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arXiv · 2208.05575

On the distribution of eigenvalues of increasing trees

Kenneth Dadedzi·Stephan Wagner
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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 $\sigma^2_{\alpha} n$ respectively. It is also shown that $\mu_{\alpha}$ and $\sigma^2_{\alpha}$ 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 $\sigma^2_0$ can be determined explicitly by means of generating functions. Analogous results are also obtained for Laplacian eigenvalues and binary increasing trees.

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Kenneth Dadedzi, Stephan Wagner. 2022-08-10. On the distribution of eigenvalues of increasing trees. https://arxiv.org/abs/2208.05575

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