On the distribution of eigenvalues of increasing trees

Abstract

We prove that the multiplicity of a fixed eigenvalue $\alpha$ in a random recursive tree on $n$ vertices satisfies a central limit theorem with mean and variance asymptotically equal to ${\mu }_{\alpha }n$ and ${\sigma }_{\alpha }^{2}n$ respectively. It is also shown that ${\mu }_{\alpha }$ and ${\sigma }_{\alpha }^2$ are positive for every totally real algebraic integer. The proofs are based on a general result on additive tree functionals due to Holmgren and Janson. In the case of the eigenvalue $0$, the constants ${\mu }_0$ and ${\sigma }_0^2$ can be determined explicitly by means of generating functions. Analogous results are also obtained for Laplacian eigenvalues and binary increasing trees.