Triangular Honeycomb Obtuse Knight Graph

Author

Eric W. Weisstein
December 17, 2024

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

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

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

Source

Wagon (2014)

Figure

4

g=TriangularHoneycombObtuseKnightGraph[4]

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\nobtuse knight graph",
LabelStyle->Directive[Black,FontFamily->"Times",14,Italic]]

5

g=TriangularHoneycombObtuseKnightGraph[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\nobtuse knight graph",
LabelStyle->Directive[Black,FontFamily->"Times",14,Italic]]

Combined

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

Layouts

Graphs.m

GraphicsRow[StyleGraphs@Table[StyleGraphs[TriangularHoneycombObtuseKnightGraph[n]],{n,5}],Dividers->All,ImageSize->500]

Curved (doubled) edges

GraphicsRow[StyleGraphs@Table[TriangularHoneycombObtuseKnightGraph[n]/.g_Graph:>AdjacencyGraph[AdjacencyMatrix[g],EdgeShapeFunction->"Line",DirectedEdges->True,VertexCoordinates->GraphEmbedding[g]],{n,5}],Dividers->All,ImageSize->500]

Edges

GraphicsRow[StyleGraphs@Table[Graph@@Through[{VertexList,EdgeList}[TriangularHoneycombObtuseKnightGraph[n]]],{n,5}],Dividers->All,ImageSize->500]

GraphData default

Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"#"],{n,6}]

GraphicsRow[StyleGraphs@Table[GraphData[{"TriangularHoneycombObtuseKnight",n}],{n,6}],Dividers->All,ImageSize->500]

GraphData "TriangularGrid"

GraphicsRow[StyleGraphs@Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"Graph","TriangularGrid"][[1]],{n,6}],Dividers->All,ImageSize->500]

Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"Graph","TriangularGrid"][[1]],{n,20}]

GraphData non-degenerate

GraphicsRow[Table[StyleGraphs[
Pick[
GraphData[{"TriangularHoneycombObtuseKnight",n},"Graph","All"],
FreeQ[#,"Degenerate"]&/@GraphData[{"TriangularHoneycombObtuseKnight",n},"EmbeddingClasses"]
][[1]]],{n,6}],Dividers->All,ImageSize->500]

Classes

GraphClassesThatAre["TriangularHoneycombObtuseKnight"]

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

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

Embeddings

<<MathWorld`Graphs`

{"TriangularHoneycombObtuseKnight", 1} == "SingletonGraph"

Names

GraphData[{"TriangularHoneycombObtuseKnight",1},"StandardName"]

GraphData[{"TriangularHoneycombObtuseKnight",1},"StandardNames"]

{"TriangularHoneycombObtuseKnight", 2} == {"Empty", 3}

Names

GraphData[{"TriangularHoneycombObtuseKnight",2},"StandardName"]

GraphData[{"TriangularHoneycombObtuseKnight",2},"StandardNames"]

{"TriangularHoneycombObtuseKnight", 3} == {"Empty", 6}

Names

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

GraphData[{"TriangularHoneycombObtuseKnight",3},"StandardNames"]

{"TriangularHoneycombObtuseKnight", 4} == "3P3+K1"

Names

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

GraphData[{"TriangularHoneycombObtuseKnight",4},"StandardNames"]

{"TriangularHoneycombObtuseKnight", 5}

Names

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

GraphData[{"TriangularHoneycombObtuseKnight",5},"StandardNames"]

Primary

GraphData[{"TriangularHoneycombObtuseKnight",5}]//StyleGraphs[#,ImageSize->Small]&

All

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

Inexact

Select[GraphData[{"TriangularHoneycombObtuseKnight",5},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"TriangularHoneycombObtuseKnight",5},"EdgeList"]]

Default embedding

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

GraphData[{"TriangularHoneycombObtuseKnight",5},"ConnectedComponents"]

g=MyGraphUnion@@{RotateGraph[,Pi/2],}

RecognizeGraph[g]

Degenerate

GraphData[{"TriangularHoneycombObtuseKnight",5},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"TriangularHoneycombObtuseKnight",5},"Graph","Degenerate"]

Configuration

GraphData[{"TriangularHoneycombObtuseKnight",5},"Configuration"]

GraphData[{"TriangularHoneycombObtuseKnight",5},"Graph","Configuration"]//StyleGraphs

Perspective

GraphData[{"TriangularHoneycombObtuseKnight",5},"Graph","Perspective"]//StyleGraphs

HoneycombToroidal construction

GraphData[{"TriangularHoneycombObtuseKnight",5},"NotationRules","HoneycombToroidal"]

