Pell Graph

Author

Eric W. Weisstein
August 1, 2023

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

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

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

Table

TextGrid[pell=Sort[{GraphData[#,"NotationRules","Pell"],#}&/@GraphData["Pell"]],Dividers->All]

Graphic

GraphData

GraphicsGrid[Partition[Table[r=RecognizeGraph[g=PellGraph[n]];Show[StyleGraphs@If[r==={}||GraphData[r,"Embeddings"]==={},g,GraphData[r]],PlotLabel->Style[HoldForm[n]==n,Directive[14,Black,FontFamily->"Times"]]],{n,0,5}],UpTo[6]],Dividers->All,ImageSize->700,Alignment->Top]

Munarini 2019

Note different indexing of Pell numbers

Munarini, E.  Pell Graphs.  Disc. Math. 342, 2415–2428, 2019.

https://re.public.polimi.it/retrieve/e0c31c11-517b-4599-e053-1705fe0aef77/11311-1088084_Munarini.pdf

Theorems

Subgraph induced by center is a Fibonacci cube

Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

RecognizeGraph/@(Subgraph[#,GraphCenter[#]]&/@pellgraphs)

Subgraph of the hypercube

Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

Do[Print[{n,IsomorphicSubgraphQ[PellGraph[n],HypercubeGraph[2n-1]]//Timing}],{n,8}]

Embeddings

<<MathWorld`Graphs`

{"Pell", 0} == "SingletonGraph"

Names

GraphData[{"Pell",0},"StandardName"]

GraphData[{"Pell",0},"StandardNames"]

{"Pell", 1} == {"Path", 2}

Names

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

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

{"Pell", 2} == "BannerGraph"

Names

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

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

{"Pell", 3}

Names

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

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

Primary

GraphData[{"Pell",3}]//StyleGraphs[#,ImageSize->Small]&

All

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

Inexact

Select[GraphData[{"Pell",3},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Pell",3},"EdgeList"]]

Default embedding

GraphPlot[GraphData[{"Pell",3},"Edges"],GraphLayout->"LayeredDigraphEmbedding"]

RotateGraph[Graph@GraphData[{"Pell",3},"Edges"],Pi/2]

s=1/Sqrt[2];
v = Join[{{0, 0}, {s,s},{-s,s},{0,2s}},{#,#2+1}&@@@{{0, 0}, {s,s},{-s,s},{0,2s}},{{s,-s},{2s,0},{-s,-s},{-2s,0}}];
g = Graph[Range[Length[v]], UndirectedEdge @@@ {e[1,2,4,3,1,5,6,8,7,5],{4,8},e[7,3,12,11,1,9,10,2,6]}, VertexCoordinates -> v, VertexLabels -> Automatic]

UnitDistanceGraphEmbeddingQ[g]

RecognizeGraph[g]

s=1/Sqrt[2];
v = Join[{{0, 0}, {s,s},{-s,s},{0,2s}},{#,#2+1}&@@@{{0, 0}, {s,s},{-s,s},{0,2s}},{{s,-s},{s,0},{-s,-s},{-s,0}}];
g = Graph[Range[Length[v]], UndirectedEdge @@@ {e[1,2,4,3,1,5,6,8,7,5],{4,8},e[7,3,12,11,1,9,10,2,6]}, VertexCoordinates -> v, VertexLabels -> Automatic]

RecognizeGraph@g

gs=UnlabelGraph/@{,,,,}

Degenerate

GraphData[{"Pell",3},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Pell",3},"Graph","Degenerate"]

Bilateral

GraphData[{"Pell",3},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Pell",3},"Graph","Circular"]//StyleGraphs

Circular construction

Length[hc=FindHamiltonianCycle[GraphData[{"Pell",3}],All][[All,All,1]]]

CircularEmbeddingFromHamiltonianCycle[GraphData[{"Pell",3}],#]&/@hc

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph@CircularEmbeddingFromHamiltonianCycle[GraphData[{"Pell",3}],#]&/@hc],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

Length[quad=DeleteDuplicatesBy[Union@Select[bilat,BilateralGraphEmbeddingQ[#,Axis->"X"]&],CircularCanonicalGraphEmbedding[#,4]&]]

quad

Planar

GraphData[{"Pell",3},"Planar"]

