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@4sphere Are you looking for https://oeis.org/A071881 ?

@4sphere In the case of d=2, this amounts to finding how many distinct graphs arise as the 1-skeleton of a Delaunay cellulation? In turn, I think this can be reduced to determining which graphs are inscribable in the sphere. Adding a point at infinity connected to all of the outermost points of the Daulaunay graph will give an inscribable polyhedron, thought of as the convex hull of ideal points in the upper half-space model of hyperbolic space.

@4sphere The Delaunay triangulation of points in $n$ dimensions is determined by the order type of their lift to $n+1$ dimensions, where "lift" = set final coordinate equal to sum of squares of input coordinates, and "order type" = orientation of each simplex. The number of these order types is $n^{(d+1{)}^{2}n(1+o(1))}$: see Goodman & Pollack, "Allowable Sequences and Order Types in Discrete and Computational Geometry", https://doi.org/10.1007/978-3-642-58043-7_6

So that's at least a rough upper bound, exponential in $O(n\log n)$ and much smaller than the trivial $2^{n(n-1)/2}$ bound on the number of possible adjacency matrices.