3D Unit-Distance Graphs

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In two dimensions, the Harborth graph has four unit-length edges meeting at each vertex. In other words, it has valency four. In three dimensions, the octahedron has valency four. The icosahedron has valency five. What about valency six?
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Contributed by: Ed Pegg Jr (October 2015)
Open content licensed under CC BY-NC-SA
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In 2D, whether the Harborth graph is the smallest graph with valency four is an unsolved question. In 3D, for valency 6–10, the methods in this Demonstration can generate unit-distance graphs with 16, 24, 36, 64, and 96 vertices, but there are likely smaller such graphs. For valency 10, use tetrahedron × tetrahedron × octahedron.
Permanent Citation
"3D Unit-Distance Graphs"
http://demonstrations.wolfram.com/3DUnitDistanceGraphs/
Wolfram Demonstrations Project
Published: October 6 2015