GeneralizedPetersen

GraphData[{"TriangularHoneycombObtuseKnight",5},"Graph","GeneralizedPetersen"]//StyleGraphs

IGraph

GraphData[{"TriangularHoneycombObtuseKnight",5},"Graph","IGraph"]//StyleGraphs

Bilateral

GraphData[{"TriangularHoneycombObtuseKnight",5},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"TriangularHoneycombObtuseKnight",5},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"TriangularHoneycombObtuseKnight",5},"Planar"]

GraphData[{"TriangularHoneycombObtuseKnight",5},"Graph","Planar"]//StyleGraphs

Planar construction

PlanarGraphEmbedding[GraphData[{"TriangularHoneycombObtuseKnight",5}],Method->#]&/@{"TutteLayout","DavidsonHarelLayout"}

Table[PlanarGraphEmbedding[GraphData[{"TriangularHoneycombObtuseKnight",5}]],10]

PlanarGraphEmbeddings

(plemb=PlanarGraphEmbeddings[GraphData[{"TriangularHoneycombObtuseKnight",5}]])//Timing

Length[plemb]

MinimalCrossing

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

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

MinimalLocalCrossing

GraphData[{"TriangularHoneycombObtuseKnight",5},"LocalCrossingNumber"]

GraphData[{"TriangularHoneycombObtuseKnight",5},"Graph","MinimalLocalCrossing"]//StyleGraphs

Circulant

GraphData[{"TriangularHoneycombObtuseKnight",5},"Circulant"]

GraphData[{"TriangularHoneycombObtuseKnight",5},"Graph","Circulant"]//StyleGraphs

Complement graph

compl=GraphData[{"TriangularHoneycombObtuseKnight",5},"ComplementGraph","Name"]

Line graph

GraphData[{"TriangularHoneycombObtuseKnight",5},"Line"]

LineGraphQ[GraphData[{"TriangularHoneycombObtuseKnight",5}]]//Timing

RootGraph

roots=Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"TriangularHoneycombObtuseKnight",5}&]

RootGraph[GraphData[{"TriangularHoneycombObtuseKnight",5}]]

RecognizeGraph[%]

Halin

GraphData[{"TriangularHoneycombObtuseKnight",5},"Halin"]

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

Rigid

GraphData[{"TriangularHoneycombObtuseKnight",5},"Rigid"]

GraphData[{"TriangularHoneycombObtuseKnight",5},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"TriangularHoneycombObtuseKnight",5},"Flexible"]

GraphData[{"TriangularHoneycombObtuseKnight",5},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombObtuseKnight",5},"LCF"]

GraphData[{"TriangularHoneycombObtuseKnight",5},"HamiltonianCycleCount"]

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

GraphData[{"TriangularHoneycombObtuseKnight",5},"LCFNotations"]

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"TriangularHoneycombObtuseKnight",5},"VertexList"],GraphData[{"TriangularHoneycombObtuseKnight",5},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"TriangularHoneycombObtuseKnight",5},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombObtuseKnight",5}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData[{"TriangularHoneycombObtuseKnight",5},"Matchstick"]

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData[{"TriangularHoneycombObtuseKnight",5},"Polyhedral"]

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

{"TriangularHoneycombObtuseKnight", 6}

Names

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

GraphData[{"TriangularHoneycombObtuseKnight",6},"StandardNames"]

Primary

GraphData[{"TriangularHoneycombObtuseKnight",6}]//StyleGraphs[#,ImageSize->Small]&

All

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

Inexact

Select[GraphData[{"TriangularHoneycombObtuseKnight",6},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"TriangularHoneycombObtuseKnight",6},"EdgeList"]]

Default embedding

GraphPlot@RotateGraph[Graph@GraphData[{"TriangularHoneycombObtuseKnight",6},"Edges"],-2.45]

v=UnitCirclePoints[8,-Pi/2];
v=Join[v,Subdivide[v[[7]],v[[3]],7][[2;;-2]],Subdivide[v[[8]],v[[2]],4][[2;;-2]]];
v=Join[v,Subdivide[{v[[10,1]],1/2},{v[[13,1]],1/2},3]];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[{1,8},1,17,13,21,20,5,19,18,10,15,1],e[7,16,3,14,13,12,19],e[7,9,10,11,20],{6,20},{4,19},e[14,21,11],e[9,18,12],{11,15},{17,12}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"TriangularHoneycombObtuseKnight",6},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"TriangularHoneycombObtuseKnight",6},"Graph","Degenerate"]

