Smallest $k$-regular unit-distance graph

Question

We can create arbitrary $k$-regular unit-distance graphs by using a "hyper-cube construction": taking a $k-1$-regular unit-distance graph, making a copy and translating it one unit distance that is not parallel to any of the existing edges, and connecting all vertices between the two copies (source).

However this is not optimal, considering in the $k=2$ case using our construction we get a rhombus looking graph while the complete graph $K_3$ is optimal. The 3-cube graph has 8 vertices while a triangular-prism looking graph has 6 vertices. What are the smallest $k$-regular unit-distance graphs?

My idea that the $k=3$ six vertex case is optimal: the four vertex case is the complete graph which is not a unit-distance graph, and the five vertex case cannot satisfy the handshaking lemma.

Answer 1

For 4, you need Generalized Quadrangle {2,1} on 9 points.

For 5, Shift and copy GQ21 for 18 points.
A smaller quintic unit distance graph might be the Clebsch graph, but that can likely be proven to not be unit distance.

At 6 you get Hamming-{3,3} (27 points)

Answer 2

What space are you in? If you are in $\Bbb R^{k-1}$ or higher, you can use a regular simplex. That is clearly minimal.