Laplacian Matrices for Extremely Balanced and Unbalanced Phylogenetic Trees

TL;DR

This paper analyzes the Laplacian matrices of fully balanced and unbalanced phylogenetic trees, revealing distinct spectral properties and degeneracies that characterize their topologies.

Contribution

It provides analytical expressions and algorithms for constructing Laplacian matrices of extreme phylogenetic trees, highlighting spectral differences.

Findings

Laplacian matrices of balanced trees show self-similar patterns and eigenvalue degeneracy.

Unbalanced trees lack eigenvalue degeneracy, indicating different spectral signatures.

The largest eigenvalue behavior is characterized analytically and numerically for both topologies.

Abstract

Phylogenetic trees are important tools in the study of evolutionary relationships between species. Measures such as the index of Sackin, Colless, and Total Cophenetic have been extensively used to quantify tree balance, one key property of phylogenies. Recently a new proposal has been introduced, based on the spectrum of the Laplacian matrix associated with the tree. In this work, we calculate the Laplacian matrix analytically for two extreme cases, corresponding to fully balanced and fully unbalanced trees. For maximally balanced trees no closed form for the Laplacian matrix was derived, but we present an algorithm to construct it. We show that Laplacian matrices of fully balanced trees display self-similar patterns that result in highly degenerated eigenvalues. Degeneracy is the main signature of this topology, since it is totally absent in fully unbalanced trees. We also establish some analytical and numerical results about the largest eigenvalue of Laplacian matrices for these topologies.