​

With[{g=TorusGraph[{50,50}]},HighlightGraph[g,NeighborhoodGraph[g,RandomChoice[VertexList[g]],5]]]
In[]:=
Out[]=
MeshConnectivityGraph[DiscretizeRegion[Sphere[],MaxCellMeasure0.0001],2]
In[]:=
Out[]=
​
BuckyballGraph[4,"Embedded"]
In[]:=
Out[]=
BuckyballGraph[5]//Graph3D
In[]:=
Out[]=
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
In[]:=
Out[]=
RandomGeodesicBundle[gx_,bradius_,n_Integer:1]:=With[{g=IndexGraph[gx,EdgeStyleLightGray,VertexStyleLightGray]},Module[{spf=FindShortestPath[g,All,All],vl=VertexList[g],n1,n2},​​n1=VertexList@NeighborhoodGraph[g,RandomChoice[vl],bradius];n2=VertexList@NeighborhoodGraph[g,RandomChoice[vl],bradius];HighlightGraph[g,Table[Style[PathGraph[spf[RandomChoice[n1],RandomChoice[n2]]],Thick,RandomColor[]],n]]]]
In[]:=
RandomGeodesicBundle[BuckyballGraph[8,"Embedded"],2,30]//Graph3D
In[]:=
Out[]=
RandomGeodesicBundle[BuckyballGraph[8,"Embedded"],1,30]//Graph3D
In[]:=
Out[]=
RandomGeodesicBundle[BuckyballGraph[10,"Embedded"],1,50]//Graph3D
In[]:=
Out[]=
VertexCount[BuckyballGraph[8,"Embedded"]]
In[]:=
1620
Out[]=
VertexCount[BuckyballGraph[9,"Embedded"]]
In[]:=
2000
Out[]=