The ancestral matrix of a rooted tree

TL;DR

This paper introduces the ancestral matrix of a rooted tree, explores its spectral properties, and provides bounds and combinatorial interpretations, revealing invariances in specific tree classes.

Contribution

It defines the ancestral matrix for rooted trees and analyzes its spectrum, offering bounds, combinatorial insights, and invariance results for certain tree structures.

Findings

Eigenvalue bounds in terms of tree parameters

Combinatorial interpretation of characteristic polynomial coefficients

Invariance of specific polynomial values in d-ary trees

Abstract

Given a rooted tree $T$ with leaves $v_1,v_2,\ldots,v_n$, we define the ancestral matrix $C(T)$ of $T$ to be the $n \times n$ 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.