Septic Graph

Author

Eric W. Weisstein
October 19, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/SepticGraph.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SepticGraph.html.

©2023 Wolfram Research, Inc. except for portions noted otherwise

Counts

Connected

A014377

Table[Total[Boole[SepticGraphQ/@ConnectedGraphs[n]]],{n,10}]//Timing

Table[Length@GraphData[{"Septic","Connected"},n],{n,10}]

Length/@Table[RegularConnectedGraphs[7,n],{n,13}]

Hamiltonian

Table[Length@GraphData[{"Septic","Hamiltonian"},n],{n,13}]

Length/@Table[Select[RegularConnectedGraphs[7,n],HamiltonianGraphQ],{n,13}]

Same as connected up to at least 12-nodes

Disconnected

A165877

gd=Length/@Table[GraphData[{"Disconnected","Septic"},n],{n,2,20,2}]

Length[gd]

A165877=Cases[Import["https://oeis.org/A165877/b165877.txt","Table"],{n_Integer?Positive,a_}:>a]

Take[A165877,10]-gd

SepticGraphQ/@(d18=GraphDisjointUnion[System`CompleteGraph[8],GraphData[#]]&/@GraphData[{"Connected","Septic"},10])

cc=Map[CanonicalName[ToEntity[#]]&,ConnectedGraphComponents/@d18,{2}]

All

A165628

A165877=Cases[Import["https://oeis.org/A165628/b165628.txt","Table"],{n_Integer?Positive,a_}:>a]

Table[Total[Boole[SepticGraphQ/@Graphs[n]]],{n,10}]//Timing

Table[Length@GraphData["Septic",n],{n,10}]

Figure

nice=SortBy[Select[DeleteCases[GraphData["Septic"],{"Circulant"|"DifferenceSetIncidence"|"NoncayleyTransitive"|"Nuciferous"|"VertexTransitive"|"Septic",_}],GraphData[#,"Embeddings"]=!={}&],GraphData[#,"V"]&]

GraphicsGrid[Partition[Show[SetProperty[GraphData[#],{VertexSize->{"Scaled",.04},System`VertexStyle->Red,System`EdgeStyle->Black}],PlotLabel->Style[TextCell[GraphData[#,"Name"],PageWidth->120,TextAlignment->Center],{Italic,FontFamily->"Times"}]]&/@nice,UpTo[5]],ImageSize->600]

Embeddings

<<MathWorld`Graphs`

Union[GraphData[#,"V"]&/@GraphData["Septic"]]

8

TextGrid[Sort[{FirstCase[GraphData[#,"StandardNames"],{"Septic",n_}:>n],#}&/@GraphData["Septic",8]],Dividers->All]

{"Septic", {8, 1}} == {"Complete", 8}

GraphData[{"Septic",{8,1}},"StandardName"]

10

TextGrid[Sort[{FirstCase[GraphData[#,"StandardNames"],{"Septic",n_}:>n],#}&/@GraphData["Septic",10]],Dividers->All]

{"Septic", {10, 1}}

Names

GraphData[{"Septic",{10,1}},"StandardName"]

GraphData[{"Septic",{10,1}},"StandardNames"]

Primary

GraphData[{"Septic",{10,1}}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"Septic",{10,1}},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Septic",{10,1}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Septic",{10,1}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Septic",{10,1}},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Septic",{10,1}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Septic",{10,1}},"Graph","Degenerate"]

Perspective

GraphData[{"Septic",{10,1}},"Graph","Perspective"]//StyleGraphs

Bilateral

GraphData[{"Septic",{10,1}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Septic",{10,1}},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Septic",{10,1}},"Planar"]

GraphData[{"Septic",{10,1}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Septic",{10,1}},{"RCN","CN"}]

GraphData[{"Septic",{10,1}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Septic",{10,1}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Septic",{10,1}}],TimeConstraint->10])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Septic",{10,1}}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[g,"OuterFace"->#]&/@faces

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Septic",{10,1}}],"CrossingNumber"->cn,RandomSeed->n]],{n,500}],Max[OuterFace[#]]<=GraphData[{"Septic",{10,1}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Septic",{10,1}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Septic",{10,1}},"Circulant"]

