In this paper, we deal with zero-divisor graphs of posets. We prove that the diameter of such a graph is either 1, 2 or 3 while its girth is either 3, 4 or ∞. We also characterize zero-divisor graphs of posets in terms of their diameter and girth. © 2012 Elsevier B.V. All rights reserved.

The idea of associating a graph with the zero-divisors of a commutative ring was originated by Beck. The problems concerning zero-divisor graphs have been studied extensively in the past 10 years. DeMeyer and DeMeyer presented some properties for the correspondence between zero-divisor graphs and their semigroups. It is very important to have adequate examples before the complete resolution of ...

This paper establishes a set of theorems that describe the diameter of a zero-divisor graph for a finite direct product R1 × R2 × · · · × Rn with respect to the diameters of the zero-divisor graphs of R1, R2, · · · , Rn−1 and Rn(n > 2).

Let R be a commutative ring (with 1) and let Z(R) be its set of zero-divisors. The zero-divisor graph Γ(R) has vertex set Z∗(R) = Z(R) \ {0} and for distinct x, y ∈ Z∗(R), the vertices x and y are adjacent if and only if xy = 0. In this paper, we consider the domination number and signed domination number on zero-divisor graph Γ(R) of commutative ring R such that for every 0 6= x ∈ Z∗(R), x 6= ...

In previous literature Coykendall & Maney, as well as Axtell & Stickles, have discussed the concept of irreducible divisor graphs of elements in domains and ring with zero-divisors respectively, with two different definitions. In this paper we seek to look at the irreducible divisor graphs of ring elements under a hybrid definition of the two previous ones—in hopes that this graph will reveal s...

the concept of the bipartite divisor graph for integer subsets has been considered in [‎graph combinator., ‎ 26 (2010) 95--105‎.]. ‎in this paper‎, ‎we will consider this graph for the set of character degrees of a finite group $g$ and obtain some properties of this graph‎. ‎we show that if $g$ is a solvable group‎, ‎then the number of connected components of this graph is at most $2$ and if $g...

In this paper we consider, for a finite commutative ring R, the wellstudied zero-divisor graph Γ(R) and the compressed zero-divisor graph Γc(R) of R and a newly-defined graphical structure — the zero-divisor lattice Λ(R) of R. We give results which provide information when Γ(R) ∼= Γ(S), Γc(R) ∼= Γc(S), and Λ(R) ∼= Λ(S) for two finite commutative rings R and S. We also provide a theorem which sa...

Inspired by a very recent work of A. Đurić, S. Jevđenić and N. Stopar, we introduce new definition zero-divisor graphs attached to rings that includes all the classical definitions already known in literature. We provide an interpretation such means functor call which is associated with family special equivalence relations fixed beforehand. thus recover generalize many results for framework mig...