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 an associative ring with non-zero identity and [Formula: see text] a non-zero unital left [Formula: see text]-module. The cozero-divisor graph of [Formula: see text], denoted by [Formula: see text], is a graph with vertices [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text] and [Formula: see text]. In this paper, we study some connections between the graph-theoretic properties of [Formula: see text] and algebraic-theoretic properties of [Formula: see text] and [Formula: see text]. Also, we study girth, independence number, clique number and planarity of this graph.

Let R be an associative ring with 1 = 0 which is not a domain. Let A(R) * = {I ⊆ R | I is a left or right ideal of R and l.ann(I) ∪ r.ann(I) = 0} \ {0}. The total graph of annihilating one-sided ideals of R, denoted by Ω(R), is a graph with the vertex set A(R) * and two distinct vertices I and J are adjacent if l.ann(I + J) ∪ r.ann(I + J) = 0. In this paper, we study the relations between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose graphs are disconnected. Also, we study diameter, girth, independence number, domination number and planarity of this graph.

Let [Formula: see text] be an associative ring with identity. A ring [Formula: see text] is called reversible if [Formula: see text], then [Formula: see text] for [Formula: see text]. The total graph of unfaithful submodules of a module [Formula: see text] over a reversible ring [Formula: see text], denoted by [Formula: see text], is a graph whose vertices are all nonzero unfaithful submodules of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text] is unfaithful. In this paper, we determine the diameter and girth of [Formula: see text]. Also, we study some combinatorial properties of [Formula: see text] such as independence number and clique number. Moreover, we study the case that the degree of a vertex of [Formula: see text] is finite.