  In this paper, we continue the program initiated by I. Beck's now classical
paper concerning zero-divisor graphs of commutative rings. After the success of
much research regarding zero-divisor graphs, many authors have turned their
attention to studying divisor graphs of non-zero elements in the ring, the so
called irreducible divisor graph. In this paper, we construct several different
associated irreducible divisor graphs of a commutative ring with unity using
various choices for the definition of irreducible and atomic in the literature.
We continue pursuing the program of exploiting the interaction between
algebraic structures and associated graphs to further our understanding of both
objects. Factorization in rings with zero-divisors is considerably more
complicated than integral domains; however, we find that many of the same
techniques can be extended to rings with zero-divisors. This allows us to not
only find graph theoretic characterizations of many of the finite factorization
properties that commutative rings may possess, but also understand graph
theoretic properties of graphs associated with certain commutative rings
satisfying nice factorization properties.
