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 \( \alpha(T) \).