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 smooth complex-valued function $E_G$ on $C_n(\mathbb{R}^3)$, where the latter is the configuration space of $n$ distinct points in $\mathbb{R}^3$, such that
- $E_G(\rho \mathbf{x}) = E_G(\mathbf{x})$ for any $\mathbf{x} \in C_n(\mathbb{R}^3)$ and any $\rho \in SO(3)$,
- $E_G(-\mathbf{x}) = \overline{E_G(\mathbf{x})}$ for any $\mathbf{x} \in C_n(\mathbb{R}^3)$,
- $E_G(\sigma \mathbf{x}) = E_G(\mathbf{x})$ for any $\mathbf{x} \in C_n(\mathbb{R}^3)$ and any $\sigma \in Aut(G)$ (by $Aut(G)$ I mean the group of all permutations of $V$ which preserve the set of edges $E$, i.e. the group of symmetries of $G$), and $Aut(G)$ acts by permuting the $n$ points in the configuration $\mathbf{x}$,
- if $G = G_1 \coprod G_2$ is the disjoint union of graphs $G_1$ and $G_2$, then $$ E_G(\mathbf{x}) = E_{G_1}(\mathbf{x}) E_{G_2}(\mathbf{x}), \text{ and}$$
- there exists a constant $c_G \geq 1$ depending on $G$, such that $$\Re(E_G(\mathbf{x})) \geq c_G, \text{ for any $\mathbf{x} \in C_n(\mathbb{R}^3)$ ?}$$
I have constructed a family of smooth complex-valued maps $E_G$ on configuration spaces, associated to finite simple graphs and satisfying properties 1-4. As for property 5, I only checked it numerically for some small graphs and proved it for a small number of graphs.
I don't know... Is it worth publishing? If $G$ is the complete graph on $n$ vertices, then my constructed $E_G$ is actually, up to a factor, the Atiyah-Sutcliffe determinant (related to the Atiyah problem on configurations of points).
Could someone please respond for instance if similar questions were asked before in the literature, whether in Mathematics or in Physics, and point to the relevant literature?
