An Unrooted Binary Tree (UBT) is a connected acyclic graph with internal vertices of degree 3. The Path-Length Ma-trix (PLM) of a UBT with n leaves encodes, for each pair of leaves i and j, the number of edges in the unique path between i and j. PLMs arise naturally in clustering and computational biolo-gy. A key example is the Balanced Minimum Evolution Problem (BMEP), an NP-hard optimization problem on PLMs for phylogenetic inference. From a topological standpoint, UBTs correspond to the facets of the simplicial complex ob-tained by intersecting the tropical Grassmannian with a unit hypersphere. In this talk, we deal with UBTs from a purely combinatorial standpoint. Building upon prior work, we show that one can characterize the PLMs of UBTs for n ≤ 11. We also describe spectral properties of PLMs by tracing connections between some tree substructures and specific eigenvalues of the associated PLMs.

Combinatorial Optimization, Unrooted Binary Trees, Balanced Minimum Evolution