The Zero-Divisor Graph Problem

Let S be a commutative semigroup with 0 element. A nonzero element a in S is called a zero-divisor if there exists a nonzero element b in S so that ab=0. The zero-divisor graph Г(S) is the simple graph whose vertices are given by the nonzero zero-divisors of S where two distinct vertices, a and b, are adjacent in case ab=0 in S. We will begin with a discussion of the history of this construction and foundational results, including definitions of terms and examples. We conclude with some very recent results, other zero divisor graph constructions, and some open questions.