On the distribution of eigenvalues of increasing trees

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 2n 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 02 can be determined explicitly by means of generating functions. Analogous results are also obtained for Laplacian eigenvalues and binary increasing trees. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).