GraphData[{"Pell",3},"Graph","Planar"]//StyleGraphs

MinimalCrossing

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

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

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Pell",3}],TimeConstraint->1])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Pell",3}],"CrossingNumber"->cn])//PolyhedralGraphQ

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Pell",3}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Pell",3},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Pell",3}],"CrossingNumber"->cn,RandomSeed->i],{i,1000}]//Union

GraphPlot@RotateGraph[,-Pi/6]

v={{0,0},{1,0},{1,1},{0,1},{0,1/6},{1,1/6},{1/3,2/6},{1/3,3/6},{1/3,4/6},{1/3,5/6},{2/3,4/6},{2/3,5/6}};
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,6,3,4,5,1],e[4,10,12,3],e[10,9,11,12],e[11,6,5,9,8,7,5]},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Pell",3},"Circulant"]

GraphData[{"Pell",3},"Graph","Circulant"]//StyleGraphs

Line graph

GraphData[{"Pell",3},"Line"]

LineGraphQ[GraphData[{"Pell",3}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Pell",3}&]

RootGraph[GraphData[{"Pell",3}]]

Halin

GraphData[{"Pell",3},"Halin"]

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

Rigid

GraphData[{"Pell",3},"Rigid"]

GraphData[{"Pell",3},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Pell",3},"Flexible"]

GraphData[{"Pell",3},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Pell",3},"LCF"]

GraphData[{"Pell",3},"HamiltonianCycleCount"]

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

GraphData[{"Pell",3},"LCFNotations"]

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Pell",3},"VertexList"],GraphData[{"Pell",3},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Pell",3},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

RotateGraph[,-Pi/2]

UnitDistance by FindUnitDistanceEmbedding

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

FindUnitDistanceEmbedding[GraphData[{"Pell",3}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

gs={,,,,,};

Matchstick

GraphData[{"Pell",3},"Matchstick"]

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData[{"Pell",3},"Polyhedral"]

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

{"Pell", 4}

Names

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

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

Primary

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

All

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

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData[{"Pell",4},"EdgeList"]]

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

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

Bilateral

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

Circular

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

Circular construction

Length[hc=FindHamiltonianCycle[GraphData[{"Pell",4}],All][[All,All,1]]]

Planar

GraphData[{"Pell",4},"Planar"]

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

GraphData[{"Pell",4},"Polyhedral"]

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

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

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Pell",4},"Circulant"]

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

Line graph

GraphData[{"Pell",4},"Line"]

LineGraphQ[GraphData[{"Pell",4}]]//Timing

RootGraph

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

RootGraph[GraphData[{"Pell",4}]]

Halin

GraphData[{"Pell",4},"Halin"]

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

Rigid

GraphData[{"Pell",4},"Rigid"]

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

Flexible

GraphData[{"Pell",4},"Flexible"]

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

LCF

GraphData[{"Pell",4},"LCF"]

GraphData[{"Pell",4},"HamiltonianCycleCount"]

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

GraphData[{"Pell",4},"LCFNotations"]

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Pell",4},"VertexList"],GraphData[{"Pell",4},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

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

Matchstick

GraphData[{"Pell",4},"Matchstick"]

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData[{"Pell",4},"Polyhedral"]

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

{"Pell", 5}

Names

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

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

Primary

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

All

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

Inexact

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

Embedding types

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

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

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

Bilateral

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

Circular

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

Circular construction 10^2

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

Length[hc=FindHamiltonianCycle[GraphData[{"Pell",5}],10^2][[All,All,1]]]//Timing

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph@CircularEmbeddingFromHamiltonianCycle[GraphData[{"Pell",5}],#]&/@hc],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

Length[quad=DeleteDuplicatesBy[Union@Select[bilat,BilateralGraphEmbeddingQ[#,Axis->"X"]&],CircularCanonicalGraphEmbedding[#,4]&]]

quad

Circular construction 10^3

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

Length[hc=FindHamiltonianCycle[GraphData[{"Pell",5}],10^3][[All,All,1]]]//Timing

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph@CircularEmbeddingFromHamiltonianCycle[GraphData[{"Pell",5}],#]&/@hc],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

