Castelnuovo-Mumford Regularity and Combinatorial Invariants of Trees

TL;DR

This work derives explicit combinatorial bounds on the Castelnuovo-Mumford regularity of edge ideals of trees in terms of order $n$, diameter $d$, and pendant vertices $p$, and extends these results to multi-whiskered trees. Leveraging a blend of graph-theoretic and homological techniques, it shows that for trees the regularity lies between $\left\lfloor\frac{n-p+d+5}{6}\right\rfloor$ and $\min\{n-p,\,\left\lfloor\frac{2n-p}{3}\right\rfloor\}$, and that reg$(S/I(T))=\operatorname{im}(T)$ due to chordality, yielding corresponding bounds on the induced matching number. For multi-whiskered trees $T_{\mathbf{a}}$, the authors establish an improved upper bound $\operatorname{reg}(S/I(T_{\mathbf{a}})) \le \min\left\{\left\lceil\frac{2n-d-1}{2}\right\rceil,\,\left\lfloor\frac{2n+p-2}{3}\right\rfloor\right\}$, and observe that reg$(S/I(T_{\mathbf{a}}))=\alpha(T)$ (via induced matching/independence correspondences) in this class. Consequently, the paper connects algebraic complexity to concrete graph invariants, providing both bounds and structural insight, and outlining directions to characterize equality cases and broaden the interplay between combinatorics and homological algebra.

Abstract

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 \( α(T) \).

Figures (1)

• Figure 1: An example of a tree where every support vertex has degree 2.

Theorems & Definitions (18)

• Lemma 2.1: dao
• Lemma 2.2: HOA
• Lemma 2.3: hanvan
• Lemma 2.4: Mujahid
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Corollary 3.4