Some Properties of Path and Closed Neighborhood Ideals

Abstract

In this paper, we investigate algebraic properties of two classes of monomial ideals associated with graphs, namely path ideals and closed neighborhood ideals. Our primary focus is on the strong persistence property and several homological invariants, including depth and the $v$-number. We first prove that the path ideal of length $n-1$ associated with a path graph on $n\geq4$ vertices attached to an arbitrary graph satisfies the strong persistence property. We then study closed neighborhood ideals and establish the strong persistence property for graphs containing a leaf vertex. Furthermore, we identify a broader class of graphs satisfying this property, namely those whose vertex set admits a decomposition $V(G)=C\sqcup\{v\}\sqcup X,$ where $C$ is a nonempty clique, every vertex of $C$ is adjacent to each vertex of $X\cup\{v\}$, and the vertex $v$ has no neighbors in $X$. In addition, we investigate algebraic invariants of ideals associated with Helm graphs. Specifically, we compute the depth of the edge ideal and determine the Krull dimension, the depth, and the $v$-number of the closed neighborhood ideal of the Helm graph. These results contribute to a deeper understanding of the interplay between graph-theoretic structures and the algebraic properties of their associated monomial ideals.