de Caenʹs inequality and bounds on the largest Laplacian eigenvalue of a graph Original Research Article

In this paper we determine the extremal graphs for which equality in de Caenʹs inequality holds and then apply the inequality to give an upper bound for the largest Laplacian eigenvalue λ1(G) of a graph. In addition, we give two other types of upper bound for λ1(G) and determine the extremal graphs which achieve the bounds.