GraphData[{"Septic",{10,1}},"Graph","Circulant"]//StyleGraphs

Line graph

GraphData[{"Septic",{10,1}},"Line"]

LineGraphQ[GraphData[{"Septic",{10,1}}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Septic",{10,1}}&]

RootGraph[GraphData[{"Septic",{10,1}}]]

Halin

GraphData[{"Septic",{10,1}},"Halin"]

GraphData[{"Septic",{10,1}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Septic",{10,1}},"Rigid"]

GraphData[{"Septic",{10,1}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Septic",{10,1}},"Flexible"]

GraphData[{"Septic",{10,1}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Septic",{10,1}},"LCF"]

GraphData[{"Septic",{10,1}},"HamiltonianCycleCount"]

GraphData[{"Septic",{10,1}},"LCFSignature"]

GraphData[{"Septic",{10,1}},"LCFNotations"]

GraphData[{"Septic",{10,1}},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute all

Length[FindHamiltonianCycle[GraphData[{"Septic",{10,1}}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Septic",{10,1}}],All])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,1}]]}])//Timing

Length[gs]

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,XXX}]]}])//Timing

Length[gs]

UnitDistance

GraphData[{"Septic",{10,1}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Septic",{10,1}}],Debug->True]//Timing

GraphData[{"Septic",{10,1}},"Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Septic",{10,1}},"VertexList"],GraphData[{"Septic",{10,1}},"EdgeList"]],MaxIterations->500]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Septic",{10,1}},"Graph",#]&/@{"All","UnitDistance"})

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Septic",{10,1}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Septic",{10,1}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

Matchstick

GraphData[{"Septic",{10,1}},"Matchstick"]

GraphData[{"Septic",{10,1}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Septic",{10,1}},"Edges"]]

3D All

GraphData[{"Septic",{10,1}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Septic",{10,1}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Septic",{10,1}},"Polyhedral"]

GraphData[{"Septic",{10,1}},"Graph",{"3D","Polyhedron"}]

{"Septic", {10, 2}}

Names

GraphData[{"Septic",{10,2}},"StandardName"]

GraphData[{"Septic",{10,2}},"StandardNames"]

Primary

GraphData[{"Septic",{10,2}}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"Septic",{10,2}},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Septic",{10,2}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Septic",{10,2}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Septic",{10,2}},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Septic",{10,2}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Septic",{10,2}},"Graph","Degenerate"]

Perspective

GraphData[{"Septic",{10,2}},"Graph","Perspective"]//StyleGraphs

Bilateral

GraphData[{"Septic",{10,2}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Septic",{10,2}},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Septic",{10,2}},"Planar"]

GraphData[{"Septic",{10,2}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Septic",{10,2}},{"RCN","CN"}]

GraphData[{"Septic",{10,2}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Septic",{10,2}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Septic",{10,2}}],TimeConstraint->10])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Septic",{10,2}}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[g,"OuterFace"->#]&/@faces

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Septic",{10,2}}],"CrossingNumber"->cn,RandomSeed->n]],{n,500}],Max[OuterFace[#]]<=GraphData[{"Septic",{10,2}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Septic",{10,2}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Septic",{10,2}},"Circulant"]

GraphData[{"Septic",{10,2}},"Graph","Circulant"]//StyleGraphs

Line graph

GraphData[{"Septic",{10,2}},"Line"]

LineGraphQ[GraphData[{"Septic",{10,2}}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Septic",{10,2}}&]

RootGraph[GraphData[{"Septic",{10,2}}]]

Halin

GraphData[{"Septic",{10,2}},"Halin"]

GraphData[{"Septic",{10,2}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Septic",{10,2}},"Rigid"]

GraphData[{"Septic",{10,2}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Septic",{10,2}},"Flexible"]

GraphData[{"Septic",{10,2}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Septic",{10,2}},"LCF"]

GraphData[{"Septic",{10,2}},"HamiltonianCycleCount"]

GraphData[{"Septic",{10,2}},"LCFSignature"]

