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

Abstract

Let λ 1(G) denote the largest eigenvalue of the adjacency matrix and let μ 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 Δ≠0 then Δλ"1(G)Δ and Δ+1μ"1(G)2Δ.Thus the gap between the maximum and minimum value of λ 1(G) and μ 1(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 Θ(Δ). Namely, we show that if a tree T has the largest vertex degree Δ thenλ"1(T) <2 Δ-1 and μ"1(T)<Δ+2 Δ-1.New bounds are an improvement for Δ3.