k-neighborhood ideals of graphs
Abstract
In this paper, we introduce and investigate the k-neighborhood ideal of a graph, a natural generalization of the closed neighborhood ideal. Let G be a simple graph on the vertex set [n], and let S=K[x1,…,xn] be the polynomial ring over a field K. For a vector k=(k1,…,kn)∈ ℕn satisfying 1≤ ki≤ degG(i)+1 for all i, the k-neighborhood ideal of G is defined as the squarefree monomial ideal NIk(G)=∑i=1n (xW: W⊆ NG[i], |W|=ki) of S, where xW=∏i∈ W xi. We study homological invariants and properties of NIk(G) focusing on its Castelnuovo-Mumford regularity, projective dimension and Cohen-Macaulayness. Special attention is devoted to the case where the vector k is the degree-vector of the graph, i.e., ki=degG(i) for all vertices i, and to the case where NIk(G) coincides with the edge ideal of a graph. In these settings, we provide combinatorial characterizations and bounds for the regularity and projective dimension of NIk(G) for several classes of graphs, and further investigate the Cohen-Macaulay property of these ideals.
Related
- The v-number of t-closed neighborhood ideals
- On the closed neighborhood ideal of the square of broom and double broom graphs
- Stabilization index of V-number of powers of edge ideals of graphs
- v-numbers of powers of edge ideals of cubic circulant graphs
- Edge ideals of gapfree graphs with matching number $3$
BibTeX
@misc{moradi2026kneighborhood,
title = {{$\textbfk$-neighborhood ideals of graphs}},
author = {Somayeh Moradi and Leila Sharifan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.08159},
eprint = {2606.08159},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.08159},
}
RIS
TY - UNPB
TI - k-neighborhood ideals of graphs
AU - Somayeh Moradi
AU - Leila Sharifan
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2606.08159
UR - https://arxiv.org/abs/2606.08159
AB - In this paper, we introduce and investigate the k-neighborhood ideal of a graph, a natural generalization of the closed neighborhood ideal. Let G be a simple graph on the vertex set [n], and let S=K[x_1,…,x_n] be the polynomial ring over a field K. For a vector k=(k_1,…,k_n)∈ ℕ^n satisfying 1≤ k_i≤ deg_G(i)+1 for all i, the k-neighborhood ideal of G is defined as the squarefree monomial ideal NI_k(G)=∑_i=1^n (x_W: W⊆ N_G[i], |W|=k_i) of S, where x_W=∏_i∈ W x_i. We study homological invariants and properties of NI_k(G) focusing on its Castelnuovo-Mumford regularity, projective dimension and Cohen-Macaulayness. Special attention is devoted to the case where the vector k is the degree-vector of the graph, i.e., k_i=deg_G(i) for all vertices i, and to the case where NI_k(G) coincides with the edge ideal of a graph. In these settings, we provide combinatorial characterizations and bounds for the regularity and projective dimension of NI_k(G) for several classes of graphs, and further investigate the Cohen-Macaulay property of these ideals.
ER -