发表机构
Manipal Institute of Technology, Manipal Academy of Higher Education; School of Mathematics and Statistics, Shandong University of Technology; Centre for Research Impact & Outcome, Chitkara University Institute of Engineering and TechnologyChitkara University（马尼帕尔高等教育局马尼帕尔技术学院; 山东理工大学数学与统计学院; 奇塔克拉大学工程与技术学院研究影响与成果中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究零因子图的重心和匹配细分的度量基，给出匹配细分任意大小的精确维度公式，表明小匹配细分可降低定位成本。
AI 中文摘要
在本文中，我们研究了$\nmathbb Z_{pq}$的零因子图的重心和部分匹配细分的度量基及相关度量性质，其中$p$和$q$是不同的奇素数且$q>p$。我们首先回顾零因子图自然划分为两个素数类，然后详细刻画当$q\geq 2p-1$时，$BS(\Gamma(\mathbb Z_{pq}))$中构成度量基的那些子集。证明通过区分闭邻域、细分顶点行和禁止孪生配置的作用来展开。然后我们研究通过对$\Gamma(\mathbb Z_{pq})$的选定边进行细分而获得的$M$-细分图。除了对$p-3$和$p-2$条边的细分的下界之外，我们证明了任意大小$r$（$0\leq r\leq p-2$）的匹配细分的精确公式，即$\dim(G_r)=p+q-r-4$。文中包含若干推论，以说明小的匹配细分如何降低原始零因子网络的定位成本。
英文摘要
In this paper, we study metric bases and related metric properties for barycentric and partial matching subdivisions of the zero-divisor graph of $\mathbb Z_{pq}$, where $p$ and $q$ are distinct odd primes with $q>p$. We first recall the natural partition of the zero-divisor graph into the two prime classes and then give a detailed characterization of those subsets of $BS(Γ(\mathbb Z_{pq}))$ that form metric bases when $q\geq 2p-1$. The proof is expanded by separating the role of closed neighborhoods, rows of subdivision vertices, and forbidden twin configurations. We then investigate $M$-subdivision graphs obtained by subdividing selected edges of $Γ(\mathbb Z_{pq})$. In addition to the lower bounds for subdivisions of $p-3$ and $p-2$ edges, we prove an exact formula for matching subdivisions of arbitrary size $r$, $0\leq r\leq p-2$, namely $\dim(G_r)=p+q-r-4$. Several consequences are included to illustrate how a small matching subdivision can reduce the localization cost of the original zero-divisor network.
