Spectral structure of path-length density matrices of unrooted binary trees

The Path-Length Density Matrix (PLDM) of an Unrooted Binary Tree (UBT) is the real symmetric matrix with zero diagonal and off-diagonal entries 2−τij , where τij denotes the number of edges on the unique path between leaves i and j. Here, UBTs are trees with n ≥ 3 leaves and all internal vertices of degree 3, and their PLDMs arise naturally in computational phylogenetics, network design, and information theory. In this article, we study PLDMs of UBTs and investigate how their spectra reflect the structure of the underlying trees. We prove that the smallest eigenvalue of any PLDM of a UBT is−1/4, with multiplicity equal to the number of leaf pairs sharing a common parent. We also show that specific eigenvalues characterize the presence of particular subtree configurations. For UBTs of minimum diameter, we derive closed-form expressions for the eigenvalues; for UBTs of maximum diameter, we relate the spectrum to an auxiliary Kac–Murdock–Szeg˝o-type matrix and obtain asymptotic results for the empirical spectral distribution and the edge spectral densities. These results show that the spectrum of a PLDM encodes explicit structural information about the underlying tree and suggest that spectral methods may be useful for optimization over this matrix class.