A decreasing sequence of upper bounds on the largest Laplacian eigenvalue of a graph Original Research Article

Let G be a simple graph. In this paper, we obtain a sequence (bp)p=1∞ of upper bounds on the largest eigenvalue λ1(G) of the Laplacian matrix of G. Then, we show that this sequence converges to λ1(G) and that (b2p)p=0∞ is a monotone strictly decreasing sequence except if G is a complete graph or G is a star graph or G is a regular complete bipartite graph. For these graphs, bp=λ1(G) for all p. The bounds b1 and b2 are discussed.