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CirculantSexticToHexagonalToroidal[{a_,b_,mod_}]:=Module[{triangles,graph},triangles=Union[Sort/@Flatten[Table[Mod[{{0,b,a+b},{a,0,a+b}}+n,mod,1],{n,1,mod}],1]];If[Length[triangles]2mod,graph=Graph[UndirectedEdge@@@Select[Subsets[triangles,{2}],Length[Intersection@@#]2&]],Return["not valid"]];If[ConnectedGraphQ[graph]&&Length[FindCycle[graph,{3},All]]<4mod,CanonicalGraph[graph],"not valid"]];
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CirculantSexticToHexagonalToroidal[{20,37,300}]
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Graph3D[%]
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CirculantSexticToHexagonalToroidal[{17,11,300}]
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Graph3D[%]
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Graph3D[TorusGraph[{30,30}]]
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Graph3D[TorusGraph[{10,10,10}]]
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