Length[quad=DeleteDuplicatesBy[Union@Select[bilat,BilateralGraphEmbeddingQ[#,Axis->"X"]&],CircularCanonicalGraphEmbedding[#,4]&]]

quad

Circular construction 10^4

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

Length[hc=FindHamiltonianCycle[GraphData[{"Pell",5}],10^4][[All,All,1]]]//Timing

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph@CircularEmbeddingFromHamiltonianCycle[GraphData[{"Pell",5}],#]&/@hc],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

Length[quad=DeleteDuplicatesBy[Union@Select[bilat,BilateralGraphEmbeddingQ[#,Axis->"X"]&],CircularCanonicalGraphEmbedding[#,4]&]]

quad

Circular construction

Length[hc=FindHamiltonianCycle[GraphData[{"Pell",5}],All][[All,All,1]]]//Timing

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph@CircularEmbeddingFromHamiltonianCycle[GraphData[{"Pell",5}];PellGraph[5],#]&/@hc],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

Length[quad=DeleteDuplicatesBy[Union@Select[bilat,BilateralGraphEmbeddingQ[#,Axis->"X"]&],CircularCanonicalGraphEmbedding[#,4]&]]

quad

Planar

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

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

MinimalCrossing

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

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

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Pell",5}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Pell",5}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Pell",5}],TimeConstraint->60])//AbsoluteTiming

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

l={{{"Pell",5},{0,151}}};

PropertyReplace[l,"CrossingNumber"]

RenameGraphFiles[]

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Pell",5}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Pell",5},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Pell",5}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

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

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

Line graph

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

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

RootGraph

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

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

Halin

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

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

Rigid

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

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

Flexible

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

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

LCF

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

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

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

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

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

g=Annotate[,ImageSize->Automatic]

Matchstick

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

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

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

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

{"Pell", 6}

Names

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

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

Primary

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

All

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

Inexact

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

Embedding types

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

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

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

Bilateral

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

Circular

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

Circular construction

Length[hc=FindHamiltonianCycle[GraphData[{"Pell",6}],All][[All,All,1]]]

Planar

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

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Pell",6}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Pell",6},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Pell",6}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

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

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

Line graph

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

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

RootGraph

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

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

Halin

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

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

Rigid

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

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

Flexible

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

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

LCF

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

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

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

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

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

Matchstick

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

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

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

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

{"Pell", 7}

Names

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

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

Primary

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

All

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

Inexact

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

Embedding types

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

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

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

Bilateral

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

Circular

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

Circular construction 100

Length[hc=FindHamiltonianCycle[GraphData[{"Pell",7}],100][[All,All,1]]]//Timing

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph@CircularEmbeddingFromHamiltonianCycle[GraphData[{"Pell",7}];PellGraph[7],#]&/@hc],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

Length[quad=DeleteDuplicatesBy[Union@Select[bilat,BilateralGraphEmbeddingQ[#,Axis->"X"]&],CircularCanonicalGraphEmbedding[#,4]&]]

quad

Circular construction

Length[hc=FindHamiltonianCycle[GraphData[{"Pell",7}],All][[All,All,1]]]

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph@CircularEmbeddingFromHamiltonianCycle[GraphData[{"Pell",7}],#]&/@hc],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

Length[quad=DeleteDuplicatesBy[Union@Select[bilat,BilateralGraphEmbeddingQ[#,Axis->"X"]&],CircularCanonicalGraphEmbedding[#,4]&]]

quad

Planar

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

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

RenameGraphFiles[]

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Pell",7}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Pell",7},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Pell",7}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

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

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

Line graph

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

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

RootGraph

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

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

Halin

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

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

Rigid

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

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

Flexible

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

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

LCF

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

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

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

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

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Pell",7},"VertexList"],GraphData[{"Pell",7},"EdgeList"]],MaxIterations->1000]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->1000]&/@Complement@@(GraphData[{"Pell",7},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Pell",7}],MaxIterations->500,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Pell",7}],MaxIterations->500,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

Matchstick

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

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

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

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

{"Pell",  8}

Names

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

GraphData[{"Pell",8},"StandardNames"]

Primary

GraphData[{"Pell",8}]//StyleGraphs[#,ImageSize->Small]&

All

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

Inexact

Select[GraphData[{"Pell",8},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Pell",8},"EdgeList"]]

Default embedding

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

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

RecognizeGraph[g]

