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Powers of edge ideals of edge-weighted trees

Jiaxin Li, Guangjun Zhu, Shiya Duan

Abstract

This paper gives exact formulas for the regularity of edge ideals of edge-weighted integrally closed trees. In addition, we provide some linear upper bounds on the regularity of powers of such ideals.

Powers of edge ideals of edge-weighted trees

Abstract

This paper gives exact formulas for the regularity of edge ideals of edge-weighted integrally closed trees. In addition, we provide some linear upper bounds on the regularity of powers of such ideals.
Paper Structure (6 sections, 24 theorems, 34 equations, 10 figures)

This paper contains 6 sections, 24 theorems, 34 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notions of simple graphs
  4. Notions from commutative algebra
  5. Regularity of the edge ideal of an edge-weighted integrally closed tree
  6. Regularity of powers of the edge ideal of an edge-weighted integrally closed tree

Key Result

Lemma 2.1

(HT) Let $0 \longrightarrow M \longrightarrow N \longrightarrow P \longrightarrow 0$ be a short exact sequence of finitely generated graded $S$-modules. Then The equality holds if $\operatorname{reg}(P) \neq \operatorname{reg}(M)-1$.

Figures (10)

  • Figure 1: $Caterpillar\ graph\ with\ k=4, d=1, \omega_2\ge 2$
  • Figure : The case $\omega_3\ge 2$
  • Figure : The case $\omega_1\ge 2$
  • Figure : The case $k=4,d=2$
  • Figure : The case $\omega_3\ge 2$
  • ...and 5 more figures

Theorems & Definitions (47)

  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • Definition 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Lemma 3.4
  • ...and 37 more