Configuration

GraphData[{"TriangularHoneycombObtuseKnight",6},"Configuration"]

GraphData[{"TriangularHoneycombObtuseKnight",6},"Graph","Configuration"]//StyleGraphs

Perspective

GraphData[{"TriangularHoneycombObtuseKnight",6},"Graph","Perspective"]//StyleGraphs

HoneycombToroidal construction

GraphData[{"TriangularHoneycombObtuseKnight",6},"NotationRules","HoneycombToroidal"]

GeneralizedPetersen

GraphData[{"TriangularHoneycombObtuseKnight",6},"Graph","GeneralizedPetersen"]//StyleGraphs

IGraph

GraphData[{"TriangularHoneycombObtuseKnight",6},"Graph","IGraph"]//StyleGraphs

Bilateral

GraphData[{"TriangularHoneycombObtuseKnight",6},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"TriangularHoneycombObtuseKnight",6},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"TriangularHoneycombObtuseKnight",6},"Planar"]

GraphData[{"TriangularHoneycombObtuseKnight",6},"Graph","Planar"]//StyleGraphs

MinimalCrossing

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

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

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"TriangularHoneycombObtuseKnight",6}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"TriangularHoneycombObtuseKnight",6}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"TriangularHoneycombObtuseKnight",6}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"TriangularHoneycombObtuseKnight",6}],TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"TriangularHoneycombObtuseKnight",6}],TimeConstraint->5 3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"TriangularHoneycombObtuseKnight",6}],TimeConstraint->86400])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"TriangularHoneycombObtuseKnight",6}],"CrossingNumber"->cn])//PolyhedralGraphQ

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombObtuseKnight",6}],"CrossingNumber"->cn,RandomSeed->n]],{n,500}],Max[OuterFace[#]]<=GraphData[{"TriangularHoneycombObtuseKnight",6},"V"]&]]//Timing

GraphPlot@RotateGraph[,-2Pi/6]

Table[MinimalCrossingGraph[GraphData[{"TriangularHoneycombObtuseKnight",6}],"CrossingNumber"->cn,RandomSeed->i],{i,500}]//Union//Timing

v={#/8,#2/6}&@@@{{0,0},{4,0},{8,0},{3,1},{5,1},{1,2},{2,2},{3,2},{4,2},{5,2},{6,2},{7,2},{2,3},{6,3},{1,4},{7,4},{1,5},{7,5},{0,6},{4,6},{8,6}};
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,3,21,20,19,1,5,9,4,3,12,11,10,9,8,7,6,1],e[6,15,17,20,18,16,12],e[17,14,11,16,13,7,15,14,10],e[8,13,18],{4,8},{5,10}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

LocalCrossingCount@g

GraphData[{"TriangularHoneycombObtuseKnight",6},"V"]

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombObtuseKnight",6}],"CrossingNumber"->5],Pi]

v={{3,0},{-1,1},{1,1},{3,1},{6,1},{9,1},{1,2},{2,2},{3,2},{1,3},{3,3},{4,3},{5,3},{6,3},{3,4},{5,4},{2+1/4,5},{3,5},{2+1/2,6},{3,6},{3,8}};
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[9,8,7,2,3,4,5,6,1,2,17,18,19,21,10],e[7,10,8,11,12,13,16,14,21],e[4,9,11,15,18,20,6],{4,14},{19,20},{21,16},e[15,12,16],{6,13},{10,19},{15,17}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

MinimalLocalCrossing

GraphData[{"TriangularHoneycombObtuseKnight",6},"LocalCrossingNumber"]

GraphData[{"TriangularHoneycombObtuseKnight",6},"Graph","MinimalLocalCrossing"]//StyleGraphs

MinimalLocalCrossing construction

AdjacentCrossingsQ[g_Graph,cn_Integer]:=Module[{a=AdjacencyList[g],n=VertexCount[g]-cn},Max[Take[a,-cn]]>n]

(cn=CrossingNumber[GraphData[{"TriangularHoneycombObtuseKnight",6}],TimeConstraint->1])//AbsoluteTiming

Select[Table[MinimalCrossingGraph[GraphData[{"TriangularHoneycombObtuseKnight",6}],"CrossingNumber"->cn,RandomSeed->i],{i,20000}],!AdjacentCrossingsQ[#,cn]&]//Union//Timing

Select[Table[MinimalCrossingGraph[GraphData[{"TriangularHoneycombObtuseKnight",6}],"CrossingNumber"->(cn=6),RandomSeed->i],{i,20000}],!AdjacentCrossingsQ[#,cn]&]//Union//Timing

