‎In this paper we give a characterization for all commutative‎ ‎rings with $1$ whose zero-divisor graphs are $C_4$-free.‎

In this paper we study zero-divisor graphs of semirings. We show that all zero-divisor graphs of (possibly noncommutative) semirings are connected and have diameter less than or equal to 3. We characterize all acyclic zero-divisor graphs of semirings and prove that in the case zero-divisor graphs are cyclic, their girths are less than or equal to 4. We also give a description of the zero-diviso...

In this paper, we determine the structures of zero-divisor semigroups whose graph is Kn + 1, the complete graph Kn together with an end vertex. We also present a formula to calculate the number of non-isomorphic zero-divisor semigroups corresponding to the complete graph Kn, for all positive integer n.

Let R be a noncommutative ring. The zero-divisor graph of R, denoted by Γ(R), is the (directed) graph with vertices Z(R)∗ = Z(R)− {0}, the set of nonzero zero-divisors of R, and for distinct x, y ∈ Z(R)∗, there is an edge x → y if and only if xy = 0. In this paper we investigate the zero-divisor graph of triangular matrix rings over commutative rings. Mathematics Subject Classification: 16S70; ...