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Phylogenetic trees are important tools in the study of evolutionaryrelationships between species. Measures such as the index of Sackin, Colless,and Total Cophenetic have been extensively used to quantify tree balance, onekey property of phylogenies. Recently a new proposal has been introduced, basedon 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 maximallybalanced trees no closed form for the Laplacian matrix was derived, but wepresent an algorithm to construct it. We show that Laplacian matrices of fullybalanced trees display self-similar patterns that result in highly degeneratedeigenvalues. Degeneracy is the main signature of this topology, since it istotally absent in fully unbalanced trees. We also establish some analytical andnumerical results about the largest eigenvalue of Laplacian matrices for thesetopologies.

Laplacian Matrices for Extremely Balanced and Unbalanced Phylogenetic Trees