Select[Table[MinimalCrossingGraph[GraphData[{"TriangularHoneycombObtuseKnight",6}],"CrossingNumber"->(cn=7),RandomSeed->i],{i,20000}],!AdjacentCrossingsQ[#,cn]&]//Union//Timing

Select[Table[MinimalCrossingGraph[GraphData[{"TriangularHoneycombObtuseKnight",6}],"CrossingNumber"->(cn=8),RandomSeed->i],{i,20000}],!AdjacentCrossingsQ[#,cn]&]//Union//Timing

Circulant

GraphData[{"TriangularHoneycombObtuseKnight",6},"Circulant"]

GraphData[{"TriangularHoneycombObtuseKnight",6},"Graph","Circulant"]//StyleGraphs

Complement graph

compl=GraphData[{"TriangularHoneycombObtuseKnight",6},"ComplementGraph","Name"]

Line graph

GraphData[{"TriangularHoneycombObtuseKnight",6},"Line"]

LineGraphQ[GraphData[{"TriangularHoneycombObtuseKnight",6}]]//Timing

RootGraph

roots=Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"TriangularHoneycombObtuseKnight",6}&]

RootGraph[GraphData[{"TriangularHoneycombObtuseKnight",6}]]

RecognizeGraph[%]

Halin

GraphData[{"TriangularHoneycombObtuseKnight",6},"Halin"]

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

Rigid

GraphData[{"TriangularHoneycombObtuseKnight",6},"Rigid"]

GraphData[{"TriangularHoneycombObtuseKnight",6},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"TriangularHoneycombObtuseKnight",6},"Flexible"]

GraphData[{"TriangularHoneycombObtuseKnight",6},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombObtuseKnight",6},"LCF"]

GraphData[{"TriangularHoneycombObtuseKnight",6},"HamiltonianCycleCount"]

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

GraphData[{"TriangularHoneycombObtuseKnight",6},"LCFNotations"]

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"TriangularHoneycombObtuseKnight",6},"VertexList"],GraphData[{"TriangularHoneycombObtuseKnight",6},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"TriangularHoneycombObtuseKnight",6},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombObtuseKnight",6}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

GraphData[{"GeneralizedHexagon",{2,1}},"Graph","UnitDistance"]

(iso1=Select[FindIsomorphicSubgraph[,,All],BilateralGraphEmbeddingQ])//Timing

(iso2=Select[FindIsomorphicSubgraph[,,All],BilateralGraphEmbeddingQ])//Timing

(iso3=Select[FindIsomorphicSubgraph[,,All],BilateralGraphEmbeddingQ])//Timing

Join[iso1,iso2,iso3]

gs={,,,,,,,};

Matchstick

GraphData[{"TriangularHoneycombObtuseKnight",6},"Matchstick"]

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData[{"TriangularHoneycombObtuseKnight",6},"Polyhedral"]

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

{"TriangularHoneycombObtuseKnight", 7}

Names

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

GraphData[{"TriangularHoneycombObtuseKnight",7},"StandardNames"]

Primary

GraphData[{"TriangularHoneycombObtuseKnight",7}]//StyleGraphs[#,ImageSize->Small]&

All

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

Inexact

Select[GraphData[{"TriangularHoneycombObtuseKnight",7},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"TriangularHoneycombObtuseKnight",7},"EdgeList"]]

Default embedding

GraphPlot[GraphData[{"TriangularHoneycombObtuseKnight",7},"Edges"],GraphLayout->"HighDimensionalEmbedding"]

g=GraphicsToGraph@

v=Join[
CirclePoints[{1,Pi/2},3],CirclePoints[{1,Pi/2+Pi/(7/2)},3],CirclePoints[{1,Pi/2-Pi/(7/2)},3],
CirclePoints[1/5,3],CirclePoints[3/4,3],
CirclePoints[{3/4,Pi/2+π/10},3],CirclePoints[{3/4,Pi/2-π/10},3],
CirclePoints[{7/8,Pi/2+π/60},3],CirclePoints[{7/8,Pi/2-π/60},3],
{{0,0}}];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,4],e[7,1,16,14,19,1,22,16,12,5,20,28],e[9,10,19,25,11,22],e[20,2,5,21,6,19],e[8,12,21,27],e[21,13,18,24,3,27,11,26,2,23,11,24],e[7,18,3,21],e[2,17,9,18],e[6,3,9],e[4,20,15,17,10,6],e[8,2],{18,28},{20,26},{23,17},{7,16},{4,19},{8,17},e[4,28,7],{8,16},{1,25}},VertexCoordinates->v,VertexLabels->Automatic,VertexSize->.2,BaseStyle->Directive[FontFamily->"Times",12]]

