Let [Formula: see text] be a commutative ring with identity, and let [Formula: see text] be the set of zero-divisors of [Formula: see text]. The essential graph of [Formula: see text] is defined as the graph [Formula: see text] with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if ann[Formula: see text] is an essential ideal. It is proved that [Formula: see text] is connected with diameter at most three and with girth at most four, if [Formula: see text] contains a cycle. Furthermore, rings with complete or star essential graphs are characterized. Also, we study the affinity between essential graph and zero-divisor graph that is associated with a ring. Finally, we show that the essential graph associated with an Artinian ring is weakly perfect, i.e. its vertex chromatic number equals its clique number.

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Abstract. Let R be a commutative ring with unity. The annihilator ideal graph of R, denoted by Γ Ann (R), is a graph whose vertices are all non-trivial ideals of R and two distinct vertices I and J are adjacent if and only if I ∩ Ann(J) = {0} or J ∩ Ann(I) = {0}. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose annihilator ideal graphs are totally disconnected. Also, we study diameter, girth, clique number and chromatic number of this graph. Moreover, we study some relations between annihilator ideal graph and zero-divisor graph associated with R. Among other results, it is proved that for a Noetherian ring R if Γ Ann (R) is triangle free, then R is Gorenstein.

Let [Formula: see text] be a commutative ring with identity, and let [Formula: see text] be the set of zero-divisors of [Formula: see text]. The annihilator graph of [Formula: see text] is defined as the graph AG[Formula: see text] with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if ann[Formula: see text]. 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.