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 dv, mv, and Nv the degree of v, the average 2-degree of v, and the set of neighbors of v, respectively. Then dvmv is the 2-degree of v. Let ��, ����, ��, and �ġ� denote the maximum degree, second largest degree, minimum degree, and second smallest degree of vertices of G, respectively. Obviously, we have ���� < �� and �ġ� > ��. A graph is d-regular if �� = �� = d.The complement graph Gc 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 LG of G is defined by V(LG) = E(G), where any two vertices in LG are adjacent if and only if they are adjacent as edges of G. Let X be a nonnegative square matrix. The spectral radius ��(X) of X is the maximum eigenvalue of X. Denote by B the adjacency matrix of LG, then ��(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 ��(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 ��(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�C4].In the recent thirty years, the researchers obtained many good upper bounds for ��(G) [5�C8]. 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 Gc.At the end of this section, we introduce some lemmas which will be used later on. Lemma 1 (see [9]) ��Let M = (mij)n��n be an irreducible nonnegative matrix with spectral radius ��(M), and let Ri(M) be the ith row sum of M; that is, Ri(M) = ��jmij.