I don’t know of any other UDGs for which exceeds , but in the diploma work of Carsten Schade, which I used in my thesis, there is an example for a UDG with 27 vertices and 81 edges, where . It seems not to be a rigid graph, but still, I didn’t manage to add a single edge, neither a new vertex with degree at least 5 (the latter doesn’t seem that surprising, but given how symmetric the graph is, first it seemed plausible that the center could have degree 6 for some unit distance representation of the graph), the best I managed is to add 3 vertices with degree 4, but that isn’t good enough. Still, it gave better bounds for and as the ones given by Schade, but they still lack one edge each to achieve . All these can be found in the folder https://drive.google.com/drive/folders/1UvgH4ASqMVyOuWc9Y8brI3-zNJB3pyZr

In the GeoGebra file Schade27.ggb, B is the movable vertex, and the center of the drawing is fixed, although I made the assumption that the three cube graphs on the sides of Schade27.png are axially symmetric (I am still not sure, if it is necessarily true).
If you are interested in the diploma work of Schade (written in German), I can send it to you privately (it was sent to me privately too by his supervisor, so I guess I am not allowed to make it public).

Note: I don’t (yet) have a mathematically precise proof for the above, but they are very clear from the GeoGebra files.

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