GENERALIZATIONS OF THE ZERO-DIVISOR GRAPH

Let R R be a commutative ring with 1 ≠ 0 1≠0 and Z ( R ) Z(R) its set of zero-divisors. The zero-divisor graph of R R is the (simple) graph Γ ( R ) Γ(R) with vertices Z ( R ) ∖ { 0 } Z(R)∖{0}, and distinct vertices x x and y y are adjacent if and only if x y = 0 xy=0. In this paper, we consider generalizations of Γ ( R ) Γ(R) by modifying the vertices or adjacency relations of Γ ( R ) Γ(R). In particular, we study the extended zero-divisor graph ¯¯¯ Γ ( R ) ​Γ ​ ​​ (R), the annihilator graph A G ( R ) AG(R), and their analogs for ideal-based and congruence-based graphs.