Some spectral properties of starlike trees

M. LepoviŠ, I. Gutman

A tree is said to be starlike if exactly one of its vertices has degree greater than two. We show that almost all starlike trees are hyperbolic, and determine all exceptions. If $k$ is the maximal vertex degree of a starlike tree and $\lambda_1$ is its largest eigenvalue, then $\sqrt{k} \leq \lambda_1 < k/\sqrt{k-1}$ . A new way to characterize integral starlike trees is put forward.