Given a (non commutative) ring $R$, we construct a (directed) graph $G_0(R)$ with vertex set $Z(R)\backslash \{0\}$, the zero divisors of $R$ except for $0$. And an edge from $x$ to $y$ whenever $xy=0$. This is called the zero divisor graph of $R$.

My question is, what are the known obstructions to a graph being a zero divisor graph for some ring $R$?

(Edit) Following the reference by M. Sapir below, I found several articles which give partial answers. By this result of Dolžan and Oblak we learn that there are restrictions on the diameter and girth (they work over semirings). In the case of commutative rings, one knows by a result of Belshoff and Chapman, which zero-divisor graphs are planar, and there are similar results for projective zero-divisor graphs and other genera. Another restriction is that the number of edges must be even.

It seems that a characterisation of the set of zero-divisor graphs is open and complicated enough that is only worth studying over special subclasses of graphs. Because of this I would like to focus just on the second question below, which I have the feeling should have an easy counter-example.

I'd also like to ask the same question for a weaker notion of graphs associated to a ring. Let $G_{k}(R)$ be defined for every ring $R$ and $k\in R$ as above but with edges going from $x$ to $y$ whenever $xy=k$ (we can start with the vertex set being all of $R$ and then throw away the isolated vertices).

What graphs can be written as $G_k(R)$ for some pair $(R,k)$?

Are there graphs which are isomorphic to $G_k(R)$ for some $k\neq 0$ but are not isomorphic to any zero divisor graph?