RecognizeGraph@g

Degenerate

GraphData[{"TriangularHoneycombObtuseKnight",7},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"TriangularHoneycombObtuseKnight",7},"Graph","Degenerate"]

Configuration

GraphData[{"TriangularHoneycombObtuseKnight",7},"Configuration"]

GraphData[{"TriangularHoneycombObtuseKnight",7},"Graph","Configuration"]//StyleGraphs

Perspective

GraphData[{"TriangularHoneycombObtuseKnight",7},"Graph","Perspective"]//StyleGraphs

HoneycombToroidal construction

GraphData[{"TriangularHoneycombObtuseKnight",7},"NotationRules","HoneycombToroidal"]

GeneralizedPetersen

GraphData[{"TriangularHoneycombObtuseKnight",7},"Graph","GeneralizedPetersen"]//StyleGraphs

IGraph

GraphData[{"TriangularHoneycombObtuseKnight",7},"Graph","IGraph"]//StyleGraphs

Bilateral

GraphData[{"TriangularHoneycombObtuseKnight",7},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"TriangularHoneycombObtuseKnight",7},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"TriangularHoneycombObtuseKnight",7},"Hamiltonian"]

GraphData[{"TriangularHoneycombObtuseKnight",7},"HamiltonianCycleCount"]

Planar

GraphData[{"TriangularHoneycombObtuseKnight",7},"Planar"]

GraphData[{"TriangularHoneycombObtuseKnight",7},"Graph","Planar"]//StyleGraphs

MinimalCrossing

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

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

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"TriangularHoneycombObtuseKnight",7}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"TriangularHoneycombObtuseKnight",7}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"TriangularHoneycombObtuseKnight",7}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"TriangularHoneycombObtuseKnight",7}],TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"TriangularHoneycombObtuseKnight",7}],TimeConstraint->5 3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"TriangularHoneycombObtuseKnight",7}],TimeConstraint->86400])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"TriangularHoneycombObtuseKnight",7}],"CrossingNumber"->cn])//PolyhedralGraphQ

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"TriangularHoneycombObtuseKnight",7}],"CrossingNumber"->cn,RandomSeed->n]],{n,500}],Max[OuterFace[#]]<=GraphData[{"TriangularHoneycombObtuseKnight",7},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"TriangularHoneycombObtuseKnight",7}],"CrossingNumber"->cn,RandomSeed->i],{i,500}]//Union//Timing

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

MinimalLocalCrossing

GraphData[{"TriangularHoneycombObtuseKnight",7},"LocalCrossingNumber"]

GraphData[{"TriangularHoneycombObtuseKnight",7},"Graph","MinimalLocalCrossing"]//StyleGraphs

MinimalLocalCrossing construction

AdjacentCrossingsQ[g_Graph,cn_Integer]:=Module[{a=AdjacencyList[g],n=VertexCount[g]-cn},Max[Take[a,-cn]]>n]

(cn=CrossingNumber[GraphData[{"TriangularHoneycombObtuseKnight",7}],TimeConstraint->1])//AbsoluteTiming

Select[Table[MinimalCrossingGraph[GraphData[{"TriangularHoneycombObtuseKnight",7}],"CrossingNumber"->cn,RandomSeed->i],{i,20000}],!AdjacentCrossingsQ[#,cn]&]//Union//Timing

Select[Table[MinimalCrossingGraph[GraphData[{"TriangularHoneycombObtuseKnight",7}],"CrossingNumber"->(cn=21),RandomSeed->i],{i,5000}],!AdjacentCrossingsQ[#,cn]&]//Union//Timing

Circulant

GraphData[{"TriangularHoneycombObtuseKnight",7},"Circulant"]

GraphData[{"TriangularHoneycombObtuseKnight",7},"Graph","Circulant"]//StyleGraphs

Complement graph

compl=GraphData[{"TriangularHoneycombObtuseKnight",7},"ComplementGraph","Name"]

Line graph

GraphData[{"TriangularHoneycombObtuseKnight",7},"Line"]

LineGraphQ[GraphData[{"TriangularHoneycombObtuseKnight",7}]]//Timing

RootGraph

roots=Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"TriangularHoneycombObtuseKnight",7}&]

RootGraph[GraphData[{"TriangularHoneycombObtuseKnight",7}]]

