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

Let λ1(G) denote the largest eigenvalue of the adjacency matrix and let μ1(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 Δ≠0 then Thus the gap between the maximum and minimum value of λ1(G) and μ1(G) in the class of graphs with fixed Δ is Θ(Δ). In this note we show that in the class of trees with fixed Δ this gap is just . Namely, we show that if a tree T has the largest vertex degree Δ then New bounds are an improvement for Δ 3.