​

In[]:=
With[{g=TorusGraph[{50,50}]},HighlightGraph[g,NeighborhoodGraph[g,RandomChoice[VertexList[g]],5]]]
Out[]=
In[]:=
MeshConnectivityGraph[DiscretizeRegion[Sphere[],MaxCellMeasure0.0001],2]
Out[]=
​
In[]:=
BuckyballGraph[4,"Embedded"]
Out[]=
In[]:=
BuckyballGraph[5]//Graph3D
Out[]=
In[]:=
RandomGeodesic[gx_,n_Integer:1]:=With[{g=IndexGraph[gx,EdgeStyleLightGray,VertexStyleLightGray]},With[{spf=FindShortestPath[g,All,All],vl=VertexList[g]},HighlightGraph[g,Table[Style[PathGraph[spf[RandomChoice[vl],RandomChoice[vl]]],Thick,RandomColor[]],n]]]]
In[]:=
RandomGeodesic[BuckyballGraph[8,"Embedded"],20]//Graph3D
Out[]=