Bounds on the (Laplacian) spectral radius of graphs

Abstract

The spectral radius of a graph is the largest eigenvalue of adjacency matrix of the graph and its Laplacian spectral radius is the largest eigenvalue of the Laplacian matrix which is the difference of the diagonal matrix of vertex degrees and the adjacency matrix. Some sharp bounds are obtained for the (Laplacian) spectral radii of connected graphs. As consequences, some (sharp) upper bounds of the Nordhaus–Gaddum type are also obtained for the sum of (Laplacian) spectral radii of a connected graph and its connected complement.