On the second Laplacian eigenvalues of trees of odd order

We study the largest value and the second largest value of the second Laplacian eigenvalue λ2(T) among all trees T of a given odd order 2t + 1 (the case of the even order 2t was already settled by Zhang and Li in [Xiao-dong Zhang, Jiong-sheng Li, The two largest Laplacian eigenvalues of Laplacian matrices of trees, J. Univ. Sci. Technol. China 28 (5) (1998) 513–518]). We show that the largest value of λ2(T) among all trees T of order 2t + 1 (t ⩾ 4) is , while the second largest value of λ2(T) is the second largest root of the cubic equation x3-(2t+3)x2+(t2+3t+3)x-(2t+1)=0. We also determine the unique tree T1 of order 2t + 1 whose λ2(T1) reaches this largest value, and determine the unique tree T2 of order 2t + 1 whose λ2(T2) reaches this second largest value.