Could someone identify this graph for me? I apologise for the lack of technical vocabulary, I hope I can describe this clearly enough nonetheless. The graph is formed by the vertices and edges of a two-dimensional hexagonal grid, plus all internal diagonals of each hexagonal patch appearing as edges. Thus each node has 12 edges connecting to it. The entire thing is circularised in both dimensions, with hexagonal patches identified as follows (imagine the hexagons oriented so two of the sides are vertical, and they are arranged in horizontal rows): the patch you reach by travelling three patches "north east" (or south west) is the same patch you started at; and the patch you reach by travelling to the adjacent patch down-and-to-the-right, followed by three patches to the right, is also the same patch you started at.
There are 24 vertices and 144 edges to the graph. It has diameter 2.
I added an illustration that shows the underlying grid and the identification of edges of the whole thing.