RecognizeGraph[%]

Halin

GraphData[{"TriangularHoneycombObtuseKnight",7},"Halin"]

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

Rigid

GraphData[{"TriangularHoneycombObtuseKnight",7},"Rigid"]

GraphData[{"TriangularHoneycombObtuseKnight",7},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"TriangularHoneycombObtuseKnight",7},"Flexible"]

GraphData[{"TriangularHoneycombObtuseKnight",7},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"TriangularHoneycombObtuseKnight",7},"LCF"]

GraphData[{"TriangularHoneycombObtuseKnight",7},"HamiltonianCycleCount"]

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

GraphData[{"TriangularHoneycombObtuseKnight",7},"LCFNotations"]

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"TriangularHoneycombObtuseKnight",7},"VertexList"],GraphData[{"TriangularHoneycombObtuseKnight",7},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"TriangularHoneycombObtuseKnight",7},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

FindUnitDistanceEmbedding[GraphData[{"TriangularHoneycombObtuseKnight",7}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData[{"TriangularHoneycombObtuseKnight",7},"Matchstick"]

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData[{"TriangularHoneycombObtuseKnight",7},"Polyhedral"]

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

Properties

Acyclic

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

AdjacencyMatrixCount

seq=Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"AdjacencyMatrixCount"],{n,20}]

Arboricity

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

ArcTransitive

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

ArcTransitivity

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

AutomorphismCount

Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"AutomorphismCount"],{n,20}]

BalabanIndex

Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"BalabanIndex"],{n,4}]//FullSimplify

Bandwidth

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

BridgeCount

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

BurningNumber

seq=Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"BurningNumber"],{n,4,20}]

CharacteristicPolynomial

(poly=Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"CharacteristicPolynomial"][x],{n,k0=4,20}])//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[{"TriangularHoneycombObtuseKnight",n},"ChordlessCycleCount"],{n,4,10}]

FindSequenceFunction[seq,n]

ChromaticInvariant

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

FindSequenceFunction[Drop[seq,2],n]

ChromaticNumber

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

ChromaticPolynomial

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

FindSequenceFunction[Drop[seq,2],n]

Classes

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

CliqueCount

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

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

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

CliqueCoveringNumber

A251533

seq=Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"CliqueCoveringNumber"],{n,4,20}]

FindLinearRecurrence[seq]

FindSequenceFunction[DeleteMissing[seq],n-3]

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

Do[Print[{n,CliqueCoveringNumber[TriangularHoneycombObtuseKnightGraph[n]]//Timing}],{n,20}]

Do[Print[{n,CliqueCoveringNumber[TriangularHoneycombObtuseKnightGraph[n],Method->"ComplementChromaticNumber"]//Timing}],{n,20}]

CliqueNumber

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

CliquePolynomial

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

ComplementOddChordlessCycleCount

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

FindSequenceFunction[Take[seq,{3,5}],n]

Connected

seq=Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"Connected"],{n,4,20}]

ConnectedComponentCount

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

ConnectedDominationNumber

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

FindSequenceFunction[seq,n]

ConnectedInducedSubgraphCount

A290784

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

FindSequenceFunction[DeleteMissing[seq],n]

ConnectedInducedSubgraphPolynomial

(poly=Factor[Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"ConnectedInducedSubgraphPolynomial"],{n,k0=3,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,6}]

FullSimplify[poly2-poly]

CrossingNumber

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

CycleCount

seq=Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"CycleCount"],{n,4,10}]

CyclePolynomial

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

CyclomaticNumber

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

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

DetourIndex

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

FindSequenceFunction[DeleteMissing@Drop[seq,5],n-5]

DetourMatrix

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

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

DetourPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"DetourPolynomial"],{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]

Diameter

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

DistanceMatrix

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

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

DistancePolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"DistancePolynomial"][x],{n,k0=1,20}])//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]

DistinguishingNumber

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

DomaticNumber

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

Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"DominatingSetCount"],{n,9}]

Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"MinimalDominatingSetCount"],{n,9}]

Table[Floor[Divide@@GraphData[{"TriangularHoneycombBishop",n},{"VertexCount","DominationNumber"}]],{n,9}]

DominatingSetCount

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

FactorInteger[{1,3,21,1435,6865267}]

DominationNumber

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

DominationPolynomial

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

FindSequenceFunction[seq,n]-2

gf=x FindGeneratingFunction[seq,x]

SeriesCoefficient[x((x-2 x +2 (x^4))/(1-3 x+2 (x^2))),{x,0,n}]//FullSimplify

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

EdgeConnectivity

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

EdgeCount

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

FindSequenceFunction[Drop[seq,3],n]//Factor

6Binomial[n+1,2]

