Triangular Honeycomb Acute Knight Graph

Author

Eric W. Weisstein
December 10, 2024

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

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

©2024 Wolfram Research, Inc. except for portions noted otherwise

Figure

4

g=TriangularHoneycombAcuteKnightGraph[4,VertexLabels->Automatic]

EuclideanDistance@@GraphEmbedding[g][[{1,5}]]

EdgeLengths[g]//Tally

v=GraphEmbedding[g];

edges=List@@@EdgeList[g];

g4=Graphics[{
FaceForm[LightGray],EdgeForm[Black],Polygon[CirclePoints[#,{1/Sqrt[3],Pi/2},6]]&/@v,
Thick,Line[v,edges],
FaceForm[White],Disk[v,.1]
},
PlotLabel->"4\[Hyphen]triangular honeycomb\nacute knight graph",
LabelStyle->Directive[Black,FontFamily->"Times",14,Italic]]

5

g=TriangularHoneycombAcuteKnightGraph[5]

EdgeLengths[g]//Tally

v=GraphEmbedding[g];

edges=List@@@EdgeList[g];

g5=Graphics[{
FaceForm[LightGray],EdgeForm[Black],Polygon[CirclePoints[#,{1/Sqrt[3],Pi/2},6]]&/@v,
Thick,Line[v,edges],
FaceForm[White],Disk[v,.1]
},
PlotLabel->"5\[Hyphen]triangular honeycomb\nacute knight graph",
LabelStyle->Directive[Black,FontFamily->"Times",14,Italic]]

Combined

GraphicsRow[{g4,g5},ImageSize->600,Alignment->Top]

Figure

Graphs.m

GraphicsRow[gs=Table[TriangularHoneycombAcuteKnightGraph[n]//StyleGraphs,{n,5}],Dividers->All,ImageSize->500]

"TriangularGrid"

GraphicsRow[Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Graph","TriangularGrid"][[1]]//StyleGraphs,{n,5}],Dividers->All,ImageSize->500]

Default

GraphicsRow[Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Graph"]//StyleGraphs,{n,5}],Dividers->All,ImageSize->500]

Classes

GraphClassesThatAre["TriangularHoneycombAcuteKnight"]

Intersection@@(GraphData[#,"Classes"]&/@GraphData["TriangularHoneycombAcuteKnight"])

StringReplace[StringRiffle[GraphData[#,"Description"]&/@%,", "]," graph"->""]

Special cases

RecognizeGraph/@TriangularHoneycombAcuteKnightGraph/@Range[10]

With[{cl="TriangularHoneycombAcuteKnight"},TextGrid[Sort[{GraphData[#,"NotationRules",cl],GraphData[#,"Name"]}&/@DeleteCases[GraphData[cl],{cl,_}]],Dividers->All]]

Embeddings

<<MathWorld`Graphs`

{"TriangularHoneycombAcuteKnight", 3} == {"LadderRung", 3}

GraphData[{"TriangularHoneycombAcuteKnight",3},"StandardName"]

All

GraphData[{"TriangularHoneycombAcuteKnight",3},"Graph","All"]//StyleGraphs

Halin

GraphData[{"TriangularHoneycombAcuteKnight",3},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombAcuteKnight",3},"LCFSignature"]

GraphData[{"TriangularHoneycombAcuteKnight",3},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",3},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"TriangularHoneycombAcuteKnight",3} ],Debug->True]//Timing

GraphData[{"TriangularHoneycombAcuteKnight",3},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",3},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",3} ],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance construction

g1=Graph[Range[6],UnitEdges[v=CirclePoints[6]/2],VertexCoordinates->v]

RecognizeGraph[g]

g2=TriangularHoneycombAcuteKnightGraph[3]

UnitDistanceGraphEmbeddingQ[g2]

gs={g1,g2}

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",3},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",3} ],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData[{"TriangularHoneycombAcuteKnight",3},"Graph","Matchstick"]//StyleGraphs

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",3},{"RCN","CN"}]

GraphData[{"TriangularHoneycombAcuteKnight",3},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",3},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",3} ],"CrossingNumber"->-1]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",3},"Edges"]]

3D All

GraphData[{"TriangularHoneycombAcuteKnight",3},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",3},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"TriangularHoneycombAcuteKnight",3},"Graph",{"3D","Polyhedron"}]

{"TriangularHoneycombAcuteKnight", 4} == "2C3+K1,3"

GraphData[{"TriangularHoneycombAcuteKnight",4},"StandardName"]

All

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

Halin

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

LCF

GraphData[{"TriangularHoneycombAcuteKnight",4},"LCFSignature"]

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

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",4},"UnitDistance"]

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

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

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",4},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

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

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",4},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",4}],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

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

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",4},{"RCN","CN"}]

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

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",4},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",4}],"CrossingNumber"->-1]]

{,}

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

MinimalCrossing construction triangular grid

g0=GraphData[{"TriangularHoneycombAcuteKnight",4},"Graph","TriangularGrid"][[1]]

v0=GraphEmbedding[g0]

v={2#/(Sqrt[3]),#2}/2&@@@v0

g=Annotate[g0,{VertexCoordinates->v,VertexLabels->Automatic}]

EdgeLengths[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",4},"Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

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

{"TriangularHoneycombAcuteKnight", 5} == "ThreeDarts"

GraphData[{"TriangularHoneycombAcuteKnight",5},"StandardName"]

All

GraphData[{"TriangularHoneycombAcuteKnight",5},"Graph","All"]//StyleGraphs

Halin

GraphData[{"TriangularHoneycombAcuteKnight",5},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombAcuteKnight",5},"LCFSignature"]

GraphData[{"TriangularHoneycombAcuteKnight",5},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",5},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"TriangularHoneycombAcuteKnight",5}],Debug->True]//Timing

GraphData[{"TriangularHoneycombAcuteKnight",5},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",5},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",5}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",5},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",5}],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

gs={,,,,};

Matchstick

GraphData[{"TriangularHoneycombAcuteKnight",5},"Graph","Matchstick"]//StyleGraphs

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",5},{"RCN","CN"}]

GraphData[{"TriangularHoneycombAcuteKnight",5},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",5},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",5}],"CrossingNumber"->-1]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",5},"Edges"]]

3D All

GraphData[{"TriangularHoneycombAcuteKnight",5},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",5},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"TriangularHoneycombAcuteKnight",5},"Graph",{"3D","Polyhedron"}]

{"TriangularHoneycombAcuteKnight", 6}

GraphData[{"TriangularHoneycombAcuteKnight",6},"StandardName"]

All

GraphData[{"TriangularHoneycombAcuteKnight",6},"Graph","All"]//StyleGraphs

Halin

GraphData[{"TriangularHoneycombAcuteKnight",6},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombAcuteKnight",6},"LCFSignature"]

