Perron value and moment of rooted trees

Abstract

The Perron value ρ(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 μ(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 μ(T) is “almost” an upper bound for ρ(T) and the ratio μ(T)/ρ(T) is unbounded but at most linear in the order of T. Furthermore, 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.