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 implemented in the Wolfram Language as GraphData["TruncatedDodecadodecahedralGraph"].
Truncated Dodecadodecahedral Graph
See also
Truncated DodecadodecahedronExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Truncated Dodecadodecahedral Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TruncatedDodecadodecahedralGraph.html