EdgeCoverCount

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

EdgeCoverNumber

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

FindSequenceFunction[seq,n]

x FindGeneratingFunction[seq,x]

EdgeCoverPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"EdgeCoverPolynomial"][x],{n,k0=1,20}])//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]

EulerianCycleCount

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

FaceCount

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

FlowPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"FlowPolynomial"][x],{n,k0=1,20}])//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]

FractionalChromaticNumber

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

RSolve[{a[n]==a[n-1]+1/a[n-1],a[1]==1},a[n],n]

RecurrenceTable[{a[n]==a[n-1]+1/a[n-1],a[1]==1},a,{n,20}]

FractionalEdgeChromaticNumber

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

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

Table[3 2^(n-3)-1,{n,20}]

Girth

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

Gonality

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

HamiltonConnected

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

HamiltonDecompositionCount

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

Hamiltonian

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

HamiltonianCycleCount

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

HamiltonianNumber

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

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

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

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

HamiltonianPathCount

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

HamiltonianWalkCount

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

HararyIndex

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

FindSequenceFunction[seq,n]//FunctionExpand

FindGeneratingFunction[seq,x]

HexagonCount

A290392

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

seq=Table[CycleCount[TriangularHoneycombObtuseKnightGraph[n],6],{n,30}]

FindLinearRecurrence[seq]

FindLinearRecurrence[Drop[seq,4]]

FindLinearRecurrence[Drop[seq,6]]

FindLinearRecurrence[Drop[seq,8]]

Join[{0,0,0,0,4,13,98,415},LinearRecurrence[{3,-3,1},{1151,2471,4385},30]]

FindSequenceFunction[seq,n]

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

ToString[gf,InputForm]

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

Factor[16001-4323 n+297 (n^2)]

Table[Piecewise[{{0,n<=4},{4,n==5},{13,n==6},{98,n==7},{415,n==8},{16001-4323n+297n^2,n>8}},0],{n,20}]

Series[(((x^5) (-4-x-71 (x^2)-156 (x^3)-187 (x^4)-165 (x^5)-10 (x^6)))/(((-1+x))^3)),{x,0,20}]

CoefficientList[Series[(((x^4) (-4-x-71 (x^2)-156 (x^3)-187 (x^4)-165 (x^5)-10 (x^6)))/(((-1+x))^3)),{x,0,20}],x]

IdiosyncraticPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"IdiosyncraticPolynomial"][x,y],{n,k0=1,20}])//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]

IndependenceNumber

A251532

seq=Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"IndependenceNumber"],{n,4,20}]

Do[Print[{n,{Length[ivs=FindIndependentVertexSet[TriangularHoneycombObtuseKnightGraph[n]][[1]]],ivs}//Timing}],{n,21}]

seq={1,3,6,7,7,9,15,18,21,21,27,31,36,42,42,47,54,62,70,70,77,85,95};

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

IndependencePolynomial

(poly=Factor[Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"IndependencePolynomial"],{n,k0=1,20}])//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]

IndependenceRatio

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

FindSequenceFunction[seq,n]

IndependentEdgeSetCount

A287228

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

FindLinearRecurrence[DeleteMissing[seq]]

FindLinearRecurrence[Rest@DeleteMissing[seq]]

FindSequenceFunction[DeleteMissing[seq],n]

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

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

IndependentVertexSetCount

A287229

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

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

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

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

IntersectionNumber

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

KirchhoffIndex

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

FindSequenceFunction[seq,n]

KirchhoffSumIndex

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

LaplacianMatrix

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

LaplacianPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"LaplacianPolynomial"][x],{n,k0=1,20}])//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]

LeafCount

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

LongestCycleCount

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

LovaszNumber

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

MatchingGeneratingPolynomial

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

MatchingNumber

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

FindLinearRecurrence[Drop[seq,3]]

FindSequenceFunction[Drop[seq,3],n]

gf=x FindGeneratingFunction[seq,x]

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

MatchingPolynomial

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

MaximalCliqueCount

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

FindSequenceFunction[seq,n]//FunctionExpand

SeriesCoefficient[FindGeneratingFunction[seq,x],{x,0,n}]

Table[(1/6) (3-9 (2^(n-1))+7 (3^(n-1))),{n,20}]

MaximalIndependentEdgeSetCount

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

MaximalIndependentVertexSetCount

A290617

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

FindLinearRecurrence[DeleteMissing@seq]

