Authors: Stevanović, Dragan 
Title: Bounding the largest eigenvalue of trees in terms of the largest vertex degree
Journal: Linear Algebra and Its Applications
Volume: 360
First page: 35
Last page: 42
Issue Date: 1-Feb-2003
Rank: M22
ISSN: 0024-3795
DOI: 10.1016/S0024-3795(02)00442-1
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.
Keywords: Adjacency matrix | Laplacian matrix | Largest eigenvalue | Tree
Publisher: Elsevier

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