GraphData[{"TriangularHoneycombAcuteKnight",6},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",6},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"TriangularHoneycombAcuteKnight",6}],Debug->True]//Timing

GraphData[{"TriangularHoneycombAcuteKnight",6},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",6},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",6}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",6},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",6}],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

gs={,,};

Matchstick

GraphData[{"TriangularHoneycombAcuteKnight",6},"Graph","Matchstick"]//StyleGraphs

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",6},{"RCN","CN"}]

GraphData[{"TriangularHoneycombAcuteKnight",6},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",6},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",6}],"CrossingNumber"->-1]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",6},"Edges"]]

3D All

GraphData[{"TriangularHoneycombAcuteKnight",6},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",6},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"TriangularHoneycombAcuteKnight",6},"Graph",{"3D","Polyhedron"}]

{"TriangularHoneycombAcuteKnight", 7}

GraphData[{"TriangularHoneycombAcuteKnight",7},"StandardName"]

All

GraphData[{"TriangularHoneycombAcuteKnight",7},"Graph","All"]//StyleGraphs

Halin

GraphData[{"TriangularHoneycombAcuteKnight",7},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombAcuteKnight",7},"LCFSignature"]

GraphData[{"TriangularHoneycombAcuteKnight",7},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",7},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"TriangularHoneycombAcuteKnight",7}],Debug->True]//Timing

GraphData[{"TriangularHoneycombAcuteKnight",7},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",7},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",7}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",7},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",7}],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

gs={,,};

Matchstick

GraphData[{"TriangularHoneycombAcuteKnight",7},"Graph","Matchstick"]//StyleGraphs

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",7},{"RCN","CN"}]

GraphData[{"TriangularHoneycombAcuteKnight",7},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",7},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

cc=ConnectedGraphComponents@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",7}],"CrossingNumber"->-1]]

GraphPlot@cc[[1]]

cc1=First@ConnectedGraphComponents@GraphData[{"TriangularHoneycombAcuteKnight",7},"Graph","Matchstick"][[1]]

cc2=RotateGraph[cc[[2]],{12,9}]

g=MyGraphUnion@@(IndexGraph/@{cc1,cc2,cc2})

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",7},"Edges"]]

3D All

GraphData[{"TriangularHoneycombAcuteKnight",7},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",7},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"TriangularHoneycombAcuteKnight",7},"Graph",{"3D","Polyhedron"}]

{"TriangularHoneycombAcuteKnight", 8}

GraphData[{"TriangularHoneycombAcuteKnight",8},"StandardName"]

All

GraphData[{"TriangularHoneycombAcuteKnight",8},"Graph","All"]//StyleGraphs

Halin

GraphData[{"TriangularHoneycombAcuteKnight",8},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombAcuteKnight",8},"LCFSignature"]

GraphData[{"TriangularHoneycombAcuteKnight",8},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",8},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"TriangularHoneycombAcuteKnight",8}],Debug->True]//Timing

GraphData[{"TriangularHoneycombAcuteKnight",8},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",8},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",8}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",8},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",8}],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

gs={,,};

Matchstick

GraphData[{"TriangularHoneycombAcuteKnight",8},"Graph","Matchstick"]//StyleGraphs

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",8},{"RCN","CN"}]

GraphData[{"TriangularHoneycombAcuteKnight",8},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",8},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",8}],"CrossingNumber"->-1]]

cc=ConnectedGraphComponents@GraphData[{"TriangularHoneycombAcuteKnight",8}]

IsomorphicGraphQ@@cc

e[l__]:=Sequence@@Partition[{l},2,1]
s=Sqrt[3]/2;
v={{0,0},{s,-1/2},{s,1/2},{2s,0},{0,1},{0,2},{s,3/2},{0,3}};
v=Union[v,{-#,#2}&@@@v];
g1=Graph[Range[Length[v]],UnitEdges[v],VertexCoordinates->v,VertexLabels->Automatic]

g=MyGraphUnion[g1,g1,g1,"Scaling"->False,Offset->1]

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",8},"Edges"]]

3D All

GraphData[{"TriangularHoneycombAcuteKnight",8},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",8},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"TriangularHoneycombAcuteKnight",8},"Graph",{"3D","Polyhedron"}]

{"TriangularHoneycombAcuteKnight", 9}

GraphData[{"TriangularHoneycombAcuteKnight",9},"StandardName"]

All

GraphData[{"TriangularHoneycombAcuteKnight",9},"Graph","All"]//StyleGraphs

Halin

GraphData[{"TriangularHoneycombAcuteKnight",9},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombAcuteKnight",9},"LCFSignature"]

GraphData[{"TriangularHoneycombAcuteKnight",9},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",9},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"TriangularHoneycombAcuteKnight",9}],Debug->True]//Timing

GraphData[{"TriangularHoneycombAcuteKnight",9},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",9},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",9}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",9},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",9}],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

gs={,};

Matchstick

GraphData[{"TriangularHoneycombAcuteKnight",9},"Graph","Matchstick"]//StyleGraphs

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",9},{"RCN","CN"}]

GraphData[{"TriangularHoneycombAcuteKnight",9},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",9},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",9}],"CrossingNumber"->-1]]

g1=RotateGraph[First@ConnectedGraphComponents@GraphData[{"TriangularHoneycombAcuteKnight",9}],{36,3}]

s=Sqrt[3]/2;
v={{0,0},{2s,0},{s,1/2},{0,1},{2s,1},{0,2},{0,3},{0,4},{s,3/2},{s,5/2}};
v=Union[v,{-#,#2}&@@@v];
g1=Graph[Range[Length[v]],UnitEdges[v],VertexCoordinates->v,VertexLabels->Automatic]

g=MyGraphUnion[g1,g1,g1,Offset->1,"Scaling"->False]

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",9},"Edges"]]

3D All

GraphData[{"TriangularHoneycombAcuteKnight",9},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",9},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"TriangularHoneycombAcuteKnight",9},"Graph",{"3D","Polyhedron"}]

{"TriangularHoneycombAcuteKnight", 10}

GraphData[{"TriangularHoneycombAcuteKnight",10},"StandardName"]

All

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

Halin

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

LCF

GraphData[{"TriangularHoneycombAcuteKnight",10},"LCFSignature"]

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

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",10},"UnitDistance"]

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

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

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",10},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

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

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",10},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",10}],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

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

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",10},{"RCN","CN"}]

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

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",10},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",10}],"CrossingNumber"->-1]]

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",10},"Edges"]

cc=ConnectedGraphComponents@GraphData[{"TriangularHoneycombAcuteKnight",10},"Edges"]

