐺)≤𝛿^′−1+𝛿^′+1²+8𝑚−4𝛽𝛿−𝛿^′−4𝑛𝛿^′2; (24) the equality holds if and only if 𝐺 is a regular graph.

Proof. According to the proof of Theorem 6, we have 𝑅_𝑣𝐴²=2𝑚−𝑑_𝑣−_{𝑢̸∼𝑣}𝑑_𝑢≤2𝑚+𝛿^′−1𝑑_𝑣−𝛽𝛿−𝑛−1−𝛽𝛿^′. (25)
Thus 𝑅_𝑣𝐴²−𝛿^′−1𝐴≤2𝑚−𝛽𝛿−𝑛−1−𝛽𝛿^′. (26)
From Lemma 3, we have 𝜌²(𝐴)−𝛿^′−1𝜌(𝐴)−2𝑚+𝛽𝛿+𝑛−1−𝛽𝛿^′≤0. (27)
Solving this quadratic inequality, we obtain that upper bound (24) holds.
If equality in (24) holds, then all equalities in the argument must hold. By the similar discussion of Theorem 6, the equality holds if and only if 𝐺 is a regular graph.

2.3. Numerical Examples

In this section, we will present two graphs to illustrate that our some new bounds are better than other bounds in some sense. Let Figures 2 and 3 be graphs of orders 7 and 8.

Figure 2 

Graph of order 7.

Figure 3 

Graph of order 8.

The estimated value of each upper bound is listed in Table 1. Obviously, from Table 1, bound (24) is the best in all known upper bounds for Figure 2 and bound (14) is the best for Figure 3. Furthermore, bound (18) is the best except (13) and (24) for Figure 2. Hence, commonly, these upper bounds are incomparable.

Table 1 

Estimated value of each upper bound.

3. Bounds of the Nordhaus-Gaddum Type

3.1. Previous Results

In this part, we mainly discuss the upper bounds on the sum of Laplacian spectral radius of a connected graph 𝐺 and its complement 𝐺^𝑐, which is called the upper bound of the Nordhaus-Gaddum type. For convenience, let 𝜎(𝐺)=𝜇(𝐺)+𝜇𝐺^𝑐. (28)

The following are some classic upper bounds of Nordhaus-Gaddum type. The coarse bound 𝜇(𝐺)≤2Δ easily implies the simplest upper bound on 𝜎(𝐺): 𝜎(𝐺)≤2(𝑛−1)+2(Δ−𝛿). (29)

In particular, if both 𝐺 and 𝐺^𝑐 are connected and irregular, Shi [20] gave a better upper bound as follows: 𝜎(𝐺)2≤2𝑛−1−2𝑛²().−𝑛+2Δ−𝛿 (30)

Liu et al. [21] proved that 𝜎(𝐺)≤𝑛−2+(Δ−𝜔)²+𝑛²+4(Δ−𝛿)(𝑛−1)^1/2, (31) where 𝜔=𝑛−𝛿−1.

Shi [20] gives another upper bound 𝜎(𝐺)≤2(𝑛−1)(2𝜔−𝛿)+(Δ+𝛿)²−Δ+𝛿^1/2. (32)

To learn other bounds of the Nordhaus-Gaddum type, see references [22, 23]. In order to state the main result of this section, we first give an upper bound for the Laplacian spectral radius.

3.2. Laplacian Spectral Radius

Here we give a new upper bound for the Laplacian spectral radius. For convenience, let 𝑓()=𝛿𝑚,Δ,𝛿Δ−2−1²+16𝑚−2𝛿(4𝑛−𝛿−2)^1/2. (33)

Theorem 8. Let 𝐺 be a simple connected graph of order 𝑛 with Δ and 𝛿; then 𝜇(𝐺)≤Δ+(3/2)𝛿−1+𝑓(𝑚,Δ,𝛿)2, (34) with equality holds if and only if 𝐺 is bipartite regular.

Proof. Let 𝐾=𝑄−𝛿𝐸; then 𝑅_𝑣(𝐾)=2𝑑_𝑣−𝛿, it means that 2𝑑_𝑣=𝑅_𝑣(𝐾)+𝛿. Considering the 𝑣th row sum of matrix 𝐾², we have 𝑅_𝑣𝐾²=𝑅_𝑣𝑄²−2𝛿𝑅_𝑣(𝑄)+𝛿²=2𝑑²_𝑣+2_𝑢∼𝑣𝑑_𝑢−4𝛿𝑑_𝑣+𝛿²=2𝑑²_𝑣+22𝑚−𝑑_𝑣−_{𝑢̸∼𝑣,𝑢≠𝑣}𝑑_𝑢−4𝛿𝑑_𝑣+𝛿²≤2𝑑²_𝑣+22𝑚−𝑑_𝑣−𝑛−𝑑_𝑣−1𝛿−4𝛿𝑑_𝑣+𝛿²=2𝑑²_𝑣−2𝑑_𝑣−2𝛿𝑑_𝑣+4𝑚−2(𝑛−1)𝛿+𝛿²=2𝑑_𝑣−𝛿𝑑_𝑣−(2+𝛿)𝑑_𝑣+4𝑚−2(𝑛−1)𝛿+𝛿²≤Δ𝑅_𝑣(𝐾)−()𝑅2+𝛿^𝑣(𝐾)+𝛿2+4𝑚−2(𝑛−1)𝛿+𝛿²=𝛿Δ−2−1𝑅_𝑣(𝐾)𝛿+4𝑚−2𝑛𝛿+𝛿+²2. (35)
This is equivalent to the following inequality: 𝑅_𝑣𝐾²−𝛿Δ−2𝐾𝛿−1≤4𝑚−2𝑛𝛿+𝛿+²2. (36)
From Lemma 3, we obtain that 𝜌²(𝐾)−𝛿Δ−2𝜌(𝐾)𝛿−1≤4𝑚−2𝑛𝛿+𝛿+²2. (37)
By simple calculation, we get the upper bound of the spectral radius of matrix 𝐾 as follows: 𝜌(𝐾)≤Δ−(𝛿/2)−12+(Δ−(𝛿/2)−1)²+16𝑚−2𝛿(4𝑛−𝛿−2)^1/22. (38)
Since 𝜌(𝐾)=𝜌(𝑄)−𝛿, therefore from Lemma 4 we obtain that the result (34) holds.
If the spectral radius 𝜇(𝐺) achieves the upper bound in (34), then each inequality in the above proof must be equal. This implies that Δ=𝛿 for all 𝑣∈𝑉(𝐺), thus 𝐺 is regular graph. From Lemma 4 again, G is regular bipartite graph.
Conversely, it is easy to verify that equality in (34) holds for regular bipartite graphs.

