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

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.


Introduction
The graphs in this paper are simple and undirected. Let be a simple graph with vertices and edges. For V ∈ , denote by V , V , and V the degree of v, the average 2-degree of V, and the set of neighbors of v, respectively. Then V V is the 2degree of V. Let Δ, Δ , , and denote the maximum degree, second largest degree, minimum degree, and second smallest degree of vertices of G, respectively. Obviously, we have Δ < Δ and > . A graph is d-regular if Δ = = .
The complement graph of 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 . The line graph of is defined by ( ) = ( ), where any two vertices in are adjacent if and only if they are adjacent as edges of G.
Let be a nonnegative square matrix. The spectral radius ( ) of is the maximum eigenvalue of . Denote by the adjacency matrix of , then ( ) is the spectral radius of . Let ( ) and ( ) denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Then the matrix ( ) = ( )− ( ) is called the Laplacian matrix of a graph . Obviously, it is symmetric and positive semidefinite.
Similarly, the quasi-Laplacian matrix is defined as ( ) = ( ) + ( ), which is a nonnegative irreducible matrix. The largest eigenvalue of the Laplacian matrix, denoted by ( ), 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 ( ) 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][2][3][4].
In the recent thirty years, the researchers obtained many good upper bounds for ( ) [5][6][7][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 2 The Scientific World Journal is the sum of Laplacian spectral radius of a connected graph and its complement . At the end of this section, we introduce some lemmas which will be used later on.
Lemma 1 (see [9]). Let = ( ) × be an irreducible nonnegative matrix with spectral radius ( ), and let ( ) be the th row sum of M; that is, ( ) = ∑ . Then Moreover, if the row sums of are not all equal, then both inequalities are strict.
Lemma 2 (see [10]). Let = [ , ] be a connected graph with vertices; then The equality holds if and only if 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.

3)
Here V ( ( )) is the vth row sum of matrix ( ). Moreover, if the row sums of ( ) are not all equal, then both inequalities are strict.
Lemma 4 (see [11]). Let be a simple connected graph with vertices and let ( ) be the largest eigenvalue of the quasi-Laplacian matrix of graph . Then with equality holds if and only if 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.

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][13][14][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: The equality holds if and only if is regular graph or a star plus of 2 , or a complete graph plus a regular graph with smaller degree of vertices.
Hu [17] obtained an upper bound with simple form as follows: The equality holds if and only if is − 2 regular graph.
In 2005, Xu [18] proved that The equality holds if and only if is regular graph or a star graph.
Using the average 2-degree of the vertices, the researchers got more upper bounds.
Cao's [16] another upper bound: The equality holds if and only if is a regular graph or a semiregular bipartite graph.
Similarly, Abrham and Zhang [19] proved that The equality holds if and only if 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: The equality holds if and only if is regular graph.
The equality holds if and only if is regular graph.
The equality holds if and only if is regular graph.
The equality holds if and only if is regular graph.
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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 is a triangle's vertex in a graph, then is incident with this triangle. Denote by the number of the triangles associated with the vertex . For example, in Figure 1, we have = 3 and Let ∩ V be the set of the common adjacent points of vertex and v; then | ∩ V | present the cardinality of ∩ V . Now, some new and sharp upper and lower bounds for the spectral radius will be given.

Theorem 5. Let be a simple connected graph with vertices. Then
the equality holds if and only if is a regular graph.
is a diagonal matrix and is the adjacency matrix of the line graph. Denote = −1 , then and have the same eigenvalues. Since is a simple connected graph, it is easy to obtain that is nonnegative and irreducible matrix. The ( V, )th entry of is equal to here ∼ V implies that and V are adjacent in graph. Hence, the Vth row sum V ( ) of is From Lemmas 1 and 2, we have It means that (14) holds and the equality in (14) holds if and only if is a regular graph.
In a graph, let and represent the number of vertices with the maximum degree and minimum degree, respectively. Then, we get the following results. Theorem 6. Let be a simple connected graph with vertices.
the equality holds if and only if is a regular graph.
Proof. Since V ( 2 ) is exactly the number of walks of length 2 in with a starting point V, thus Therefore, from Lemmas 1 and 3, if Δ ≤ − 1 − , we have V ≤ − 1 − for any V ∈ ( ). Then 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 ∈ ( ) It means that V = − 1 and = , or = = ; this shows that the graph is regular. Conversely, if is k-regular, it is not difficult to check that ( ) attains the upper bound by direct calculation.

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Similarly for the lower bound, if Δ ≤ − 1 − , we have It means that (19) holds and the equality in (19) holds if and only if is a regular graph by similar discussion. (24)

the equality holds if and only if is a regular graph.
Proof. According to the proof of Theorem 6, we have 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 is a regular graph.

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.
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.

Previous Results.
In this part, we mainly discuss the upper bounds on the sum of Laplacian spectral radius of a connected graph and its complement , which is called the upper bound of the Nordhaus-Gaddum type. For convenience, let ( ) = ( ) + ( ) . (28) The following are some classic upper bounds of Nordhaus-Gaddum type. The coarse bound ( ) ≤ 2Δ easily implies the simplest upper bound on ( ): In particular, if both and are connected and irregular, Shi [20] gave a better upper bound as follows: Liu et al. [21] proved that Shi [20] gives another upper bound 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. Proof. Let = − ; then V ( ) = 2 V − , it means that 2 V = V ( ) + . Considering the Vth row sum of matrix 2 , we have

Laplacian Spectral
This is equivalent to the following inequality: From Lemma 3, we obtain that By simple calculation, we get the upper bound of the spectral radius of matrix as follows: Since ( ) = ( )− , therefore from Lemma 4 we obtain that the result (34) holds.
If the spectral radius ( ) achieves the upper bound in (34), then each inequality in the above proof must be equal. This implies that Δ = for all V ∈ ( ), thus 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.

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.
Here, the symbol * represents the right hand of the above inequality. Then we can assert that ( ) is an increasing function for ≤ * , and it implies that ( ) ≤ ( * ). Therefore, we have 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 and are regular bipartite. But it is impossible for a connected graph, this implies that the Laplacian spectral radius of either or fails to achieve its upper bound and so does the sum. Hence the inequality in (39) is strict.

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.
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.