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The $\v$-number of generalized binomial edge ideals of some graphs

Yi-Huang Shen, Guangjun Zhu

Abstract

Let \(G\) be a finite connected simple graph, and let \(\calJ_{K_m,G}\) denote its generalized binomial edge ideal. By investigating the colon ideals of \(\calJ_{K_m,G}\), we derive a formula for the local \(\v\)-number of \(\calJ_{K_m,G}\) with respect to the empty cut set. Furthermore, we classify graphs for which this generalized binomial edge ideal has \(\v\)-numbers 1 or 2. When \(G\) is a connected closed graph, we compute the local \(\v\)-number of \(\calJ_{K_2,G}\) by generalizing the work of Dey et al. Additionally, under the condition that \(G\) is Cohen--Macaulay, we derive formulas for the \(\v\)-number of \(\calJ_{K_m,G}\) and \(\calJ_{K_2,G}^k\), and show that the \(\v\)-number of \(\calJ_{K_2,G}^k\) is a linear function of \(k\).

The $\v$-number of generalized binomial edge ideals of some graphs

Abstract

Let be a finite connected simple graph, and let denote its generalized binomial edge ideal. By investigating the colon ideals of , we derive a formula for the local -number of with respect to the empty cut set. Furthermore, we classify graphs for which this generalized binomial edge ideal has -numbers 1 or 2. When is a connected closed graph, we compute the local -number of by generalizing the work of Dey et al. Additionally, under the condition that is Cohen--Macaulay, we derive formulas for the -number of and , and show that the -number of is a linear function of .

Paper Structure

This paper contains 8 sections, 23 theorems, 71 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. $\operatorname{v}$-number of generalized binomial edge ideals
  4. Local $\operatorname{v}$-numbers of binomial edge ideals of connected closed graphs
  5. $\operatorname{v}$-number of generalized binomial edge ideals of Cohen--Macaulay closed graphs
  6. Local $\operatorname{v}$-number
  7. Realization of $\operatorname{v}(\mathcal{J}_{K_m,G})$
  8. $\operatorname{v}$-number of powers of binomial edge ideals of connected Cohen--Macaulay closed graphs

Key Result

Lemma 3.1

Let $G$ be a simple graph on the vertex set $[n]$. Then for any $i \in [m]$ and $j \in [n]$, we have $(\mathcal{J}_{K_m,G} : x_{i,j}) = \mathcal{J}_{K_m,G_j}$, where $G_j$ is the completion graph of $G$ with respect to the vertex $j$.

Figures (4)

  • Figure 1: The associated graph $L(T)$
  • Figure 2: A Cohen--Macaulay closed graph
  • Figure 3: The associated graph $L(T)$
  • Figure 4: The associated graph $L(T)$ for the optimal cut set $T$

Theorems & Definitions (53)

  • Definition 2.1
  • Definition 2.2: MR2863365 and MR3290687
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 43 more