RecognizeGraph/@cc

s=Sqrt[3]/2;
v={{0,0},{0,1},{0,2},{0,3},{0,4},{s,-1/2},{s,1/2},{s,3/2},{s,5/2},{2s,0},{2s,1},{3s,-1/2}};
v=Union[v,{-#,#2}&@@@v];
g1=Graph[Range[Length[v]],UnitEdges[v],VertexCoordinates->v,VertexLabels->Automatic]

s=Sqrt[3]/2;
v={{0,0},{0,1},{0,2},{0,3},{0,4},{s,1/2},{s,3/2},{s,5/2},{s,7/2},{2s,0},{2s,1},{2s,2},{3s,1/2},{-s,1/2},{-s,3/2},{-s,5/2},{-2s,0},{-2s,1}};
g2=Graph[Range[Length[v]],UnitEdges[v],VertexCoordinates->v,VertexLabels->Automatic]

g=MyGraphUnion[g1,g2,g2,Offset->1,"Scaling"->False]

RecognizeGraph[g]

RectilinearCrossingCount[g]

MinimalCrossing construction triangular grid

GraphData[{"TriangularHoneycombAcuteKnight",10}]

GraphEmbedding[%]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",10},"Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

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

{"TriangularHoneycombAcuteKnight", 11}

GraphData[{"TriangularHoneycombAcuteKnight",11},"StandardName"]

All

GraphData[{"TriangularHoneycombAcuteKnight",11},"Graph","All"]//StyleGraphs

Halin

GraphData[{"TriangularHoneycombAcuteKnight",11},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombAcuteKnight",11},"LCFSignature"]

GraphData[{"TriangularHoneycombAcuteKnight",11},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",11},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"TriangularHoneycombAcuteKnight",11} ],Debug->True]//Timing

GraphData[{"TriangularHoneycombAcuteKnight",11},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",11},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",11} ],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",11},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",11} ],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData[{"TriangularHoneycombAcuteKnight",11},"Matchstick"]

GraphData[{"TriangularHoneycombAcuteKnight",11},"Graph","Matchstick"]//StyleGraphs

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",11},{"RCN","CN"}]

GraphData[{"TriangularHoneycombAcuteKnight",11},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",11},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",11} ],"CrossingNumber"->-1]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",11},"Edges"]]

3D All

GraphData[{"TriangularHoneycombAcuteKnight",11},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",11},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"TriangularHoneycombAcuteKnight",11},"Graph",{"3D","Polyhedron"}]

{"TriangularHoneycombAcuteKnight", 12}

GraphData[{"TriangularHoneycombAcuteKnight",12},"StandardName"]

All

GraphData[{"TriangularHoneycombAcuteKnight",12},"Graph","All"]//StyleGraphs

Halin

GraphData[{"TriangularHoneycombAcuteKnight",12},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombAcuteKnight",12},"LCFSignature"]

GraphData[{"TriangularHoneycombAcuteKnight",12},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",12},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"TriangularHoneycombAcuteKnight",12} ],Debug->True]//Timing

GraphData[{"TriangularHoneycombAcuteKnight",12},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",12},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",12} ],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",12},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",12} ],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick construction

cc=ConnectedGraphComponents[GraphData[{"TriangularHoneycombAcuteKnight",12}]]

IsomorphicGraphQ@@cc

Table[cc[[1]],3]

c=IndexGraph[cc[[3]]]

g=MyGraphUnion[c,c,c,"Scaling"->False,Offset->1]

RecognizeGraph[g]

EmbeddingClasses[g,Debug->True]

Matchstick

GraphData[{"TriangularHoneycombAcuteKnight",12},"Matchstick"]

GraphData[{"TriangularHoneycombAcuteKnight",12},"Graph","Matchstick"]//StyleGraphs

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",12},{"RCN","CN"}]

GraphData[{"TriangularHoneycombAcuteKnight",12},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",12},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",12} ],"CrossingNumber"->-1]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",12},"Edges"]]

3D All

GraphData[{"TriangularHoneycombAcuteKnight",12},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",12},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"TriangularHoneycombAcuteKnight",12},"Graph",{"3D","Polyhedron"}]

{"TriangularHoneycombAcuteKnight", 13}

GraphData[{"TriangularHoneycombAcuteKnight",13},"StandardName"]

All

GraphData[{"TriangularHoneycombAcuteKnight",13},"Graph","All"]//StyleGraphs

Halin

GraphData[{"TriangularHoneycombAcuteKnight",13},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombAcuteKnight",13},"LCFSignature"]

GraphData[{"TriangularHoneycombAcuteKnight",13},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",13},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"TriangularHoneycombAcuteKnight",13} ],Debug->True]//Timing

GraphData[{"TriangularHoneycombAcuteKnight",13},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",13},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",13} ],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",13},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",13} ],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData[{"TriangularHoneycombAcuteKnight",13},"Graph","Matchstick"]//StyleGraphs

Matchstick construction

cc=ConnectedGraphComponents[GraphData[{"TriangularHoneycombAcuteKnight",13}]]

IsomorphicGraphQ@@cc

c1=IndexGraph[cc[[1]]]

c2=IndexGraph[cc[[2]]]

g=MyGraphUnion[c1,c2,c2,"Scaling"->False,Offset->1]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",13},{"RCN","CN"}]

GraphData[{"TriangularHoneycombAcuteKnight",13},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",13},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",13} ],"CrossingNumber"->-1]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",13},"Edges"]]

3D All

GraphData[{"TriangularHoneycombAcuteKnight",13},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",13},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"TriangularHoneycombAcuteKnight",13},"Graph",{"3D","Polyhedron"}]

{"TriangularHoneycombAcuteKnight",  14}

GraphData[{"TriangularHoneycombAcuteKnight",14},"StandardName"]

All

GraphData[{"TriangularHoneycombAcuteKnight",14},"Graph","All"]//StyleGraphs

Halin

GraphData[{"TriangularHoneycombAcuteKnight",14},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombAcuteKnight",14},"LCFSignature"]

GraphData[{"TriangularHoneycombAcuteKnight",14},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",14},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"TriangularHoneycombAcuteKnight",14} ],Debug->True]//Timing

GraphData[{"TriangularHoneycombAcuteKnight",14},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",14},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",14} ],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",14},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",14} ],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData[{"TriangularHoneycombAcuteKnight",14},"Graph","Matchstick"]//StyleGraphs

Matchstick construction

cc=ConnectedGraphComponents[GraphData[{"TriangularHoneycombAcuteKnight",14}]]

IsomorphicGraphQ@@cc

Table[cc[[1]],3]

c=IndexGraph[cc[[3]]]

g=MyGraphUnion[c,c,c,"Scaling"->False,Offset->1]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",14},{"RCN","CN"}]

GraphData[{"TriangularHoneycombAcuteKnight",14},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",14},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",14} ],"CrossingNumber"->-1]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",14},"Edges"]]