Degenerate

GraphData[{"Pell",8},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Pell",8},"Graph","Degenerate"]

Bilateral

GraphData[{"Pell",8},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Pell",8},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"Pell",8},"Hamiltonian"]

Length[hc=FindHamiltonianCycle[GraphData[{"Pell",8}],All][[All,All,1]]]

Planar

GraphData[{"Pell",8},"Planar"]

GraphData[{"Pell",8},"Graph","Planar"]//StyleGraphs

MinimalCrossing

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

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

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Pell",8}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Pell",8}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Pell",8}],TimeConstraint->3600])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Pell",8}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Pell",8},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Pell",8}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Pell",8},"Circulant"]

GraphData[{"Pell",8},"Graph","Circulant"]//StyleGraphs

Line graph

GraphData[{"Pell",8},"Line"]

LineGraphQ[GraphData[{"Pell",8}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Pell",8}&]

RootGraph[GraphData[{"Pell",8}]]

Halin

GraphData[{"Pell",8},"Halin"]

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

Rigid

GraphData[{"Pell",8},"Rigid"]

GraphData[{"Pell",8},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Pell",8},"Flexible"]

GraphData[{"Pell",8},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Pell",8},"LCF"]

GraphData[{"Pell",8},"HamiltonianCycleCount"]

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

GraphData[{"Pell",8},"LCFNotations"]

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Pell",8},"VertexList"],GraphData[{"Pell",8},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Pell",8},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Pell",8}],MaxIterations->500,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Pell",8}],MaxIterations->500,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

Matchstick

GraphData[{"Pell",8},"Matchstick"]

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData[{"Pell",8},"Polyhedral"]

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

{"Pell", 9}

Names

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

GraphData[{"Pell",9},"StandardNames"]

Primary

GraphData[{"Pell",9}]//StyleGraphs[#,ImageSize->Small]&

All

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

Inexact

Select[GraphData[{"Pell",9},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Pell",9},"EdgeList"]]

Default embedding

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

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

RecognizeGraph[g]

Degenerate

GraphData[{"Pell",9},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Pell",9},"Graph","Degenerate"]

Bilateral

GraphData[{"Pell",9},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Pell",9},"Graph","Circular"]//StyleGraphs

Circular construction 10

GraphData[{"Pell",9},"Hamiltonian"]

FindHamiltonianCycle[GraphData[{"Pell",9}]]//Timing

Length[hc=FindHamiltonianCycle[GraphData[{"Pell",9}],All][[All,All,1]]]

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph@CircularEmbeddingFromHamiltonianCycle[GraphData[{"Pell",9}],#]&/@hc],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

Length[quad=DeleteDuplicatesBy[Union@Select[bilat,BilateralGraphEmbeddingQ[#,Axis->"X"]&],CircularCanonicalGraphEmbedding[#,4]&]]

quad

Circular construction

Length[hc=FindHamiltonianCycle[GraphData[{"Pell",9}],All][[All,All,1]]]

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph@CircularEmbeddingFromHamiltonianCycle[GraphData[{"Pell",9}],#]&/@hc],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

Length[quad=DeleteDuplicatesBy[Union@Select[bilat,BilateralGraphEmbeddingQ[#,Axis->"X"]&],CircularCanonicalGraphEmbedding[#,4]&]]

quad

Planar

GraphData[{"Pell",9},"Planar"]

GraphData[{"Pell",9},"Graph","Planar"]//StyleGraphs

MinimalCrossing

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

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

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Pell",9}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Pell",9}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Pell",9}],TimeConstraint->3600])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Pell",9}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Pell",9},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Pell",9}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Pell",9},"Circulant"]

GraphData[{"Pell",9},"Graph","Circulant"]//StyleGraphs

Line graph

GraphData[{"Pell",9},"Line"]

LineGraphQ[GraphData[{"Pell",9}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Pell",9}&]

RootGraph[GraphData[{"Pell",9}]]

Halin

GraphData[{"Pell",9},"Halin"]

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

Rigid

GraphData[{"Pell",9},"Rigid"]

GraphData[{"Pell",9},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Pell",9},"Flexible"]

GraphData[{"Pell",9},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Pell",9},"LCF"]

GraphData[{"Pell",9},"HamiltonianCycleCount"]

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

GraphData[{"Pell",9},"LCFNotations"]

