Coloring of the annihilator graph of a commutative ring

Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The annihilator graph of R is defined as the graph AG(R) with the vertex set Z(R)* = Z(R)\{0}, and two distinct vertices x and y are adjacent if and only if ann R(xy) not equal ann R(x) boolean OR ann R(y). In this paper, we study annihilator graphs of rings with equal clique number and chromatic number. For some classes of rings, we give an explicit formula for the clique number of annihilator graphs. Among other results, bipartite annihilator graphs of rings are characterized. Furthermore, some results on annihilator graphs with finite clique number are given.