3D All

GraphData[{"TriangularHoneycombAcuteKnight",14},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",14},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"TriangularHoneycombAcuteKnight",14},"Graph",{"3D","Polyhedron"}]

{"TriangularHoneycombAcuteKnight", 15}

GraphData[{"TriangularHoneycombAcuteKnight",15},"StandardName"]

All

GraphData[{"TriangularHoneycombAcuteKnight",15},"Graph","All"]//StyleGraphs

Halin

GraphData[{"TriangularHoneycombAcuteKnight",15},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombAcuteKnight",15},"LCFSignature"]

GraphData[{"TriangularHoneycombAcuteKnight",15},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",15},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"TriangularHoneycombAcuteKnight",15} ],Debug->True]//Timing

GraphData[{"TriangularHoneycombAcuteKnight",15},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",15},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",15} ],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",15},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",15} ],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData[{"TriangularHoneycombAcuteKnight",15},"Graph","Matchstick"]//StyleGraphs

Matchstick construction

cc=ConnectedGraphComponents[GraphData[{"TriangularHoneycombAcuteKnight",15}]]

IsomorphicGraphQ@@cc

c=IndexGraph[cc[[3]]]

g=MyGraphUnion[c,c,c,"Scaling"->False,Offset->1]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",15},{"RCN","CN"}]

GraphData[{"TriangularHoneycombAcuteKnight",15},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",15},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",15} ],"CrossingNumber"->-1]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",15},"Edges"]]

3D All

GraphData[{"TriangularHoneycombAcuteKnight",15},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",15},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"TriangularHoneycombAcuteKnight",15},"Graph",{"3D","Polyhedron"}]

{"TriangularHoneycombAcuteKnight", 16}

GraphData[{"TriangularHoneycombAcuteKnight",16},"StandardName"]

All

GraphData[{"TriangularHoneycombAcuteKnight",16},"Graph","All"]//StyleGraphs

Halin

GraphData[{"TriangularHoneycombAcuteKnight",16},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombAcuteKnight",16},"LCFSignature"]

GraphData[{"TriangularHoneycombAcuteKnight",16},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",16},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"TriangularHoneycombAcuteKnight",16} ],Debug->True]//Timing

GraphData[{"TriangularHoneycombAcuteKnight",16},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",16},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",16} ],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",16},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",16} ],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData[{"TriangularHoneycombAcuteKnight",16},"Graph","Matchstick"]//StyleGraphs

Matchstick construction

cc=ConnectedGraphComponents[GraphData[{"TriangularHoneycombAcuteKnight",16}]]

IsomorphicGraphQ@@cc

c1=IndexGraph[cc[[1]]]

c2=IndexGraph[cc[[2]]]

g=MyGraphUnion[c1,c2,c2,"Scaling"->False,Offset->1]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",16},{"RCN","CN"}]

GraphData[{"TriangularHoneycombAcuteKnight",16},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",16},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",16} ],"CrossingNumber"->-1]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",16},"Edges"]]

3D All

GraphData[{"TriangularHoneycombAcuteKnight",16},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",16},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"TriangularHoneycombAcuteKnight",16},"Graph",{"3D","Polyhedron"}]

{"TriangularHoneycombAcuteKnight", 17}

GraphData[{"TriangularHoneycombAcuteKnight",17},"StandardName"]

All

GraphData[{"TriangularHoneycombAcuteKnight",17},"Graph","All"]//StyleGraphs

Halin

GraphData[{"TriangularHoneycombAcuteKnight",17},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombAcuteKnight",17},"LCFSignature"]

GraphData[{"TriangularHoneycombAcuteKnight",17},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",17},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"TriangularHoneycombAcuteKnight",17} ],Debug->True]//Timing

GraphData[{"TriangularHoneycombAcuteKnight",17},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",17},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",17} ],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",17},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",17} ],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData[{"TriangularHoneycombAcuteKnight",17},"Graph","Matchstick"]//StyleGraphs

Matchstick construction

cc=ConnectedGraphComponents[GraphData[{"TriangularHoneycombAcuteKnight",17}]]

IsomorphicGraphQ@@cc

c=IndexGraph[cc[[3]]]

g=MyGraphUnion[c,c,c,"Scaling"->False,Offset->1]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",17},{"RCN","CN"}]

GraphData[{"TriangularHoneycombAcuteKnight",17},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",17},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",17} ],"CrossingNumber"->-1]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",17},"Edges"]]

3D All

GraphData[{"TriangularHoneycombAcuteKnight",17},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",17},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"TriangularHoneycombAcuteKnight",17},"Graph",{"3D","Polyhedron"}]

{"TriangularHoneycombAcuteKnight", 18}

GraphData[{"TriangularHoneycombAcuteKnight",18},"StandardName"]

All

GraphData[{"TriangularHoneycombAcuteKnight",18},"Graph","All"]//StyleGraphs

Halin

GraphData[{"TriangularHoneycombAcuteKnight",18},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombAcuteKnight",18},"LCFSignature"]

GraphData[{"TriangularHoneycombAcuteKnight",18},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",18},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"TriangularHoneycombAcuteKnight",18} ],Debug->True]//Timing

GraphData[{"TriangularHoneycombAcuteKnight",18},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",18},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",18} ],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",18},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",18} ],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData[{"TriangularHoneycombAcuteKnight",18},"Graph","Matchstick"]//StyleGraphs

Matchstick construction

cc=ConnectedGraphComponents[GraphData[{"TriangularHoneycombAcuteKnight",18}]]

IsomorphicGraphQ@@cc

c=IndexGraph[cc[[3]]]

g=MyGraphUnion[c,c,c,"Scaling"->False,Offset->1]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",18},{"RCN","CN"}]

GraphData[{"TriangularHoneycombAcuteKnight",18},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",18},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",18} ],"CrossingNumber"->-1]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",18},"Edges"]]

3D All

GraphData[{"TriangularHoneycombAcuteKnight",18},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",18},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"TriangularHoneycombAcuteKnight",18},"Graph",{"3D","Polyhedron"}]

{"TriangularHoneycombAcuteKnight", 19}

GraphData[{"TriangularHoneycombAcuteKnight",19},"StandardName"]

All

GraphData[{"TriangularHoneycombAcuteKnight",19},"Graph","All"]//StyleGraphs

Halin

GraphData[{"TriangularHoneycombAcuteKnight",19},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombAcuteKnight",19},"LCFSignature"]

GraphData[{"TriangularHoneycombAcuteKnight",19},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",19},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"TriangularHoneycombAcuteKnight",19} ],Debug->True]//Timing

