Perron value and moment of rooted trees

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 Kemenys 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.