GraphData[{"Septic",{10,2}},"LCFNotations"]

GraphData[{"Septic",{10,2}},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute all

Length[FindHamiltonianCycle[GraphData[{"Septic",{10,2}}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Septic",{10,2}}],All])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,1}]]}])//Timing

Length[gs]

UnitDistance

GraphData[{"Septic",{10,2}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Septic",{10,2}}],Debug->True]//Timing

GraphData[{"Septic",{10,2}},"Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Septic",{10,2}},"VertexList"],GraphData[{"Septic",{10,2}},"EdgeList"]],MaxIterations->500]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Septic",{10,2}},"Graph",#]&/@{"All","UnitDistance"})

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Septic",{10,2}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Septic",{10,2}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

Matchstick

GraphData[{"Septic",{10,2}},"Matchstick"]

GraphData[{"Septic",{10,2}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Septic",{10,2}},"Edges"]]

3D All

GraphData[{"Septic",{10,2}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Septic",{10,2}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Septic",{10,2}},"Polyhedral"]

GraphData[{"Septic",{10,2}},"Graph",{"3D","Polyhedron"}]

{"Septic", {10, 3}} == {"Circulant", {10, {1, 2, 4, 5}}}

GraphData[{"Septic",{10,3}},"StandardName"]

{"Septic", {10, 4}}

Names

GraphData[{"Septic",{10,4}},"StandardName"]

GraphData[{"Septic",{10,4}},"StandardNames"]

Primary

GraphData[{"Septic",{10,4}}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"Septic",{10,4}},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Septic",{10,4}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Septic",{10,4}},"EdgeList"]]

Default embedding

GraphPlot[Graph[GraphData[{"Septic",{10,4}},"Edges"]],GraphLayout->"HighDimensionalEmbedding"]

v=Join[CirclePoints[7],CirclePoints[1/4,3]];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,3,4,5,6,7,1,3,5,7,10,4,8,1,6,4,2,7,9,1,10,5,9,3,8,7],e[5,8,2,10],e[3,10,6,8],{4,9},e[6,9,2]},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Septic",{10,4}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Septic",{10,4}},"Graph","Degenerate"]

Perspective

GraphData[{"Septic",{10,4}},"Graph","Perspective"]//StyleGraphs

Bilateral

GraphData[{"Septic",{10,4}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Septic",{10,4}},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Septic",{10,4}},"Planar"]

GraphData[{"Septic",{10,4}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Septic",{10,4}},{"RCN","CN"}]

GraphData[{"Septic",{10,4}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Septic",{10,4}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Septic",{10,4}}],TimeConstraint->10])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Septic",{10,4}}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[g,"OuterFace"->#]&/@faces

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Septic",{10,4}}],"CrossingNumber"->cn,RandomSeed->n]],{n,500}],Max[OuterFace[#]]<=GraphData[{"Septic",{10,4}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Septic",{10,4}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Septic",{10,4}},"Circulant"]

GraphData[{"Septic",{10,4}},"Graph","Circulant"]//StyleGraphs

Line graph

GraphData[{"Septic",{10,4}},"Line"]

LineGraphQ[GraphData[{"Septic",{10,4}}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Septic",{10,4}}&]

RootGraph[GraphData[{"Septic",{10,4}}]]

Halin

GraphData[{"Septic",{10,4}},"Halin"]

GraphData[{"Septic",{10,4}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Septic",{10,4}},"Rigid"]

GraphData[{"Septic",{10,4}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Septic",{10,4}},"Flexible"]

GraphData[{"Septic",{10,4}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Septic",{10,4}},"LCF"]

GraphData[{"Septic",{10,4}},"HamiltonianCycleCount"]

GraphData[{"Septic",{10,4}},"LCFSignature"]

GraphData[{"Septic",{10,4}},"LCFNotations"]

GraphData[{"Septic",{10,4}},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute all

Length[FindHamiltonianCycle[GraphData[{"Septic",{10,4}}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Septic",{10,4}}],All])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,1}]]}])//Timing

Length[gs]

UnitDistance

GraphData[{"Septic",{10,4}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Septic",{10,4}}],Debug->True]//Timing

