Abstract

Let λ1 (G) ≥ λ2 (G) ≥ ⋯ ≥ λn (G) = 0 be the Laplacian eigenvalues of a simple undirected graph G. Let s ≥ 2 and t ≥ 2 be integers and let Ts, t be the rooted tree of three levels and order st + 1 such that the vertex root has degree s, the vertices in level 2 have degree t and the s (t - 1) pendants vertices are in level 3. We prove thatλs (Ts, t) = max {λs (T) : T is a tree of order st + 1} = frac(1, 2) fenced(t + 1 + sqrt(t2 + 2 t - 3)) .This result solves a conjecture due to Shao et al. in [J.-y. Shao, L. Zhang, X.-y. Yuan, On the second Laplacian eigenvalue of trees of odd order, Linear Algebra Appl. 419 (2006) 475-485]. © 2007 Elsevier Inc. All rights reserved.