http://users.cecs.anu.edu.au/~bdm/data/graphs.html
In[]:=
ConnectedGraphs[2]={
};
ConnectedGraphs[3]=
,
;
In[]:=
ConnectedGraphs[4]=
,
,
,
,
,
;
In[]:=
ConnectedGraphs[5]=
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
;
In[]:=
ConnectedGraphs[6]=
;
In[]:=
ConnectedGraphs[7]=
;
In[]:=
ConnectedGraphs[8]=
;
In[]:=
Table[Length[ConnectedGraphs[i]],{i,8}]
Out[]=
{1,1,1,6,21,112,853,11117}
In[]:=
Select[ConnectedGraphs[6],AllTrue[Values[ResourceFunction["GraphNeighborhoodVolumes"][#]],MatchQ[#,{1,4,___}]&]&]
Out[]=

,

In[]:=
Select[ConnectedGraphs[7],AllTrue[Values[ResourceFunction["GraphNeighborhoodVolumes"][#]],MatchQ[#,{1,4,___}]&]&]
Out[]=
{}
In[]:=
Select[ConnectedGraphs[8],AllTrue[Values[ResourceFunction["GraphNeighborhoodVolumes"][#]],MatchQ[#,{1,4,___}]&]&]
Out[]=

,
,
,
,

In[]:=
allgraphs=Join@@Table[ConnectedGraphs[i],{i,2,8}];
In[]:=
Select[allgraphs,AllTrue[Values[ResourceFunction["GraphNeighborhoodVolumes"][#]],MatchQ[#,{1,4,8,___}]&]&]
Out[]=

,

In[]:=
VertexCount/@%
Out[]=
{8,8}

Distance Regular

In[]:=
Intersection[GraphData["DistanceRegular"],GraphData["Cubic"],GraphData["Connected"]]
Out[]=
{BiggsSmithGraph,CoxeterGraph,CubicalGraph,DesarguesGraph,DodecahedralGraph,Foster090A,HeawoodGraph,LeviGraph,PappusGraph,PetersenGraph,TetrahedralGraph,Tutte12Cage,UtilityGraph}
In[]:=
GraphData/@SortBy[Intersection[GraphData["DistanceRegular"],GraphData["Connected"]],GraphData[#,"EdgeCount"]&]​​
In[]:=
First[Values[ResourceFunction["GraphNeighborhoodVolumes"][#,{1}]]]&/@%107
Out[]=

Regular Graphs

In[]:=
GraphData["Regular",7]
Out[]=
{C4+C3,{Circulant,{7,{1,2}}},{Complete,7},{Cycle,7},{Empty,7},{Quartic,{7,1}}}
In[]:=
GraphData/@GraphData["Regular",7]
Out[]=

,
,
,
,
,

In[]:=
Select[GraphData/@GraphData["Regular",7],AllTrue[Values[ResourceFunction["GraphNeighborhoodVolumes"][#]],MatchQ[#,{1,4,8,___}]&]&]
Out[]=
{}
In[]:=
Values[ResourceFunction["GraphNeighborhoodVolumes"][#]]&/@(GraphData/@GraphData["Regular",7])
Out[]=
{{{1,3},{1,3,4},{1,3,4},{1,3},{1,3},{1,3,4},{1,3,4}},{{1,5,7},{1,5,7},{1,5,7},{1,5,7},{1,5,7},{1,5,7},{1,5,7}},{{1,7},{1,7},{1,7},{1,7},{1,7},{1,7},{1,7}},{{1,3,5,7},{1,3,5,7},{1,3,5,7},{1,3,5,7},{1,3,5,7},{1,3,5,7},{1,3,5,7}},{{1},{1},{1},{1},{1},{1},{1}},{{1,5,7},{1,5,7},{1,5,7},{1,5,7},{1,5,7},{1,5,7},{1,5,7}}}
In[]:=
Values[ResourceFunction["GraphNeighborhoodVolumes"][#]]&/@(GraphData/@GraphData["Regular",9])
Out[]=
In[]:=
Values[ResourceFunction["GraphNeighborhoodVolumes"][#]]&/@(GraphData/@GraphData["Regular",11])
Out[]=
{{{1,3,4},{1,3,4},{1,3,4},{1,3,4},{1,3,4},{1,3,4},{1,3,4},{1,3,4},{1,3},{1,3},{1,3}},{{1,5,11},{1,5,11},{1,5,11},{1,5,11},{1,5,11},{1,5,11},{1,5,11},{1,5,11},{1,5,11},{1,5,11},{1,5,11}},{{1,3,5},{1,3,5},{1,3,5},{1,3,5},{1,3,5},{1,3},{1,3},{1,3},{1,3},{1,3},{1,3}},{{1,3,5,6},{1,3,5,6},{1,3,5,6},{1,3,5,6},{1,3,5,6},{1,3,5,6},{1,3,5},{1,3,5},{1,3,5},{1,3,5},{1,3,5}},
⋯538⋯
,{{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11}},{{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11}},{{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11}},{{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11},{1,7,11}}}
Full expression not available
(
original memory size:
0.6 MB)