Let L be a lattice with the least element 0. An element x ∈ L is a zero divisor if x∧ y = 0 for some y ∈ L∗ = L \ {0}. The set of all zero divisors is denoted by Z(L). We associate a simple graph Γ(L) to L with vertex set Z(L)∗ = Z(L) \ {0}, the set of non-zero zero divisors of L and distinct x, y ∈ Z(L)∗ are adjacent if and only if x ∧ y = 0. In this paper, we obtain certain properties and dia...

A divisor graph G is an ordered pair (V, E) where V ⊂ and for all u = v ∈ V , uv ∈ E if and only if u | v or v | u. A graph which is isomorphic to a divisor graph is also called a divisor graph. In this note, we will prove that for any n 1 and 0 m n 2 then there exists a divisor graph of order n and size m. We also present a simple proof of the characterization of divisor graphs which is due to...

The main objective of this paper is to calculate the metric dimension and resolving-domination number zero-divisor graphs direct product finite fields. Let $F_1, \ldots, F_n(n \geqslant 2)$ be We consider graph $\Gamma(R)$ ring $R=F_1 \times \cdots F_n$ $(n determine $\beta(\Gamma(R))$ $\gamma_r(\Gamma(R))$ If order each field two, then we prove that $\beta(\Gamma(R)) \leqslant n-1$ $\gamma_r(\...

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; ...

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.