On the \(s\)th Laplacian eigenvalue of trees of order \(st+1\) (Q2370787)

The paper deals with Laplacian eigenvalues of a rooted tree of three levels and order \(st+1\). Let \(L({\mathcal G})\) be the \(n \times n\) Laplacian matrix of an undirected graph on \(n\) vertices, \({\mathcal G}\), and \(\lambda_1({\mathcal G})\geq \lambda_2({\mathcal G}) \geq \cdots \geq \lambda_n({\mathcal G})=0\) its eigenvalues. Let \(s \geq 2\) and \(t \geq 2\) be given integers. The author denotes by \({\mathcal F}_{s,t}\) 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 \(s(t-1)\) pendant vertices are in level 3. In the main result of this paper the author proves that \[ \lambda_s({\mathcal F}_{s,t})= \max\{ \lambda_s({\mathcal F}): {\mathcal F} \text{ is a tree of order }st+1 \}= (1/2)\left(t+1+\sqrt{t^2+2t-3}\right). \] This result solves a conjecture due to \textit{J.-Y. Shao} et al. in [Linear Algebra Appl. 419, 475--485 (2006; Zbl 1110.05063)].