Generalizations and Variations of the Zero-Divisor Graph

We explore generalizations and variations of the zero-divisor graph on commutative rings with identity. A zero-divisor graph is a graph whose vertex set is the nonzero zero-divisors of a ring, wherein two distinct vertices are adjacent if their product is zero. Variations of the zero-divisor graph are created by changing the vertex set, the edge condition, or both. The annihilator graph and the extended zero-divisor graph are both variations that change the edge condition, whereas the compressed graph and ideal-based graph change the vertex set. By combining these concepts, we define and investigate graphs where both the vertex set and edge condition are changed such as compressed annihilator graphs and ideal-based extended zero-divisor graphs.

We then generalize these variations by defining congruence-based versions of the annihilator graph and extended zero-divisor graph. Many of the previous graphs are shown to fit this more general framework. The congruence-based version of a graph has a vertex set equal to the equivalence classes of some multiplicative congruence relation, with the edge condition determined separately. We prove several foundational properties for these and additionally look at the relationships between and within the families of congruence-based graphs.