Perron value and moment of rooted trees

TL;DR

This paper compares the Perron value and the moment of rooted trees, revealing their relationship and introducing new concepts like Perron entropy and neckbottle matrix to analyze spectral and combinatorial weights.

Contribution

It introduces the Perron entropy and neckbottle matrix, and analyzes how these weights relate, providing bounds and new insights into the spectral properties of rooted trees.

Findings

The moment is almost an upper bound for the Perron value.

The ratio of moment to Perron value is unbounded but at most linear in tree size.

Operations on rooted trees affect the Perron value and moment in specific ways.

Abstract

The Perron value $\rho(T)$ of a rooted tree $T$ has a central role in the study of the algebraic connectivity and characteristic set, and it can be considered a weight of spectral nature for $T$. A different, combinatorial weight notion for $T$ - the moment $\mu(T)$ - emerges from the analysis of Kemeny's constant in the context of random walks on graphs. In the present work, we compare these two weight concepts showing that $\mu(T)$ is "almost" an upper bound for $\rho(T)$ and the ratio $\mu(T)/\rho(T)$ is unbounded but at most linear in the order of $T$. To achieve these primary goals, we introduce two new objects associated with $T$ - the Perron entropy and the neckbottle matrix - and we investigate how different operations on the set of rooted trees affect the Perron value and the moment.