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Pell",9},"VertexList"],GraphData[{"Pell",9},"EdgeList"]],MaxIterations->1000]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->1000]&/@Complement@@(GraphData[{"Pell",9},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Pell",9}],MaxIterations->500,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Pell",9}],MaxIterations->500,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

Matchstick

GraphData[{"Pell",9},"Matchstick"]

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData[{"Pell",9},"Polyhedral"]

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

Properties

Acyclic

Table[GraphData[{"Pell",n},"Acyclic"],{n,0,9}]

AdjacencyMatrixCount

seq=Table[GraphData[{"Pell",n},"AdjacencyMatrixCount"],{n,0,9}]

AlgebraicConnectivity

seq=Table[GraphData[{"Pell",n},"AlgebraicConnectivity"],{n,0,9}]

poly=ReadProperty["LaplacianPolynomial",GraphNameToFileName[{"Pell",9}]<>".gz"]

Exponent[poly,x]

$MaxRootDegree=Exponent[poly,x]

spec=CharacteristicPolynomialToSpectrum[poly,x];//Timing

N[spec]

Laplacian spectral radius:

laplacianspectralradius=spec[[-1]]

ByteCount[laplacianspectralradius]

Algebraic connectivity:

algebraiconnectivity=spec[[2]]

ByteCount[algebraiconnectivity]

Laplacian spectral ratio:

?ToAlgebraicRoot

ToAlgebraicRoot[(laplacianspectralradius/algebraiconnectivity),"MaxDegree"->1024]//Timing

AlmostHamiltonian

Table[GraphData[{"Pell",n},"AlmostHamiltonian"],{n,0,9}]

Anarboricity

seq=Table[GraphData[{"Pell",n},"Anarboricity"],{n,0,9}]

Antipodal

Table[GraphData[{"Pell",n},"Antipodal"],{n,0,9}]

Apex

Table[GraphData[{"Pell",n},"Apex"],{n,0,9}]

ApexCount

seq=Table[GraphData[{"Pell",n},"ApexCount"],{n,0,9}]

Arboricity

Table[GraphData[{"Pell",n},"Arboricity"],{n,0,9}]

ArcTransitive

Table[GraphData[{"Pell",n},"ArcTransitive"],{n,0,9}]

ArcTransitivity

Table[GraphData[{"Pell",n},"ArcTransitivity"],{n,0,9}]

ArticulationVertexCount

seq=Table[GraphData[{"Pell",n},"ArticulationVertexCount"],{n,0,9}]

AutomorphismCount

seq=Table[GraphData[{"Pell",n},"AutomorphismCount"],{n,0,9}]

BalabanIndex

seq=Table[GraphData[{"Pell",n},"BalabanIndex"],{n,0,9}]

FindSequenceFunction[seq,n+1]//Factor

Bandwidth

Table[GraphData[{"Pell",n},"Bandwidth"],{n,0,9}]

Bicolorable

Table[GraphData[{"Pell",n},"Bicolorable"],{n,0,9}]

Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

Biconnected

Table[GraphData[{"Pell",n},"Biconnected"],{n,0,9}]

Bicubic

Table[GraphData[{"Pell",n},"Bicubic"],{n,0,9}]

Bipartite

Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

Table[GraphData[{"Pell",n},"Bipartite"],{n,0,9}]

BipartiteDimension

seq=Table[GraphData[{"Pell",n},"BipartiteDimension"],{n,0,9}]

Biplanar

Table[GraphData[{"Pell",n},"Biplanar"],{n,0,9}]

Block

seq=Table[GraphData[{"Pell",n},"Block"],{n,0,9}]

BlockCount

seq=Table[GraphData[{"Pell",n},"BlockCount"],{n,0,9}]

BridgeCount

seq=Table[GraphData[{"Pell",n},"BridgeCount"],{n,0,9}]

Bridged

Table[GraphData[{"Pell",n},"Bridged"],{n,0,9}]

BroadcastTime

seq=Table[GraphData[{"Pell",n},"BroadcastTime"],{n,0,9}]

BurningNumber

seq=Table[GraphData[{"Pell",n},"BurningNumber"],{n,0,9}]

CharacteristicPolynomial

(poly=Factor@Table[GraphData[{"Pell",n},"CharacteristicPolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