GraphData[{"Septic",{10,4}},"Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Septic",{10,4}},"Graph",#]&/@{"All","UnitDistance"})

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Septic",{10,4}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Septic",{10,4}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

Matchstick

GraphData[{"Septic",{10,4}},"Matchstick"]

GraphData[{"Septic",{10,4}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Septic",{10,4}},"Edges"]]

3D All

GraphData[{"Septic",{10,4}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Septic",{10,4}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Septic",{10,4}},"Polyhedral"]

GraphData[{"Septic",{10,4}},"Graph",{"3D","Polyhedron"}]

{"Septic", {10, 5}} == {"Circulant", {10, {1, 2, 3, 5}}}

GraphData[{"Septic",{10,5}},"StandardName"]

12

TextGrid[Sort[{FirstCase[GraphData[#,"StandardNames"],{"Septic",n_}:>n],#}&/@GraphData["Septic",12]],Dividers->All]

{"Septic", {12, 41}} == {"Circulant", {12, {2, 3, 4, 6}}}

GraphData[{"Septic",{12,41}},"StandardName"]

{"Septic", {12, 220}} == {"Circulant", {12, {1, 3, 5, 6}}}

GraphData[{"Septic",{12,220}},"StandardName"]

{"Septic", {12, 248}} == {"VertexTransitive", {12, 60}}

GraphData[{"Septic",{12,248}},"StandardName"]

{"Septic", {12, 271}} == {"Circulant", {12, {1, 2, 4, 6}}}

GraphData[{"Septic",{12,271}},"StandardName"]

{"Septic", {12, 1315}} == {"Cone", {7, 5}}

GraphData[{"Septic",{12,1315}},"StandardName"]

{"Septic", {12, 1340}} == {"VertexTransitive", {12, 55}}

GraphData[{"Septic",{12,1340}},"StandardName"]

{"Septic", {12, 1346}} == {"Circulant", {12, {1, 4, 5, 6}}}

GraphData[{"Septic",{12,1346}},"StandardName"]

{"Septic", {12, 1353}} == {"Circulant", {12, {1, 2, 5, 6}}}

GraphData[{"Septic",{12,1353}},"StandardName"]

{"Septic", {12, 1504}} == {"Circulant", {12, {1, 2, 3, 6}}}

GraphData[{"Septic",{12,1504}},"StandardName"]

{"Septic", {12, 1534}} == {"VertexTransitive", {12, 54}}

GraphData[{"Septic",{12,1534}},"StandardName"]

{"Septic", {12, 1539}} == {"Circulant", {12, {1, 3, 4, 6}}}

GraphData[{"Septic",{12,1539}},"StandardName"]

{"Septic", {12, 1547}} == {"VertexTransitive", {12, 51}}

GraphData[{"Septic",{12,1547}},"StandardName"]

14

TextGrid[Sort[{Cases[GraphData[#,"StandardNames"],{"Septic",n_}:>n][[-1]],#}&/@GraphData["Septic",14]],Dividers->All]

16

TextGrid[Sort[{FirstCase[GraphData[#,"StandardNames"],{"Septic",n_}:>n],#}&/@GraphData["Septic",16]],Dividers->All]

18

TextGrid[Sort[{FirstCase[GraphData[#,"StandardNames"],{"Septic",n_}:>n],#}&/@GraphData["Septic",18]],Dividers->All]

20

TextGrid[Sort[{FirstCase[GraphData[#,"StandardNames"],{"Septic",n_}:>n],#}&/@GraphData["Septic",20]],Dividers->All]

GraphData

GraphDataString

file=OpenWrite["~/septic.m"]

cc

Do[WriteString[file,GraphDataString[GraphDisjointUnion@@GraphData/@cc[[i]],
"StandardName"->{"Septic",{18,i}},
"Name"->"18\[Hyphen]septic graph "<>ToString[i],
"Classes"->{"Septic"},
"Notation"->Missing["NotAvailable"],
"Information"->"SepticGraph",
Debug->True,
TimeConstraint->10
]],{i,Length[cc]}];

Close[file]