AI 中文总结
本文研究路径理想与闭邻域理想的代数性质，证明了特定图类的这两类理想满足强持久性性质，计算了Helm图相关理想的代数不变量，深化了图结构与单项式理想的关联认识。
AI 中文摘要
本文研究与图关联的两类单项式理想——路径理想和闭邻域理想的代数性质，主要关注强持久性性质及深度、v数等同调不变量。首先证明，n≥4个顶点的路径图（附加于任意图）对应的长度为n-1的路径理想满足强持久性性质；接着研究闭邻域理想，证明含叶顶点的图的闭邻域理想满足强持久性性质，还确定了更广泛的一类满足该性质的图：其顶点集可分解为V(G)=C⊔{v}⊔X，其中C为非空团，C的每个顶点与X∪{v}的每个顶点相邻，且v在X中无邻居。此外，研究与Helm图关联的理想的代数不变量，具体计算了其边理想的深度，并确定了Helm图闭邻域理想的Krull维数、深度及v数。这些结果有助于更深入理解图论结构与其关联单项式理想代数性质之间的相互作用。
英文摘要
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.
Comments16 pages, 1 figure
