The zero-divisor graphs of MV-algebras

In this paper, we will introduce and study the zero-divisor graphs of MV-algebras. Let (A, ⊕ , ∗ , 0) be an MV-algebra, and (A, ⊙ , 0) be the associated semigroup. Define the zero-divisor graph Γ (A) of A to be the simple graph with vertices V(Γ(A))={x∈A|(∃y∈A\{0})x⊙y=0}, and edges E(Γ(A))={the edge with endsxandy|(x≠y,x,y∈A)x⊙y=0}. We show that Γ (A) is connected with diam(Γ (A)) ≤ 3 , where diam(Γ (A)) denotes the diameter of Γ (A). Moreover, we characterize A with diam(Γ (A)) equal to 0, 1, 2 or 3. Finally, using the zero-divisor graph, we classify all MV-algebras of cardinality up to seven.