Isomorphism problem and perfectness of zero-divisor graphs for finite MV-algebras

Let (A; ⊕ , , 0) be a finite MV-algebra, and (A; , 0) be the associated semigroup. The zero-divisor graph (A) of A is a simple graph with the set of vertices align* &V((A))=∈ A \0\ (∃ ~y∈ A\0\)~x~~y=0\,align* and the set of edges E( (A))={the edge with ends x and y ( x, y∈ A \0\, x≠ y) ~x~ ~y=0}. In this paper, some invariants (such as domination parameters, isomorphism problems, the chromatic number and the clique number, perfectness, etc.) of (A) are discussed. We prove that (A) (B) implies that A B when A and B are finite MV-algebras with |A|=|B| 5. Also, for a finite MV-algebra A, we obtain that the chromatic number of (A) equals the clique number of (A), and characterize the perfectness of (A).