New Bounds of the Nordhaus-Gaddum Type of the Laplacian Matrix of Graphs

The Laplacian matrix is the difference of the diagonal matrix of vertex degrees and the adjacency matrix of a graph G. In this paper, we first give two sharp upper bounds for the radius of the Laplacian spectrum of G in terms of the edge number, the vertex number, the largest degree, the second largest degree and the smallest degree of G by applying non-negative matrix theory and graph theory. Then, two upper bounds of the NordhausGaddum type are obtained for the sum of Laplacian spectral radius of a connected graph and its connected complement. Moreover, we determine all extremal graphs which achieve these upper bounds. Finally, one numerical example illustrate that our results are better than the existing results in some sense.