Castelnuovo-Mumford Regularity and Combinatorial Invariants of Trees

This work establishes combinatorial bounds on the Castelnuovo-Mumford regularity of edge ideals for trees and their multi-whiskered variants. For a tree $T$, we give bounds for the Castelnuovo-Mumford regularity of $I(T)$ in terms of the order, diameter, and number of pendant vertices. Furthermore, we present an upper bound for multi-whiskered trees $T_{\mathbf{a}}$, demonstrating that the Castelnuovo-Mumford regularity of $I(T_{\mathbf{a}})$ is bounded by the same invariants of the underlying tree $T$. A principal consequence of this work is the derivation of corresponding inequalities for two key combinatorial invariants of $T$, namely the induced matching number $\operatorname{im}(T)$ and the independence number $\alpha(T)$.