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What polyhedra are known to have two vertices adjacent if and only if they are of distance $d$ apart, for fixed $d$? For example, regular Platonic solids satisfy this condition, so I am looking for other "regular" non-Platonic solids.

Related question: has work been done to consider the chromatic number of these regular polyhedra?

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The triangular bipyramid glues two regular tetrahedra base-to-base:

          TriBiPyr
The pairwise distances among the $5$ vertices are $1$, except the two top/bot apexes are $2\sqrt{\frac{2}{3}} \approx 1.6$ apart. So I think this meets your criterion of a non-Platonic polyhedron having

two vertices adjacent if and only if they are of distance $d$ [$=1$] apart

I believe the same holds true of all the deltahedra.

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    $\begingroup$ More generally, the convex solids all of whose faces are regular polygons: prisms and antiprisims, Platonic solids, Archimedean (symmetry group acts transitively on the vertices) and Johnson solids (everything else). $\endgroup$
    – Yoav Kallus
    Commented May 1, 2016 at 18:55
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I think this problem could be useful for you, "Choombam"s are other types with your conditions:

http://artofproblemsolving.com/community/c6h111388p632753

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