  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.
