The ancestral matrix of a rooted tree

2018/09/10 by Eric Ould Dadah Andriantiana, Andriantiana, Eric O. D., Kenneth Dadedzi +3
Chemistry · Mathematics · #05C05 #05C50 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Molecular spectroscopy and chirality

paper · pdf · doi:10.48550/arxiv.1809.03364

openalex publication_date 2018/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a rooted tree T with leaves v₁,v₂,…,v_n, we define the ancestral matrix C(T) of T to be the n × n matrix for which the entry in the i-th row, j-th column is the level (distance from the root) of the first common ancestor of v_i and v_j. We study properties of this matrix, in particular regarding its spectrum: we obtain several upper and lower bounds for the eigenvalues in terms of other tree parameters. We also find a combinatorial interpretation for the coefficients of the characteristic polynomial of C(T), and show that for d-ary trees, a specific value of the characteristic polynomial is independent of the precise shape of the tree.

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