AI 中文总结
研究有限交换正则环的广义冯·诺依曼逆图\(\Gamma'_{Reg}(R)\)，通过推导其连通性、无圈性、平面性等图论性质与\(R\)代数结构关系的条件，确定其呈现特定图类的情形，提供构造算法并讨论与包含理想图的联系。
AI 中文摘要
设\(R\)为含幺环。\(R\)的广义冯·诺依曼逆图\(\Gamma_{Reg}(R)\)定义为顶点集为\(Reg(R)\)的图，其中两个不同顶点\(a,b\in R\)相邻当且仅当\(aba = a\)或\(bab = b\)。本文考虑通过将顶点集限制为\(Reg(R)\setminus\{0_R\}\)得到的约化图\(\Gamma'_{Reg}(R)\)，使得\(\Gamma_{Reg}(R)\cong K_1+\Gamma'_{Reg}(R)\)。研究了有限交换冯·诺依曼正则环的\(\Gamma'_{Reg}(R)\)的结构，建立了若干关于其图论性质与\(R\)代数结构关系的结果。推导了刻画连通性、无圈性和平面性的条件，研究了顶点度、围长和悬挂顶点等结构特征及其代数含义。还确定了\(\Gamma'_{Reg}(R)\)呈现特定图类的情形，提供了构造\(\Gamma'_{Reg}(R)\)的显式算法，并讨论了与\(R\)的包含理想图的联系。
英文摘要
Let $R$ be a ring with identity. The generalized von Neumann inverse graph of $R$, denoted by $Γ_{Reg}(R)$, is defined as the graph whose vertex set is $Reg(R)$, where two distinct vertices $a,b \in R$ are adjacent if and only if $aba=a$ or $bab=b$. In this work, we consider the reduced graph $Γ'_{Reg}(R)$ obtained by restricting the vertex set to $Reg(R)\setminus{0_R}$, so that $Γ_{Reg}(R) \cong K_1 + Γ'_{Reg}(R)$, allowing the analysis to focus on its nontrivial structure. We investigate the structure of $Γ'_{Reg}(R)$ for finite commutative von Neumann regular rings and establish several results describing its graph-theoretic properties in relation to the algebraic structure of $R$. In particular, we derive conditions that characterize connectivity, acyclicity, and planarity, and examine structural features such as vertex degrees, girth, and the existence of pendant vertices, along with their algebraic implications. We also identify circumstances under which $Γ'_{Reg}(R)$ exhibits specific graph classes, including paths, cycles, and wheels, as well as the presence of certain induced subgraphs. Furthermore, an explicit algorithm is provided to construct $Γ'_{Reg}(R)$, and connections with the inclusion ideal graph of $R$ are discussed, offering additional insight into the interplay between ring-theoretic properties and graph structures.
Comments17 pages
