Upper bound for the Laplacian eigenvalues of a graph

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In this note we give a new upper bound for the Laplacian eigenvalues of an unweighted graph. Let G be a simple graph on n vertices. Let dₘ(G) and λₘ₊₁(G) be the m-th smallest degree of G and the m+1-th smallest Laplacian eigenvalue of G respectively. Then λₘ₊₁(G)≤ dₘ(G)+m-1 for G ≠ Kₘ+(n-m)K₁. We also introduce upper and lower bound for the Laplacian eigenvalues of weighted graphs, and compare it with the special case of unweighted graphs.