On the second Laplacian eigenvalues of trees of odd order (Q865409)

In the paper, the second Laplacian eigenvalue of a tree \(T\), \(\lambda_2(T)\), is studied. It is shown that among all trees of odd order \(2t+1\) (\(t\geq 4\)) the largest value of \(\lambda_2(T)\) is \(\frac 12(t+1+\sqrt{t^2+2t-3})\), while the second largest value of \(\lambda_2(T)\) is the second largest root of the cubic equation \(x^3-(2t+3)x^2+(t^2+3t+3)x-(2t+1)=0\). Trees for which these bounds are attained are constructed.