AI 中文总结
研究有限交换环的弱零因子差图\(\DR\)，通过引入该图并为有限约化环发展结构理论，确定其在不同条件下的性质，如连通性、完美性等，还给出相关参数的封闭形式及重构定理，指出约化环假设不可轻易去掉。
AI 中文摘要
对于有限交换环\(R\)，令\(\GR\)表示其零因子图，\(\WGR\)表示其弱零因子图，后者包含前者作为生成子图。我们引入“弱零因子差图”\(\DR := \WGR - \GR\)，并为有限约化环\(R\cong\mathbb F_{q_1}\times\cdots\times\mathbb F_{q_t}\)发展了完整的结构理论。我们表明\(\DR\)在一个包含巴达维的零化子图的三阶段细化中严格位于\(\GR\)和\(\WGR\)之间，并证明不同的支撑类\(X_A\)、\(X_B\)在\(\DR\)中完全相连当且仅当\(A\cap B\neq\emptyset\)，这是后续每个结果的精确准则。因此，对于\(t\leq2\)，\(\DR\)无边，而对于每个\(t\geq3\)，\(\DR\)连通，直径为\(2\)，围长为\(3\)，与域的阶无关。我们建立了一个完整的完美性二分法——\(\DR\)恰好当\(t\in\{3,4\}\)时是完美的，\(t = 4\)时有一个基本的组合证明，对于\(t\geq5\)从不完美——并确定了\(t = 3\)和\(t = 4\)时的团数；对于一般的\(t\)，我们给出了两个不可比较的下界和两个上界，在\(t = 3\)时尖锐但不超过此范围，同时给出一个压缩论证表明一个极值族总是可以取为平移的，但仅靠这一点并不能解决问题。一个重构定理表明\(\DR\)本质上恢复了域阶的多重集，所以\(\DR\cong\mathcal D(S)\)迫使\(R\cong S\)。我们还给出了对于每个\(t\geq3\)都有效的关于度序列和最小度、支配数以及独立数和顶点覆盖数的封闭形式。最后，我们通过有限链环的赋值结构简要指出为什么不能简单地去掉约化环假设。
英文摘要
For a finite commutative ring $R$, let $\GR$ denote its zero-divisor graph and $\WGR$ its weakly zero-divisor graph, the latter containing the former as a spanning subgraph. We introduce the \emph{weak zero-divisor difference graph} $\DR:=\WGR-\GR$ and develop a complete structural theory for finite reduced rings $R\cong\mathbb F_{q_1}\times\cdots\times\mathbb F_{q_t}$. We show $\DR$ sits strictly between $\GR$ and $\WGR$ in a three-stage refinement that also contains Badawi's annihilator graph, and prove that distinct support classes $X_A,X_B$ are completely joined in $\DR$ if and only if $A\cap B\ne\emptyset$ -- an exact criterion underlying every result that follows. Consequently $\DR$ is edgeless for $t\le2$ but connected with diameter $2$ and girth $3$ for every $t\ge3$, independently of the field orders. We establish a complete perfectness dichotomy -- $\DR$ is perfect exactly when $t\in\{3,4\}$, with an elementary combinatorial proof at $t=4$, and never perfect for $t\ge5$ -- and determine its clique number exactly at $t=3$ and $t=4$; for general $t$ we give two incomparable lower bounds and two upper bounds, sharp at $t=3$ but not beyond, together with a compression argument showing an extremal family may always be taken shifted without this alone resolving the problem. A reconstruction theorem shows $\DR$ recovers the multiset of field orders intrinsically, so $\DR\cong\mathcal D(S)$ forces $R\cong S$. We further give closed forms, valid for every $t\ge3$, for the degree sequence and minimum degree, the domination number, and the independence and vertex cover numbers. Finally, we briefly indicate, via the valuation structure of finite chain rings, why the reduced-ring hypothesis cannot simply be dropped.
Comments14 pages
