On the s th Laplacian eigenvalue of trees of order st+1st+1

Let λ1(G)⩾λ2(G)⩾⋯⩾λn(G)=0λ1(G)⩾λ2(G)⩾⋯⩾λn(G)=0 be the Laplacian eigenvalues of a simple undirected graph GG. Let s⩾2s⩾2 and t⩾2t⩾2 be integers and let Ts,tTs,t be the rooted tree of three levels and order st+1st+1 such that the vertex root has degree s, the vertices in level 2 have degree t   and the s(t-1)s(t-1) pendants vertices are in level 3. We prove thatλs(Ts,t)=max{λs(T):Tis a tree of orderst+1}=12t+1+t2+2t-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].