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Depth of Binomial Edge Ideals in terms of Diameter and Vertex Connectivity

A. V. Jayanthan, Rajib Sarkar

Abstract

Let $G$ be a simple connected non-complete graph and $J_G$ be its binomial edge ideal in a polynomial ring $S$. Using certain invariants associated to graphs, say $U(G)$, Banerjee and Núñez-Betancourt gave an upper bound for the depth of $S/J_G$, and Rouzbahani Malayeri, Saeedi Madani and Kiani obtained a lower bound, say $L(G)$. Hibi and Saeedi Madani gave a structural classification of graphs satisfying $L(G)=U(G)$. In this article, we give structural classification of graphs satisfying $L(G)+1=U(G)$. We also compute the depth of $S/J_G$ for all such graphs $G$.

Depth of Binomial Edge Ideals in terms of Diameter and Vertex Connectivity

Abstract

Let be a simple connected non-complete graph and be its binomial edge ideal in a polynomial ring . Using certain invariants associated to graphs, say , Banerjee and Núñez-Betancourt gave an upper bound for the depth of , and Rouzbahani Malayeri, Saeedi Madani and Kiani obtained a lower bound, say . Hibi and Saeedi Madani gave a structural classification of graphs satisfying . In this article, we give structural classification of graphs satisfying . We also compute the depth of for all such graphs .
Paper Structure (5 sections, 17 theorems, 31 equations, 38 figures)

This paper contains 5 sections, 17 theorems, 31 equations, 38 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs with $d(G)+f(G)+1=n+2-\kappa(G)$
  4. Depth
  5. On the lower bound of depths

Key Result

Lemma 2.2

$($oh$)$ Let $G$ be a graph on $V(G)$ and $v\in V(G)$ such that $v$ is an internal vertex. Then $J_G$ can be written as

Figures (38)

  • Figure 1: $H$
  • Figure 2: $u$ is an internal vertex
  • Figure 3: $G_1\in \mathcal{D}_1$
  • Figure 4: $G_2\in \mathcal{D}\setminus \mathcal{D}_1$ with $H$ as an induced subgraph and $\{u_2\}\in \mathcal{C}(G)$
  • Figure 5: $G_3\in \mathcal{D}\setminus \mathcal{D}_1$ with $H$ is not an induced subgraph
  • ...and 33 more figures

Theorems & Definitions (41)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Example 3.2
  • Theorem 3.3
  • proof
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • ...and 31 more