CheegerConstant

seq=Table[GraphData[{"Pell",n},"CheegerConstant"],{n,0,9}]

Chordal

Table[GraphData[{"Pell",n},"Chordal"],{n,0,9}]

ChordCount

seq=Table[GraphData[{"Pell",n},"ChordCount"],{n,0,9}]

ChordlessCycleCount

seq=Table[GraphData[{"Pell",n},"ChordlessCycleCount"],{n,0,9}]

ChromaticInvariant

seq=Table[GraphData[{"Pell",n},"ChromaticInvariant"],{n,0,9}]

ChromaticNumber

seq=Table[GraphData[{"Pell",n},"ChromaticNumber"],{n,0,9}]

Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

ChromaticPolynomial

(poly=Table[GraphData[{"Pell",n},"ChromaticPolynomial"],{n,0,9}]/.f_Function:>f[x])//Column

CircuitRank

A364655

seq=Table[GraphData[{"Pell",n},"CircuitRank"],{n,0,9}]

FindSequenceFunction[seq,n+1]//FullSimplify

seq=Table[((202736+143356 (Sqrt[2])-29690 (((1-(Sqrt[2])))^n)-20994 (Sqrt[2]) (((1-(Sqrt[2])))^n)+(((1+(Sqrt[2])))^n) (86523+61181 (Sqrt[2])) (-2+n)+14845 (((1-(Sqrt[2])))^n) n+10497 (Sqrt[2]) (((1-(Sqrt[2])))^n) n)/(4 (50684+35839 (Sqrt[2])))),{n,0,20}]//Factor

coef=FindLinearRecurrence[seq]

Take[seq,Length@coef]

LinearRecurrence[{5,-6,-2,3,1},{0,0,1,7,30},20]

gf=FindGeneratingFunction[seq,x]

StringReplace[ToString[gf,InputForm]," "->""]

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

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

A000129

-Table[(((((1-(Sqrt[2])))^(1+n))-(((1+(Sqrt[2])))^(1+n)))/(2(Sqrt[2]))),{n,10}]//Expand

Table[Fibonacci[n+1,2],{n,10}]

Table[(((-(((1-(Sqrt[2])))^(1+n))+(((1+(Sqrt[2])))^(1+n))) )/(2 (Sqrt[2]))),{n,10}]//Expand

Table[1-2Fibonacci[n+1,2]+Fibonacci[n+1,2] (2+n)/2,{n,0,10}]//Expand

Factor[((2+n)/2-2)]

seq=Table[Fibonacci[n+1,2](n/2-1)+1,{n,0,29}]

StringLength[ToString[seq]]

Circumference

seq=Table[GraphData[{"Pell",n},"Circumference"],{n,0,9}]

Classes

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

ClawFree

Table[GraphData[{"Pell",n},"ClawFree"],{n,0,9}]

CliqueCount

seq=Table[GraphData[{"Pell",n},"CliqueCount"],{n,0,9}]

FindSequenceFunction[seq,n+1]//FullSimplify

CliqueCoveringNumber

seq=Table[GraphData[{"Pell",n},"CliqueCoveringNumber"],{n,0,9}]

FindSequenceFunction[seq,n+1]//FullSimplify

CliqueNumber

seq=Table[GraphData[{"Pell",n},"CliqueNumber"],{n,0,9}]

CliquePolynomial

(poly=Factor@Table[GraphData[{"Pell",n},"CliquePolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

Coarseness

seq=Table[GraphData[{"Pell",n},"Coarseness"],{n,0,9}]

CoboundaryPolynomial

(seq=Table[GraphData[{"Pell",n},"CoboundaryPolynomial"],{n,0,9}]/.f_Function:>f[x,y])//Column

ComplementChordlessCycleCount

seq=Table[GraphData[{"Pell",n},"ComplementChordlessCycleCount"],{n,0,9}]

ComplementChordlessCyclePolynomial

seq=Table[GraphData[{"Pell",n},"ComplementChordlessCyclePolynomial"],{n,0,9}]/.f_Function:>f[x]

ComplementOddChordlessCycleCount

seq=Table[GraphData[{"Pell",n},"ComplementOddChordlessCycleCount"],{n,0,9}]

