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

Abstract

Let G be a simple graph. In this paper, we obtain a sequence ( b p ) p=1 ∞ of upper bounds on the largest eigenvalue λ 1( G) of the Laplacian matrix of G. Then, we show that this sequence converges to λ 1( G) and that ( b 2 p ) 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, b p = λ 1( G) for all p. The bounds b 1 and b 2 are discussed.