Bounding the largest eigenvalue of trees in terms of the largest vertex degree

Let lambda(1) (G) denote the largest eigenvalue of the adjacency matrix and let mu(1) (G) denote the largest eigenvalue of the Laplacian matrix of a graph G. It is well known that if a graph G has the largest vertex degree Deltanot equal0 then, rootDeltaless than or equal tolambda(1)(G)less than or equal toDelta and Delta+1less than or equal tomu(1)(G)less than or equal to2Delta. Thus the gap between the maximum and minimum value of lambda(1)(G) and mu(1)(G) in the class of graphs with fixed Delta is Theta(Delta). In this note we show that in the class of trees with fixed Delta this gap is just Theta(rootDelta). Namely, we show that if a tree T has the largest vertex degree Delta then lambda(1)(T)<2rootDelta-1 and mu(1)(T)<Delta+2rootDelta-1. New bounds are an improvement for Deltagreater than or equal to3. (C) 2002 Elsevier Science Inc. All rights reserved.