On the zero-divisor graph of a commutative ring

Let R be a commutative ring and Γ(R) be its zero-divisor graph. In this paper it is shown that for any finite commutative ring R, the edge chromatic number of Γ(R) is equal to the maximum degree of Γ(R), unless Γ(R) is a complete graph of odd order. In [D.F. Anderson, A. Frazier, A. Lauve, P.S. Livingston, in: Lecture Notes in Pure and Appl. Math., Vol. 220, Marcel Dekker, New York, 2001, pp. 61–72] it has been proved that if R and S are finite reduced rings which are not fields, then Γ(R) Γ(S) if and only if R S. Here we generalize this result and prove that if R is a finite reduced ring which is not isomorphic to or to and S is a ring such that Γ(R) Γ(S), then R S.