# On the Spectrum of Finite, Rooted Homogeneous Trees

## TL;DR

This paper investigates the eigenvalues of finite rooted trees with regular branching, revealing they relate to generalized Fibonacci polynomials and establishing their limiting spectral distribution as tree depth increases.

## Contribution

It introduces a novel connection between tree spectra and generalized Fibonacci polynomials, extending to periodic branching and higher-dimensional complexes.

## Key findings

- Eigenvalues are roots of generalized Fibonacci polynomials.
- Limiting distribution of eigenvalues as tree depth grows.
- Extensions to periodic branching and simplicial complexes.

## Abstract

In this paper we study the adjacency spectrum of families of finite rooted trees with regular branching properties. In particular, we show that in the case of constant branching, the eigenvalues are realized as the roots of a family of generalized Fibonacci polynomials and produce a limiting distribution for the eigenvalues as the tree depth goes to infinity. We indicate how these results can be extended to periodic branching patterns and also provide a generalization to higher order simplicial complexes.