Description of ring varieties whose finite rings are uniquely determined by their zero-divisor graphs
The zero-divisor graph of an associative ring R is a graph whose vertices are all nonzero zero-divisors (one-sided and two-sided) of R, and two distinct vertices x and y are joined by an edge if and only if either xy =0 or yx =0. In the present paper, we give a full description of ring varieties whose finite rings are uniquely determined by their zero-divisor graphs.