23/04/2026 - 15:00 - CORE C.035 

Robert Ronco 

(National Research Council of Italy)

will give a presntation on 

On the Eigenvalues of Path-Length Density Matrices of Unrooted Binary Trees

Abstract: 

An Unrooted Binary Tree (UBT) is a tree with n ≥ 3 leaves, n − 2 internal nodes of degree three, and 2n − 3 edges. In several contexts arising from information theory and the life sciences, a UBT T can be conveniently encoded by means of a Path-Length Density Matrix (PLDM), i.e., a symmetric matrix of order nxn having null diagonal entries and off-diagonal entries equal to 2^{-tau_{ij}} , where tau_{ij} denotes the number of edges along the unique path in T between leaves i and j. In this work, we investigate the extent to which the spectrum of a PLDM reflects the topology of the underlying UBT. We prove that all eigenvalues of a PLDM of a UBT T lie in the interval [-1/4,1/2] and that specific families of rational eigenvalues can be associated with well-identified subtrees of T obtained through a recursive merging of characteristic subtopologies. For extremal tree structures, we further derive

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