ComplementOddChordlessCyclePolynomial

seq=Table[GraphData[{"Pell",n},"ComplementOddChordlessCyclePolynomial"],{n,0,9}]/.f_Function:>f[x]

Conductance

seq=Table[GraphData[{"Pell",n},"Conductance"],{n,0,9}]

Connected

seq=Table[GraphData[{"Pell",n},"Connected"],{n,0,9}]

ConnectedComponentCount

seq=Table[GraphData[{"Pell",n},"ConnectedComponentCount"],{n,0,9}]

ConnectedDominatingSetCount

seq=Table[GraphData[{"Pell",n},"ConnectedDominatingSetCount"],{n,0,9}]

FindSequenceFunction[DeleteMissing@seq,n]

ConnectedDominationNumber

seq=Table[GraphData[{"Pell",n},"ConnectedDominationNumber"],{n,0,9}]

FindSequenceFunction[DeleteMissing@seq,n]

ConnectedDominationPolynomial

(poly=Table[GraphData[{"Pell",n},"ConnectedDominationPolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

ConnectedInducedSubgraphCount

seq=Table[GraphData[{"Pell",n},"ConnectedInducedSubgraphCount"],{n,0,9}]

FindSequenceFunction[DeleteMissing@seq,n]

ConnectedInducedSubgraphPolynomial

(poly=Table[GraphData[{"Pell",n},"ConnectedInducedSubgraphPolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

Corank

A364655

seq=Table[GraphData[{"Pell",n},"Corank"],{n,0,9}]

FindSequenceFunction[seq,n+1]//FullSimplify

Table[Fibonacci[n+1,2] (n/2-1)+1,{n,0,9}]

CriticalNonplanar

Table[GraphData[{"Pell",n},"CriticalNonplanar"],{n,0,9}]

CrossingNumber

seq=Table[GraphData[{"Pell",n},"CrossingNumber"],{n,0,9}]

Cubic

Table[GraphData[{"Pell",n},"CriticalNonplanar"],{n,0,9}]

CycleCount

seq=Table[GraphData[{"Pell",n},"CycleCount"],{n,0,9}]

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

FindGeneratingFunction[seq,x]

CyclePolynomial

(poly=Table[GraphData[{"Pell",n},"CyclePolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

Cyclic

Table[GraphData[{"Pell",n},"Cyclic"],{n,0,9}]

CyclicEdgeConnectivity

Table[GraphData[{"Pell",n},"CyclicEdgeConnectivity"],{n,0,9}]

Degeneracy

seq=Table[GraphData[{"Pell",n},"Degeneracy"],{n,0,9}]

DegreeCentralities

Tally/@Table[GraphData[{"Pell",n},"DegreeCentralities"],{n,0,9}]//Column

DetourIndex

seq=Table[GraphData[{"Pell",n},"DetourIndex"],{n,0,9}]

FindSequenceFunction[seq,n+1]//Factor

DetourPolynomial

(poly=Table[GraphData[{"Pell",n},"DetourPolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

Diameter

Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

seq=Table[GraphData[{"Pell",n},"Diameter"],{n,0,9}]

Table[n+Floor[n/2],{n,0,9}]

Dimension

seq=Table[GraphData[{"Pell",n},"Dimension"],{n,0,9}]

DistanceHereditary

Inferred for n > 6:

seq=Table[GraphData[{"Pell",n},"DistanceHereditary"],{n,0,9}]

DistancePolynomial

(poly=Factor@Table[GraphData[{"Pell",n},"DistancePolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

FindLinearRecurrence[DeleteMissing@poly]//Head//Timing

DistanceRegular

Table[GraphData[{"Pell",n},"DistanceRegular"],{n,0,9}]

DistanceTransitive

Table[GraphData[{"Pell",n},"DistanceTransitive"],{n,0,9}]

DistinguishingNumber

seq=Table[GraphData[{"Pell",n},"DistinguishingNumber"],{n,0,9}]

DomaticNumber

seq=Table[GraphData[{"Pell",n},"DomaticNumber"],{n,0,9}]

DominatingSetCount

seq=Table[GraphData[{"Pell",n},"DominatingSetCount"],{n,0,9}]

DominationNumber

seq=Table[GraphData[{"Pell",n},"DominationNumber"],{n,0,9}]

