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On the closed neighborhood ideal of the square of the path graph

Anda Olteanu, Oana Olteanu

Abstract

We consider the closed neighborhood ideal of square of the path graph and study its invariants. We compute the height, the projective dimension and the Castelnuovo--Mumford regularity. We prove that these ideals are sequentially Cohen--Macaulay and characterize when they are Cohen--Macaulay.

On the closed neighborhood ideal of the square of the path graph

Abstract

We consider the closed neighborhood ideal of square of the path graph and study its invariants. We compute the height, the projective dimension and the Castelnuovo--Mumford regularity. We prove that these ideals are sequentially Cohen--Macaulay and characterize when they are Cohen--Macaulay.
Paper Structure (2 sections, 23 theorems, 85 equations, 1 figure)

This paper contains 2 sections, 23 theorems, 85 equations, 1 figure.

Table of Contents

  1. Invariants of the closed neighborhood ideals of $P_n^2$
  2. The closed neighborhood ideal of $P_n^2$ and simplicial complexes

Key Result

Proposition 1.1

Let $3\leq n\leq 6$ be an integer and $NI(P_n^2)$ be the closed neighborhood ideal. The following hold:

Figures (1)

  • Figure 1: The graphs $P_7$ and $P_7^2$

Theorems & Definitions (39)

  • Proposition 1.1
  • proof
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • Corollary 1.4
  • proof
  • Remark 1.5
  • Lemma 1.6
  • Lemma 1.7
  • ...and 29 more