The truncated dodecadodecahedral graph is the skeleton of the truncated dodecadodecahedron, which is the only uniform polyhedron for which this is the case. It is illustrated above.
It is implemented in the Wolfram Language as GraphData["TruncatedDodecadodecahedralGraph"].
While all other common named cubic graphs are known to be unit-distance, it appears to be difficult to find a unit-distance embedding for the truncated dodecadodecahedral graph due to vertices coninciding under iterative unit-distance refinement. For example, the embeddings illustrated above all have unit edge lengths but also have overlapping vertices.