FindLinearRecurrence[Drop[DeleteMissing@seq],3]

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

MaximumCliqueCount

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

FindSequenceFunction[seq,n]//FunctionExpand

SeriesCoefficient[FindGeneratingFunction[seq,x],{x,0,n}]

Table[(1/6) (3-9 (2^(n-1))+7 (3^(n-1))),{n,20}]

MaximumIndependentEdgeSetCount

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

FindLinearRecurrence[DeleteMissing@Drop[seq,3]]

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

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

MaximumIndependentVertexSetCount

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

MaximumLeafNumber

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

MeanDistance

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

FindSequenceFunction[seq,n]

FindGeneratingFunction[seq,x]

MinimalDominatingSetCount

A290874

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

FindSequenceFunction[DeleteMissing@seq,n]

MinimumCliqueCoveringCount

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

MinimumDistinguishingLabelingCount

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

MinimumDominatingSetCount

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

MinimumEdgeCoverCount

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

MolecularTopologicalIndex

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

FindSequenceFunction[seq,n]

FindGeneratingFunction[seq,x]//Factor

-36(3·(2^n)-8)(27·(2^n)-(143^n)-18)

Table[(3 2^n-8)(14 3^n7 2^n+18)/36,{n,20}]

Nonhamiltonian

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

NormalizedLaplacianMatrix

MatrixForm/@Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"NormalizedLaplacianMatrix"],{n,20}]

OddChordlessCycleCount

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

PathCount

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

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

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

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]

PathPolynomial

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

FullSimplify[poly2-poly]

PentagonCount

A290391

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

Position[seq,0][[-1,1]]

FindLinearRecurrence[Drop[seq,6]]

LinearRecurrence[{3,-3,1},{30,120,294},14]

Join[{0,0,0,0,0,0},LinearRecurrence[{3,-3,1},{30,120,294},14]]

FindSequenceFunction[Drop[seq,6],n-6]//Factor

Table[If[n<7,0,6 (292-90 n+7 (n^2))],{n,20}]

FindSequenceFunction[Drop[seq,7],n]//Factor

FindSequenceFunction[Drop[seq,2],n]

gf=(x^7) FindGeneratingFunction[Drop[seq,6],x]

ToString[gf,InputForm]

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

CoefficientList[Series[-((6 (x^6) (5+5 x+4 (x^2)))/(((-1+x))^3)),{x,0,20}],x]

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

Table[coef,{n,50}]

Perfect

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

ProjectivePlaneCrossingNumber

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

Radius

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

RankPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"RankPolynomial"][x,y],{n,k0=1,20}])//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]

RectilinearCrossingNumber

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

ReliabilityPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"ReliabilityPolynomial"][x],{n,k0=1,20}])//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]

SigmaPolynomial

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

FindSequenceFunction[seq,n]//Head

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

SquareCount

A194715

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

FindLinearRecurrence[Drop[seq,4]]

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

Table[Piecewise[{{0,n<5}},(15/2) (-6+n) (-5+n)],{n,20}]

15PolygonalNumber[Range[0,20]]

15Binomial[Range[20],2]

StabilityIndex

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

FindSequenceFunction[seq,n]

FindGeneratingFunction[seq,x]

TopologicalIndex

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

FindSequenceFunction[seq,n]

FindGeneratingFunction[seq,x]

ToroidalCrossingNumber

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

Traceable

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

TriangleCount

A001105

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

FindLinearRecurrence[seq]

FindLinearRecurrence[Drop[seq,3]]

LinearRecurrence[{3,-3,1},{2,8,18},{0,20}]

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

Table[Piecewise[{{0,n<4}},2(n-4)^2],{n,20}]

2PolygonalNumber[4,Range[0,20]]

TuttePolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"TuttePolynomial"][x,y],{n,k0=1,20}])//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]

Untraceable

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

VertexConnectivity

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

VertexCount

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

FindSequenceFunction[seq,n]

Binomial[n+1,2]

VertexCoverNumber

Related to but not the same as IndependenceNumber

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

FindSequenceFunction[seq,n]//FullSimplify

gf=x FindGeneratingFunction[seq,x]

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

Table[3 (2^(-3+n)),{n,20}]

VertexCoverPolynomial

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

VertexDegrees

seq=Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"VertexDegrees"],{n,20}]

WienerIndex

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

FindSequenceFunction[seq,n]

WienerSumIndex

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

FindSequenceFunction[seq,n]

GraphData

GraphDataClassString

GraphDataClassString["TriangularHoneycombObtuseKnight"]