The Weak Zero-Divisor Difference Graph of a Finite Commutative Ring

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.