Smarandache Vertices of the Graphs Associated to the Commutative Rings

Let R be a commutative ring with identity 1 ≠ 0. A nonzero element a in R is said to be a Smarandache zero-divisor if there exist three different nonzero elements x, y, and b (≠ a) in R such that ax = ab = by = 0, but xy ≠ 0. We will generalize this notion to the Smarandache vertex of an arbitrary simple graph and characterize the Smarandache zero-divisors of commutative rings (resp. with respect to an ideal) via their associated zero-divisor graphs. We illustrate them with examples and prove some interesting results about them.