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Are these maps, associated to finite simple graphs, interesting?

Given a finite simple graph on $n$ vertices, say $G = (V,\, E)$, where $$ V = \{ v_1, \ldots , \, v_n \} $$ and $$ E \subseteq \{ (v_a, \, v_b) \, | \, 1 \leq a < b \leq n \},$$ does there exist a ...
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