GraphData[{"TriangularHoneycombAcuteKnight",19},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",19},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",19} ],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",19},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",19} ],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData[{"TriangularHoneycombAcuteKnight",19},"Graph","Matchstick"]//StyleGraphs

Matchstick construction

cc=ConnectedGraphComponents[GraphData[{"TriangularHoneycombAcuteKnight",19}]]

IsomorphicGraphQ@@cc

c1=IndexGraph[cc[[1]]]

c2=IndexGraph[cc[[2]]]

g=MyGraphUnion[c1,c2,c2,"Scaling"->False,Offset->1]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",19},{"RCN","CN"}]

GraphData[{"TriangularHoneycombAcuteKnight",19},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",19},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",19} ],"CrossingNumber"->-1]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",19},"Edges"]]

3D All

GraphData[{"TriangularHoneycombAcuteKnight",19},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",19},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"TriangularHoneycombAcuteKnight",19},"Graph",{"3D","Polyhedron"}]

{"TriangularHoneycombAcuteKnight", 20}

GraphData[{"TriangularHoneycombAcuteKnight",20},"StandardName"]

All

GraphData[{"TriangularHoneycombAcuteKnight",20},"Graph","All"]//StyleGraphs

Halin

GraphData[{"TriangularHoneycombAcuteKnight",20},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombAcuteKnight",20},"LCFSignature"]

GraphData[{"TriangularHoneycombAcuteKnight",20},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",20},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"TriangularHoneycombAcuteKnight",20} ],Debug->True]//Timing

GraphData[{"TriangularHoneycombAcuteKnight",20},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"TriangularHoneycombAcuteKnight",20},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",20} ],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"TriangularHoneycombAcuteKnight",20},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"TriangularHoneycombAcuteKnight",20} ],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData[{"TriangularHoneycombAcuteKnight",20},"Graph","Matchstick"]//StyleGraphs

Matchstick construction

cc=ConnectedGraphComponents[GraphData[{"TriangularHoneycombAcuteKnight",20}]]

IsomorphicGraphQ@@cc

c=IndexGraph[cc[[3]]]

g=MyGraphUnion[c,c,c,"Scaling"->False,Offset->1]

RecognizeGraph[g]

MinimalCrossing

GraphData[{"TriangularHoneycombAcuteKnight",20},{"RCN","CN"}]

GraphData[{"TriangularHoneycombAcuteKnight",20},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"TriangularHoneycombAcuteKnight",20},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombAcuteKnight",20} ],"CrossingNumber"->-1]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"TriangularHoneycombAcuteKnight",20},"Edges"]]

3D All

GraphData[{"TriangularHoneycombAcuteKnight",20},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"TriangularHoneycombAcuteKnight",20},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"TriangularHoneycombAcuteKnight",20},"Graph",{"3D","Polyhedron"}]

Properties

Acyclic

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Acyclic"],{n,20}]

AdjacencyMatrixCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"AdjacencyMatrixCount"],{n,10}]

Table[((Binomial[n+1,2]!)/(Piecewise[{{1,n==1},{6,n==2},{432,n==4},{108,Mod[n,3]==1}},48])),{n,10}]

Arboricity

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Arboricity"],{n,20}]

ArcTransitive

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"ArcTransitive"],{n,20}]

ArcTransitivity

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"ArcTransitivity"],{n,20}]

AutomorphismCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"AutomorphismCount"],{n,20}]

seq[[1;;All;;3]]

seq[[2;;All;;3]]

seq[[3;;All;;3]]

Table[Piecewise[{{1,n==1},{6,n==2},{432,n==4},{108,Mod[n,3]==1}},48],{n,20}]

%==seq

BalabanIndex

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"BalabanIndex"],{n,20}]

Bandwidth

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Bandwidth"],{n,20}]

BridgeCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"BridgeCount"],{n,20}]

BurningNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"BurningNumber"],{n,20}]

CharacteristicPolynomial

(poly=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"CharacteristicPolynomial"][x],{n,k0=1,kmax=20}])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

ChordlessCycleCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"ChordlessCycleCount"],{n,20}]

FindSequenceFunction[DeleteMissing@seq,n]

FindSequenceFunction[Drop[DeleteMissing@seq,6],n]

ChromaticInvariant

Porbably 0 for n > 2

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"ChromaticInvariant"],{n,20}]

ChromaticNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"ChromaticNumber"],{n,20}]

ChromaticPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"ChromaticPolynomial"][x],{n,k0=1,kmax=20}])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

Circumference

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Circumference"],{n,20}]

FindLinearRecurrence[Drop[seq,3]]

FindSequenceFunction[Drop[seq,3],n-3]

FindSequenceFunction[Drop[seq,4],n-4]

Classes

Intersection@@(GraphData[#,"Classes"]&/@GraphData["TriangularHoneycombAcuteKnight"])

CliqueCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"CliqueCount"],{n,20}]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

Simplify[SeriesCoefficient[gf,{x,0,n}],n>0]

CliqueCoveringNumber

A287195

==Independence number

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"CliqueCoveringNumber"],{n,20}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0]

CliqueNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"CliqueNumber"],{n,20}]

CliquePolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"CliquePolynomial"][x],{n,k0=1,kmax=20}])//Column

FindSequenceFunction[#,n-3]&/@Transpose[Drop[CoefficientList[#,x]&/@poly,3]]//Factor

FullSimplify[1+(1/2) n (1+n)x+(3/2) (-2+n) (-1+n)(x^2)+(-3+n) (-2+n)(x^3)]

Table[Piecewise[{{1+x,n==1}},1+(1/2) n (1+n)x+(3/2) (-2+n) (-1+n)(x^2)+(-3+n) (-2+n)(x^3)],{n,20}]//Column

ComplementOddChordlessCycleCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"ComplementOddChordlessCycleCount"],{n,20}]

ConnectedComponentCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"ConnectedComponentCount"],{n,20}]

ConnectedComponents

GraphData[{"TriangularHoneycombAcuteKnight",4},"ConnectedComponents","Name"]

GraphData[{"TriangularHoneycombAcuteKnight",5},"ConnectedComponents","Name"]

GraphData[{"TriangularHoneycombAcuteKnight",6},"ConnectedComponents","Name"]

GraphData[{"TriangularHoneycombAcuteKnight",7},"ConnectedComponents","Name"]

GraphData[{"TriangularHoneycombAcuteKnight",8},"ConnectedComponents","Name"]

ConnectedDominationNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"ConnectedDominationNumber"],{n,20}]

FindSequenceFunction[seq,n]

ConnectedInducedSubgraphCount

A291594

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"ConnectedInducedSubgraphCount"],{n,20}]

CrossingNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"CrossingNumber"],{n,20}]

CycleCount

A289901

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"CycleCount"],{n,20}]

FindLinearRecurrence[DeleteMissing@seq]

