Definitions

In[]:=
Vols[graph_,start_:All,max_:Automatic]:=MeanAround/@Transpose[Values[ResourceFunction["GraphNeighborhoodVolumes"][graph,start,max]]]
In[]:=
RandVols[graph_,n_Integer,max_:Automatic]:=Vols[graph,"Random"n,max]
In[]:=
Dims[vols_]:=ListLinePlotTable[(Log[vols[[r+1]]]-Log[vols[[r]]])/(Log[r+1]-Log[r]),{r,Length[vols]-1}],


Trivial Structures

Square (Torus) Grid

In[]:=
Dims[RandVols[TorusGraph[{40,40}],1000]]
Out[]=
In[]:=
Vols[TorusGraph[{10,10}]]
Out[]=
{1,5,13,25,41,59,75,87,95,99,100}
In[]:=
Dims[RandVols[TorusGraph[{200,200}],1000]]
Out[]=
In[]:=
Dims[RandVols[TorusGraph[{50,50}],1000]]
Out[]=
10
20
30
40
50
0.5
1.0
1.5
2.0

Hexagonal Grid & Other Lattices

In[]:=
<<IGraphM`;​​HypergraphPlot[Catenate[Partition[#,3,2,-1]&/@MeshCells[IGLatticeMesh["Hexagonal",Polygon@CirclePoints[3,6],MeshCellLabel{2"Index"}],2][[All,1]]]]
IGraph/M 0.3.112 (May 7, 2019)
Evaluate
IGDocumentation[]
to get started.
Out[]=
In[]:=
?IGLatticeMesh
Out[]=
In[]:=
IGLatticeMesh[]
Out[]=
{Square,Hexagonal,Triangular,Trihexagonal,SmallRhombitrihexagonal,TruncatedSquare,SnubSquare,TruncatedHexagonal,ElongatedTriangular,GreatRhombitrihexagonal,SnubHexagonal,Rhombille,DeltoidalTrihexagonal,TetrakisSquare,CairoPentagonal,TriakisTriangular,PrismaticPentagonal,BisectedHexagonal,FloretPentagonal,DellaRobbiaWeave,Portugal,StackBond,Herringbone,Basketweave,PersianHexagonalWeave,Hopscotch,StretcherBond,Pinwheel,BrickworkSquare,Chickenwire,Corridor,CorridorHorizontal,Brickweave,Trellis,HeeschIsohedral,PPentomino,Chevron,Shingle,Zigzag,Kite,FalseCubic,TrihexAndHex,GlideReflection,PentagonType1,PentagonType2,PentagonType3,PentagonType4,PentagonType5,PentagonType6,PentagonType7,PentagonType8,PentagonType9,PentagonType10,PentagonType11,PentagonType12,PentagonType13,PentagonType14,PentagonType15}
​
In[]:=
GraphData["Fullerene"]
(( lots of Fullerenes ... ))
In[]:=
GraphData[{"Fullerene",{50,244}}]
Out[]=
In[]:=
GraphData[{"Fullerene",{70,4085}}]
Out[]=

Known Figures

Pseudosphere

Real-Life Graphs

From First Section

[[[[ Max says the volumes grow exponentially Nov 24 10:55 pm ]]]]