Characterizing and comparing phylogenies from their Laplacian spectrum

Phylogenetic trees are central to many areas of biology, ranging from population genetics and epidemiology to microbiology, ecology, and macroevolution. The ability to summarize properties of trees, compare different trees, and identify distinct modes of division within trees is essential to all these research areas. But despite wide-ranging applications, there currently exists no common, comprehensive framework for such analyses. Here we present a graph-theoretical approach that provides such a framework. We show how to construct the spectral density profiles of phylogenetic trees from their Laplacian graphs. Using ultrametric simulated trees as well as non-ultrametric empirical trees, we demonstrate that the spectral density successfully identifies various properties of the trees and clusters them into meaningful groups. Finally, we illustrate how the eigengap can identify modes of division within a given tree. As phylogenetic data continue to accumulate and to be integrated into various areas of the life sciences, we expect that this spectral graph-theoretical framework to phylogenetics will have powerful and long-lasting applications.