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.
Aaron Meyerowitz
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