Graph Relatives for Tessellation of the Hyperbolic Plane

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I'm trying to get into the theory about the Modular group. Among the "Paracompact hyperbolic uniform tilings in [∞,3] family" in the section "Tessellation of the hyperbolic plane" I found the Order-3 apeirogonal tiling:

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The figure reminded my on Frucht's graph, which seems to have the similar structural features.

Are there also graphs for the other tilings?