FindLinearRecurrence[Drop[DeleteMissing@seq,3]]

FindSequenceFunction[DeleteMissing[seq],n]

FindSequenceFunction[DeleteMissing[Drop[seq,3]],n-3]

gf=(x^3) FindGeneratingFunction[Drop[DeleteMissing[seq],3],x]

CyclePolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"CyclePolynomial"][x],{n,k0=1,kmax=20}])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

Cyclic

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Cyclic"],{n,20}]

CyclomaticNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"CyclomaticNumber"],{n,20}]

FindSequenceFunction[seq,n]

gf=(x^4) FindGeneratingFunction[Drop[seq,3],x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0]

DetourIndex

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"DetourIndex"],{n,20}]

DetourMatrix

MatrixForm/@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"DetourMatrix"],{n,5}]

MatrixForm/@Table[SparseArray[{i_,j_}:>Abs[i-j],n],{n,5}]

DetourPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"DetourPolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x])

Diameter

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Diameter"],{n,20}]

Dimension

Inferred for n > 10:

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Dimension"],{n,20}]

DistanceMatrix

MatrixForm/@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"DistanceMatrix"],{n,5}]

MatrixForm/@Table[SparseArray[{i_,j_}:>Abs[i-j],n],{n,5}]

DistancePolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"DistancePolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x])

DistinguishingNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"DistinguishingNumber"],{n,20}]

DomaticNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"DomaticNumber"],{n,20}]

DominatingSetCount

A289902

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"DominatingSetCount"],{n,20}]

seq={1,1,27,441,9261,421875,47177249,10546683057};

FindLinearRecurrence[seq]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

DominationNumber

A289903

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"DominationNumber"],{n,20}]

(seq=Monitor[Table[DominationNumber[TriangularHoneycombAcuteKnightGraph[n]],{n,30}],n])//Timing

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

DominationPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"DominationPolynomial"][x],{n,k0=1,kmax=20}])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

EdgeChromaticNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"EdgeChromaticNumber"],{n,20}]

EdgeConnectivity

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"EdgeConnectivity"],{n,20}]

EdgeCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"EdgeCount"],{n,20}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

3Binomial[n-1,2]

EdgeCoverCount

A289904

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"EdgeCoverCount"],{n,20}]

FindSequenceFunction[DeleteMissing@seq,n]

gf=(x^3)FindGeneratingFunction[DeleteMissing@Drop[seq,2],x]

EdgeCoverNumber

A289906

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"EdgeCoverNumber"],{n,20}]

(seq=Monitor[Table[EdgeCoverNumber[TriangularHoneycombAcuteKnightGraph[n]],{n,20}],n])//Timing

FindSequenceFunction[seq,n]

gf=(x^3) FindGeneratingFunction[Drop[seq,2],x]

EdgeCoverPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"EdgeCoverPolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

EulerianCycleCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"EulerianCycleCount"],{n,20}]

FaceCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"FaceCount"],{n,20}]

FindSequenceFunction[seq,n]

gf=(x^4) FindGeneratingFunction[Drop[seq,3],x]

SeriesCoefficient[gf,{x,0,n}]

FlowPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"FlowPolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x])

FractionalChromaticNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"FractionalChromaticNumber"],{n,20}]

FractionalEdgeChromaticNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"FractionalEdgeChromaticNumber"],{n,20}]

Girth

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Girth"],{n,20}]

Gonality

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Gonality"],{n,20}]

HamiltonConnected

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"HamiltonConnected"],{n,20}]

HamiltonDecompositionCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"HamiltonDecompositionCount"],{n,20}]

Hamiltonian

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Hamiltonian"],{n,20}]

HamiltonianCycleCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"HamiltonianCycleCount"],{n,20}]

HamiltonianNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"HamiltonianNumber"],{n,20}]

HamiltonianPathCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"HamiltonianPathCount"],{n,20}]

HamiltonianWalkCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"HamiltonianWalkCount"],{n,20}]

HararyIndex

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"HararyIndex"],{n,20}]

(seq=Monitor[Table[HararyIndex[TriangularHoneycombAcuteKnightGraph[n]],{n,20}],n])//Timing

FindSequenceFunction[seq,n]

FindSequenceFunction[Drop[seq,2],n-2]

HexagonCount

A152773

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"HexagonCount"],{n,20}]

FindLinearRecurrence[Drop[seq,5]]

FindSequenceFunction[seq,n]

FindSequenceFunction[Drop[seq,5],n-5]//Factor

Table[Piecewise[{{0,n<5}},(3/2) (n-5) (5 (n-5)-3)],{n,20}]

gf=(x^6)FindGeneratingFunction[Drop[seq,5],x]

Series[gf,{x,0,20}]

CoefficientList[Series[-((3 (x^5) (1+4 x))/(((-1+x))^3)),{x,0,20}],x]

IdiosyncraticPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"IdiosyncraticPolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x,y])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

IndependenceNumber

A287195

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"IndependenceNumber"],{n,20}]

FindLinearRecurrence[seq]

LinearRecurrence[{1,0,2,-2,0,-1,1},{1,3,3,5,9,9,12},50]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0]

Table[(1/18) ((n+3) (3n+2)-2 (n+3) Cos[((2 n π)/3)]-2 (Sqrt[3]) (n+1) Sin[((2 n π)/3)]),{n,50}]

seq=Table[(1/18) ((n+3) (3n+2)-2 (n+3) Cos[((2 n π)/3)]-2 (Sqrt[3]) (n+1) Sin[((2 n π)/3)]),{n,1,50,3}]

FindSequenceFunction[%,(n-1)/3+1]//Simplify

Table[(1/18) ((n+3) (3n+2)-2 (n+3) Cos[((2 n π)/3)]-2 (Sqrt[3]) (n+1) Sin[((2 n π)/3)]),{n,2,50,3}]

FindSequenceFunction[%,(n-2)/3+1]//FullSimplify

seq=Table[(1/18) ((n+3) (3n+2)-2 (n+3) Cos[((2 n π)/3)]-2 (Sqrt[3]) (n+1) Sin[((2 n π)/3)]),{n,3,50,3}]

FindSequenceFunction[seq,n/3]//FullSimplify

Table[Piecewise[{{n(n+3),Mod[n,3]==0},{(n+1) (n+2),Mod[n,3]==1},{(n+1) (n+4),Mod[n,3]==2}}]/6,{n,50}]

IndependencePolynomial

A244500

(poly=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"IndependencePolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x])//Column

Exponent[#,x]&/@DeleteMissing[poly]

