On the distribution of eigenvalues of increasing trees

Abstract

We prove that the multiplicity of a fixed eigenvalue α in a random recursive tree on n vertices satisfies a central limit theorem with mean and variance asymptotically equal to μαn and σα2n respectively. It is also shown that μα and σα2 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 μ0 and σ02 can be determined explicitly by means of generating functions. Analogous results are also obtained for Laplacian eigenvalues and binary increasing trees.