3.3. Bound of the Nordhaus-Gaddum Type

In this part, based on Theorem 8, an upper bound of Nordhaus-Gaddum type of Laplacian matrix will be given.

Theorem 9. Let 𝐺 be a simple graph of order 𝑛 with Δ and 𝛿; then𝜎(𝐺)≤5𝑛−Δ+𝛿−9+√22(2Δ−𝛿−2)²+8𝛿(2+𝛿)+(𝜔−Δ)(𝑛+3Δ−3𝛿−5)+32𝑛𝜔−8𝜋(3𝑛+Δ−1)^1/24 (39)here 𝜔=𝑛−𝛿−1 and 𝜋=𝑛−Δ−1. Moreover, if both 𝐺 and 𝐺^𝑐 are connected, then the upper bound is strict.

Proof. According to the relation of a graph 𝐺 and its complement, it is not difficult to obtain the invariants of 𝐺^𝑐. Denote it by Δ(𝐺^𝑐)=𝑛−𝛿−1, 𝛿(𝐺^𝑐)=𝑛−Δ−1, and 𝑚(𝐺^𝑐)=𝐶²_𝑛−𝑚. From Theorem 8, we have 𝜇𝐺^𝑐≤Δ𝐺^𝑐+(3/2)𝛿𝐺^𝑐−12+𝑓𝑚𝐺^𝑐,Δ𝐺^𝑐,𝛿𝐺^𝑐2. (40)
Let 𝑔(𝑚)=𝑓(𝑚,Δ,𝛿)+𝑓𝑚𝐺^𝑐,Δ𝐺^𝑐,𝛿𝐺^𝑐. (41)
Then the upper bound of the Nordhaus-Gaddum type of Laplacian matrix is 𝜎(𝐺)=𝜇(𝐺)+𝜇𝐺^𝑐≤5𝑛−Δ+𝛿−9+2𝑔(𝑚)4 (42)
since 𝑔^′(𝑚)=8𝑓()−8𝑚,Δ,𝛿𝑓(𝑚(𝐺^𝑐),Δ(𝐺^𝑐),𝛿(𝐺^𝑐.)) (43)
Obviously, 𝑔^′(𝑚)≥0 holds if and only if the following inequality holds: 𝑓(𝑚,Δ,𝛿)≤𝑓𝐶²_𝑛−𝑚,𝑛−𝛿−1,𝑛−Δ−1. (44)
Let 𝑚 be a variable; then solving this inequality, we have ()()𝑚≤𝑛−𝛿−Δ−1)(𝑛−3𝛿+3Δ−5+32𝑛𝑛+𝛿−1−()()1288𝛿𝛿+2−8𝑛−Δ−1)(3𝑛+Δ−1128=𝑚^∗. (45)
Here, the symbol 𝑚^∗ represents the right hand of the above inequality. Then we can assert that 𝑔(𝑚) is an increasing function for 𝑚≤𝑚^∗, and it implies that 𝑔(𝑚)≤𝑔(𝑚^∗). Therefore, we have 𝜎(𝐺)≤5𝑛−Δ+𝛿−9+2𝑔𝑚^∗4=5𝑛−Δ+𝛿−9+4𝑓𝑚^∗,Δ,𝛿4. (46)
Simplifying this expression by direct calculation, we prove that the result (39) is correct.
If equality in (39) holds, then each inequality in the above proof must be equality. From Theorem 8, we obtain that both 𝐺 and 𝐺^𝑐 are regular bipartite. But it is impossible for a connected graph, this implies that the Laplacian spectral radius of either 𝐺 or 𝐺^𝑐 fails to achieve its upper bound and so does the sum. Hence the inequality in (39) is strict.

3.4. Numerical Examples

In this section, we give some examples to illustrate that the new bound is better than other bounds for some graphs. Considering the graph of order 10 in Figure 4 and Figures 1–3, the estimated value of each upper bound of the Nordhaus-Gaddum type is given in Table 2.

Table 2 

Estimated value of each upper bound.

Figure 4 

Graph of order 10.

Clearly, from Table 2, we can see that new bound (39) is the best in all known upper bounds for all figures mentioned in this paper.

4. Conclusion

From numerical examples of Sections 2 and 3, the estimated value of new upper bounds of the spectral radius and the Nordhaus-Gaddum type of graphs are the smallest in all known upper bounds for the graphs considered in these examples. It means that our results are better than the existing upper bounds in some sense.

Acknowledgment

This work is supported by the Research Fund of Sichuan Provincial Education Department (Grant no. 11ZA159, 11ZZ020, 12ZB238, and 13ZB0108).