Crosscap two of class of graphs from commutative rings

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of all nonzero zero-divisors of [Formula: see text]. The annihilator graph of commutative ring [Formula: see text] is the simple undirected graph [Formula: see text] with vertices [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [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 [Formula: see text] is an essential ideal. In this paper, we classify all finite commutative rings with identity whose annihilator graph and essential graph have crosscap two.

When the annihilator graph of a commutative ring is planar or toroidal?

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 undirected 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) \neq ann_R(x) \cup ann_R(y). In this paper, all rings whose annihilator graphs can be embedded on the plane or torus are classified.

On perfect annihilator graphs of commutative rings

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 [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 [Formula: see text]. In this paper, we study the perfectness of annihilator graphs of a vast range of rings. Indeed, it is shown that if [Formula: see text] is reduced with finitely many minimal primes or nonreduced, then [Formula: see text] is perfect.

Annihilator graphs of MV-algebras

In this paper, we introduce the annihilator graph [Formula: see text] of an MV-algebra [Formula: see text]. We show that [Formula: see text] contains the zero-divisor graph [Formula: see text] as a spanning subgraph. We then prove that [Formula: see text] if and only if [Formula: see text]. Moreover, we obtain that the girth [Formula: see text].