A note on the \(k\)th eigenvalue of trees
In [Linear Algebra Appl. 149, 19--34 (1991; Zbl 0764.05066)], \textit{J.-Y. Shao} proved that, for a tree \(T\) on \(n\) vertices, the \(k\)th eigenvalue \(\lambda_k(T)\) satisfies \( \lambda_k(T)\leq \sqrt{\frac{n}{k} - 1}\) \((1\leq k \leq \frac{n}{k}) \) and that this bound is best possible for all \(n \not\equiv 0\pmod k\) and for \(k=1\). Here it is proved that this bound is attained if and only if there exists a vertex \(v\) of \(T\) such that \(T-v = K_{1,\frac{n}{k} - 1} \cup F\), where \(F\) is a forest and \(\lambda_{k-1}(F) = \sqrt{\frac{n}{k} - 1}\).
scientific article; zbMATH DE number 5012995