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 . 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.