I suppose we are implicitely  appending $x \neq y \land \dots$ to all relations lest we get loops. Also,  every binary relation is (vacuously) self fulfilling for the empty and 1 point graphs. Hence we should specify at least two vertices.  

Graphs with no edges are characterized by any empty relation such as $d(x) \neq d(x)$ or $d(x)d(y) \neq d(x)d(y)$ or merely $\emptyset$. For complete graphs $d(x)=d(x)$ or $d(x)d(y)=d(x)d(y)$ could work.



The relation $\Phi$ has to be symmetric if we want an undirected graph. So, for undirected graphs with at least two points,  $d(x)=d(y)^2$  would specify the one edge path.