$\textbf{k}$-neighborhood ideals of graphs

In this paper, we introduce and investigate the $\textbf{k}$-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 $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$. For a vector $\textbf{k}=(k_