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With[{g=TorusGraph[{50,50}]},HighlightGraph[g,NeighborhoodGraph[g,RandomChoice[VertexList[g]],5]]]
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MeshConnectivityGraph[DiscretizeRegion[Sphere[],MaxCellMeasure0.0001],2]
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​
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BuckyballGraph[4,"Embedded"]
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BuckyballGraph[5]//Graph3D
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RandomGeodesic[gx_,n_Integer:1]:=With[{g=IndexGraph[gx,EdgeStyleLightGray,VertexStyleLightGray]},With[{spf=FindShortestPath[g,All,All],vl=VertexList[g]},HighlightGraph[g,Table[Style[PathGraph[spf[RandomChoice[vl],RandomChoice[vl]]],Thick,RandomColor[]],n]]]]
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RandomGeodesic[BuckyballGraph[8,"Embedded"],20]//Graph3D
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