Mathematics Commons^™

Generalizations And Variations Of The Zero-Divisor Graph, Grace Elizabeth Mcclurkin

Doctoral Dissertations

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 …