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Betti numbers of normal edge rings (\bf{I})

Zexin Wang, Dancheng Lu

TL;DR

This work develops an induced-subgraph approach to compute the multi-graded Betti numbers of normal edge rings for graphs satisfying the odd-cycle condition, connecting Betti data to the canonical module and square-free initial ideals. It provides explicit computations for two families: two-ear graphs $\mathbf{G}_m$ and compact graphs, distinguishing type-1/2 (top-Betti) behavior from type-3 (second-top Betti) phenomena. The method combines Bruns–Gubeladze duality with induced-subgraph decompositions and yields exact Betti spectra by analyzing top- and second-top Betti subgraphs and their induced-subgraph data. These results not only give complete Betti descriptions for the studied classes but also suggest broader applicability of the approach to other normal edge rings, with the Betti data tied to square-free initial ideals.

Abstract

We introduce a novel approach named the {\it induced-subgraph approach} for investigating the Betti numbers of normal edge rings. Utilizing this approach, we compute all the multi-graded Betti numbers of the edge rings associated with two-ear graphs (Definition 4.1) and compact graphs (Definition 5.1). In particular, we show that for two-ear graphs and compact graphs of type 1 or 2, their multi-graded Betti numbers are always equal to the top multi-graded Betti numbers of some of their induced subgraphs. In contrast, some of the multi-graded Betti numbers of compact graphs of type 3 are not the top multi-graded Betti numbers of any of their induced subgraph. We speculate that our approach can be applicable to many other normal edge rings.

Betti numbers of normal edge rings (\bf{I})

TL;DR

This work develops an induced-subgraph approach to compute the multi-graded Betti numbers of normal edge rings for graphs satisfying the odd-cycle condition, connecting Betti data to the canonical module and square-free initial ideals. It provides explicit computations for two families: two-ear graphs and compact graphs, distinguishing type-1/2 (top-Betti) behavior from type-3 (second-top Betti) phenomena. The method combines Bruns–Gubeladze duality with induced-subgraph decompositions and yields exact Betti spectra by analyzing top- and second-top Betti subgraphs and their induced-subgraph data. These results not only give complete Betti descriptions for the studied classes but also suggest broader applicability of the approach to other normal edge rings, with the Betti data tied to square-free initial ideals.

Abstract

We introduce a novel approach named the {\it induced-subgraph approach} for investigating the Betti numbers of normal edge rings. Utilizing this approach, we compute all the multi-graded Betti numbers of the edge rings associated with two-ear graphs (Definition 4.1) and compact graphs (Definition 5.1). In particular, we show that for two-ear graphs and compact graphs of type 1 or 2, their multi-graded Betti numbers are always equal to the top multi-graded Betti numbers of some of their induced subgraphs. In contrast, some of the multi-graded Betti numbers of compact graphs of type 3 are not the top multi-graded Betti numbers of any of their induced subgraph. We speculate that our approach can be applicable to many other normal edge rings.
Paper Structure (17 sections, 37 theorems, 114 equations, 2 figures)

This paper contains 17 sections, 37 theorems, 114 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Betti numbers and Monomial ideals with regular quotients
  4. Betti numbers of monomial ideals in a monomial algebras
  5. Simple graphs and minimal cycles
  6. Conventions and notions
  7. Edge cones and canonical modules
  8. top-Betti graphs and second-Betti graphs
  9. Betti numbers of $\mathbf{G}_m$
  10. The composition of $\mathcal{N}_{\mathbf{G}_m}$
  11. The composition of $\mathcal{N}_{\mathbf{P}_m}$.
  12. The composition of $\mathcal{N}_{\mathbf{K}_{2,d}}$.
  13. Compact graphs revisited
  14. Type one
  15. type two
  16. ...and 2 more sections

Key Result

Lemma 1.1

(Lemma start) Let $H$ be an induced subgraph of a simple graph $G$. Then for $h\in \mathbb{Z}_{\geq0}^{V(G)}$ with $\mathrm{supp}(h)\subseteq V(H)$, we have

Figures (2)

  • Figure 1: The two-ear graph $\mathbf{G}_{m}$
  • Figure 2: The graph $\mathbf{C}_{(1):(1):(1,1)}$

Theorems & Definitions (62)

  • Lemma 1.1
  • Lemma 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • ...and 52 more