CoefficientList[Take[poly,10],x]//Column

poly=Import["https://oeis.org/A244500/b244500.txt","Table"][[All,2]]

coef=FindLinearRecurrence[DeleteMissing[poly]]

coef=FindLinearRecurrence[Drop[DeleteMissing[poly],2]]

coef=FindPolynomialRecurrence[DeleteMissing[poly],6]

coef=FindPolynomialRecurrence[DeleteMissing@Drop[poly,2]

coef=Factor/@FindLinearRecurrence[Drop[DeleteMissing[poly],2]]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

IndependenceRatio

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"IndependenceRatio"],{n,20}]

Table[(1/18) ((((n+3) (3n+2)-2 (n+3) Cos[((2 n π)/3)]-2 (Sqrt[3]) (n+1) Sin[((2 n π)/3)]))/(Binomial[n+1,2])),{n,20}]

FindSequenceFunction[seq,n]

IndependentEdgeSetCount

A287230

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"IndependentEdgeSetCount"],{n,20}]

FindLinearRecurrence[DeleteMissing@seq]

FindLinearRecurrence[DeleteMissing@Rest@seq]

FindSequenceFunction[DeleteMissing@seq,n]

FindSequenceFunction[Rest@DeleteMissing@seq,n-1]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

IndependentVertexSetCount

A287204

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"IndependentVertexSetCount"],{n,20}]

FindLinearRecurrence[DeleteMissing[seq]]

FindLinearRecurrence[Rest@DeleteMissing[seq]]

FindSequenceFunction[DeleteMissing[seq],n]

FindSequenceFunction[DeleteMissing@Rest@seq,n-1]

gf=x FindGeneratingFunction[DeleteMissing[seq],x]

IntersectionNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"IntersectionNumber"],{n,20}]

KirchhoffIndex

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"KirchhoffIndex"],{n,20}]

FindSequenceFunction[seq,n]

KirchhoffSumIndex

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"KirchhoffSumIndex"],{n,20}]

LaplacianMatrix

MatrixForm/@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"LaplacianMatrix"],{n,20}]

LaplacianPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"LaplacianPolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

LeafCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"LeafCount"],{n,20}]

LongestCycleCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"LongestCycleCount"],{n,20}]

FindSequenceFunction[DeleteMissing@Drop[seq,6],n-6]

FindSequenceFunction[DeleteMissing@Drop[seq,7],n-7]

LovaszNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"LovaszNumber"],{n,20}]

MatchingGeneratingPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MatchingGeneratingPolynomial"][x],{n,k0=1,kmax=20}])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

MatchingNumber

A289143

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MatchingNumber"],{n,20}]

Do[Print[{n,MatchingNumber[TriangularHoneycombAcuteKnightGraph[n]]//Timing}],{n,40}]

(seq=Monitor[Table[MatchingNumber[TriangularHoneycombAcuteKnightGraph[n]],{n,50}],n])//Timing

FindLinearRecurrence[seq]

FindLinearRecurrence[Drop[seq,3]]

FindSequenceFunction[seq,n]

FindSequenceFunction[Drop[seq,13],n-13]//FullSimplify

FindSequenceFunction[Drop[seq,2],n-2]//FullSimplify

Clear[a]

Do[a[i]=seq[[i+2]],{i,11}]

-36-11 \[FormalN]-(\[FormalN]^2)+2 \[FormalY][\[FormalN]]+2 \[FormalY][6+\[FormalN]]==0/.\[FormalN]->-2+n/.n->n-4//Simplify

a[n_]:=a[n]=(n^2-n+6-2a[n-6])/2

seq=Array[a,70]

Table[a[i]==seq[[i]],{i,13}]

RecurrenceTable[{a[n]==(n^2-n+6-2a[n-6])/2,a[1]==0,a[2]==0,a[3]==3,a[4]==3,a[5]==6,a[6]==9,a[7]==12,a[8]==18,a[9]==21,a[10]==27,a[11]==33,a[12]==39,a[13]==45},a,{n,50}]

gf=(x^3) FindGeneratingFunction[seq,x]

gf=(x^14) FindGeneratingFunction[Drop[seq,11],x]

Series[gf,{x,0,20}]

gf1=Sum[seq[[i]](x^(i+2)),{i,11}]+gf

Factor[gf1]

Series[gf1,{x,0,20}]

FullSimplify[SeriesCoefficient[gf1,{x,0,n}],n∈Integers&&n>0]

FullSimplify[Table[%50,{n,72}]-Join[{0,0},seq]]

Join[{0,0},seq]

seq2=Table[Piecewise[{{3,n==4},{12,n==7}},(1/12) (3 (-3+n+(n^2))+5 Cos[((n π)/2)]+4 (((-1))^n) (Cos[((n π)/3)]+(Sqrt[3]) Sin[((n π)/3)]) (Cos[((n π)/2)]-Sin[((n π)/2)])-5 Sin[((n π)/2)])],{n,72}]

seq2-Join[{0,0},seq]//Union

Table[Piecewise[{{3,n==4},{12,n==7}},(3 (n^2+n-3)+5 Cos[n Pi/2]+4 (-1)^n (Cos[n Pi/3]+Sqrt[3] Sin[n Pi/3]) (Cos[n Pi/2]-Sin[n Pi/2])-5 Sin[n Pi/2])/12],{n,50}]

Series[x^3*(3-6*x+6*x^2-3*x^3+3*x^5-3*x^6+3*x^8-3*x^9+3*x^11-3*x^12+x^13)/((1-x)^3*(1+x^2)*(1-x^2+x^4)),{x,0,30}]

MatchingPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MatchingPolynomial"][x],{n,k0=1,kmax=20}])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

Matchstick

Inferred for n > 10:

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Matchstick"],{n,20}]

MaximalCliqueCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MaximalCliqueCount"],{n,20}]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>1]//Factor

MaximalIndependentEdgeSetCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MaximalIndependentEdgeSetCount"],{n,20}]

FindLinearRecurrence[DeleteMissing@Drop[seq,2]]

FindSequenceFunction[DeleteMissing@seq,n]

gf=(x^3)FindGeneratingFunction[DeleteMissing@Drop[seq,2],x]

MaximalIndependentVertexSetCount

A290614

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MaximalIndependentVertexSetCount"],{n,20}]

FindLinearRecurrence[DeleteMissing@Drop[seq,2]]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

MaximumCliqueCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MaximumCliqueCount"],{n,20}]

FindSequenceFunction[seq,n]

gf=FindGeneratingFunction[seq,x]//Factor

SeriesCoefficient[gf,{x,0,n}]//FullSimplify

MaximumIndependentEdgeSetCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MaximumIndependentEdgeSetCount"],{n,20}]

seq[[1;;All;;2]]

FindLinearRecurrence[DeleteMissing@seq]

FindLinearRecurrence[DeleteMissing@Drop[seq,3]]

FindSequenceFunction[DeleteMissing@seq,n]

seq[[2;;All;;2]]

FindSequenceFunction[DeleteMissing@Drop[%,1],n]

Table[Length@FindAllMaximumIndependentEdgeSets[TriangularHoneycombAcuteKnightGraph[n]],{n,8}]//Timing

FindLinearRecurrence[DeleteMissing@Drop[seq,3]]

FindSequenceFunction[DeleteMissing@Drop[seq,3],n-4]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

MaximumIndependentVertexSetCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MaximumIndependentVertexSetCount"],{n,20}]

seq[[1;;All;;3]]

seq[[2;;All;;3]]

seq[[3;;All;;3]]

MaximumLeafNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MaximumLeafNumber"],{n,20}]

MeanDistance

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MeanDistance"],{n,20}]

MinimalDominatingSetCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MinimalDominatingSetCount"],{n,20}]

MinimalDominatingSetSignature

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MinimalDominatingSetSignature"],{n,20}]//Column

MinimumCliqueCoveringCount

Probably 1

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MinimumCliqueCoveringCount"],{n,20}]

MinimumDistinguishingLabelingCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MinimumDistinguishingLabelingCount"],{n,20}]

FindSequenceFunction[DeleteMissing@seq,n]

MinimumDominatingSetCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MinimumDominatingSetCount"],{n,20}]

MinimumEdgeCoverCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MinimumEdgeCoverCount"],{n,20}]

FindSequenceFunction[DeleteMissing@Drop[seq,2],n-2]

Table[(3^n),{n,10}]

MolecularTopologicalIndex

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MolecularTopologicalIndex"],{n,20}]

Nonhamiltonian

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Nonhamiltonian"],{n,20}]

NormalizedLaplacianMatrix

MatrixForm/@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"NormalizedLaplacianMatrix"],{n,4}]

OddChordlessCycleCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"OddChordlessCycleCount"],{n,20}]

PathCount

A290331

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"PathCount"],{n,20}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

gf=(x^3) FindGeneratingFunction[DeleteCases[seq,0|_Missing],x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0]

PathPolynomial

(poly=Factor[Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"PathPolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[DeleteCases[poly,0|_Missing]]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

PentagonCount

A028896

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"PentagonCount"],{n,20}]

FindLinearRecurrence[Drop[seq,3]]

LinearRecurrence[{3,-3,1},{0,6,18},20]

Table[3n(n+1),{n,20}]

6PolygonalNumber[Range[0,20]]

Perfect

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Perfect"],{n,20}]

ProjectivePlaneCrossingNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"ProjectivePlaneCrossingNumber"],{n,20}]

Radius

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Radius"],{n,20}]

RankPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"RankPolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x,y])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

RectilinearCrossingNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"RectilinearCrossingNumber"],{n,20}]

ReliabilityPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"ReliabilityPolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

SigmaPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"SigmaPolynomial"][x],{n,k0=1,kmax=20}])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

SpanningTreeCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"SpanningTreeCount"],{n,20}]

SquareCount

A045943

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"SquareCount"],{n,20}]

Table[3n(n+1)/2,{n,20}]

FindLinearRecurrence[seq]

FindLinearRecurrence[Drop[seq,2]]

LinearRecurrence[{3,-3,1},{0,0,3},20]

gf=(x^5) FindGeneratingFunction[Drop[seq,4],x]

Series[gf,{x,0,10}]

StabilityIndex

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"StabilityIndex"],{n,20}]

FindLinearRecurrence[Drop[seq,2]]

FindSequenceFunction[seq,n]

FindSequenceFunction[Drop[seq,2],n-2]

gf=x FindGeneratingFunction[seq,x]

TopologicalIndex

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"TopologicalIndex"],{n,20}]

ToroidalCrossingNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"ToroidalCrossingNumber"],{n,20}]

Traceable

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Traceable"],{n,20}]

TriangleCount

A002378

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"TriangleCount"],{n,20}]

FindLinearRecurrence[seq]

FindLinearRecurrence[Drop[seq,1]]

LinearRecurrence[{3,-3,1},{2,6,12},{0,20}]

Table[n(n+1),{n,-1,10}]

TuttePolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"TuttePolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x,y])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

UnitDistance

Inferred for n > 10:

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"UnitDistance"],{n,20}]

Untraceable

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"Untraceable"],{n,20}]

VertexConnectivity

Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"VertexConnectivity"],{n,20}]

VertexCount

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"VertexCount"],{n,20}]

FindSequenceFunction[seq,n]

Binomial[n+1,2]

VertexCoverNumber

Related to but not the same as IndependenceNumber

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"VertexCoverNumber"],{n,20}]

Table[(1/2) n (1+n)-(1/18) ((3+n) (2+3 n)-2 (3+n) Cos[((2 n π)/3)]-2 (Sqrt[3]) (1+n) Sin[((2 n π)/3)]),{n,20}]

FullSimplify[(1/2) n (1+n)-(1/18) ((3+n) (2+3 n)-2 (3+n) Cos[((2 n π)/3)]-2 (Sqrt[3]) (1+n) Sin[((2 n π)/3)])]

FindLinearRecurrence[seq]

seq=LinearRecurrence[{1,0,2,-2,0,-1,1},{0,0,3,5,6,12,16},40]

FullSimplify[RSolve[{a[n]==a[n-1]+2a[n-3]-2a[n-4]-a[n-6]+a[n-7],a[1]==0,a[2]==0,a[3]==3,a[4]==5,a[5]==6,a[6]==12,a[7]==16},a[n],n],n∈Integers&&n>0]

Table[(((E^((2 I n π)/3)) (114 I+132 (Sqrt[3])-9 I n+85 (Sqrt[3]) n)+(E^(-(2/3) I n π)) (255 I+9 (Sqrt[3])+(132 I-38 (Sqrt[3])) n)+(123 I+47 (Sqrt[3])) (-3+n (-1+3 n)))/(9 (123 I+47 (Sqrt[3])))),{n,10}]//N//Chop

FullSimplify[(((E^((2 I n π)/3)) (114 I+132 (Sqrt[3])-9 I n+85 (Sqrt[3]) n)+(E^(-(2/3) I n π)) (255 I+9 (Sqrt[3])+(132 I-38 (Sqrt[3])) n)+(123 I+47 (Sqrt[3])) (-3+n (-1+3 n)))/(9 (123 I+47 (Sqrt[3]))))]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0]

VertexCoverPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"VertexCoverPolynomial"][x],{n,k0=1,kmax=20}])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

VertexDegrees

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"VertexDegrees"],{n,10}]

WienerIndex

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"WienerIndex"],{n,20}]

FindSequenceFunction[seq,n]

WienerSumIndex

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"WienerSumIndex"],{n,20}]

GraphData

GraphDataClassString

GraphDataClassString["TriangularHoneycombAcuteKnight"]