Do[Print[{gn={"Pell",n},Flatten[{set=Flatten@FindDominatingSet[PellGraph[n],Method->"Mosek"],Length[set]}//Timing,1]}],{n,7}]

DominationPolynomial

(poly=Table[GraphData[{"Pell",n},"DominationPolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

Doublecross

Table[GraphData[{"Pell",n},"Doublecross"],{n,0,9}]

Eccentricities

Tally/@Table[GraphData[{"Pell",n},"Eccentricities"],{n,0,9}]//Column

EccentricityCentralities

seq=Table[GraphData[{"Pell",n},"EdgeChromaticNumber"],{n,0,9}]

EdgeBetweennessCentralities

seq=Table[GraphData[{"Pell",n},"EdgeChromaticNumber"],{n,0,9}]

EdgeChromaticNumber

seq=Table[GraphData[{"Pell",n},"EdgeChromaticNumber"],{n,0,9}]

Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

EdgeConnectivity

Table[GraphData[{"Pell",n},"EdgeConnectivity"],{n,0,9}]

EdgeCount

A364553

seq=Table[GraphData[{"Pell",n},"EdgeCount"],{n,0,9}]

Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

Table[n Fibonacci[n+1,2]/2,{n,0,20}]

FindSequenceFunction[seq,n+1]//FullSimplify

FindLinearRecurrence[seq]

LinearRecurrence[{4,-2,-4,-1},{0,1,5,18},30]

StringLength[ToString[%]]

gf=FindGeneratingFunction[seq,x]

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

EdgeCount/@pellgraphs

FindLinearRecurrence[%]

FindGeneratingFunction[%122,x]

seq={0,1,5,18,58,175,507,1428,3940,10701,28705,76230,200766};

gf=FindGeneratingFunction[seq,x]

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

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

(1/8)Table[ (-(((1-(Sqrt[2])))^n) (-2+(Sqrt[2]))+(((1+(Sqrt[2])))^n) (2+(Sqrt[2]))) n,{n,0,10}]//Expand

(1/8)Table[ ((-(((1-(Sqrt[2])))^n) (-2)-(((1-(Sqrt[2])))^n) (Sqrt[2]))+(((1+(Sqrt[2])))^n) (2)+(Sqrt[2])(((1+(Sqrt[2])))^n) ) n,{n,0,10}]//Expand

(1/8)Table[((Sqrt[2])((((1+(Sqrt[2])))^n)-(((1-(Sqrt[2])))^n))+2((((1-(Sqrt[2])))^n)+(((1+(Sqrt[2])))^n) ))n,{n,0,10}]//Expand

A000129

0,1,2,5,12

Table[((((((1+(Sqrt[2])))^n)-(((1-(Sqrt[2])))^n)))/(2(Sqrt[2]))),{n,10}]//Expand

A001333

1, 1, 3, 7, 17, 41, 99,

Table[((((((1-(Sqrt[2])))^n)+(((1+(Sqrt[2])))^n)))/2),{n,10}]//Expand

Table[(-I)^n ChebyshevT[n,I],{n,10}]

(1/8)Table[(4Fibonacci[n,2]+4(-I)^n ChebyshevT[n,I])n,{n,0,10}]//Expand

Table[n(Fibonacci[n,2]+(-I)^n ChebyshevT[n,I])/2,{n,0,20}]

Table[With[{s=Sqrt[2]},n((s+2)(1+s)^n-(s-2)(1-s)^n)/8],{n,0,20}]//Expand

EdgeCoverCount

seq=Table[GraphData[{"Pell",n},"EdgeCoverCount"],{n,0,9}]

EdgeCoverNumber

A113225

== EdgeCutCount

seq=Table[GraphData[{"Pell",n},"EdgeCoverNumber"],{n,0,9}]

gf=x FindGeneratingFunction[seq,x]//FullSimplify

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

A000129

Table[-(Sqrt[2]) (((1-(Sqrt[2])))^n)+(Sqrt[2]) (((1+(Sqrt[2])))^n),{n,10}]/4//Expand

Table[Fibonacci[n,2],{n,10}]

Table[(1-(((-1))^n)+2Fibonacci[n,2])/4,{n,10}]//Expand

seq=Table[(1+(((-1))^n)+2Fibonacci[n+1,2])/4,{n,0,20}]//Expand

gf=FindGeneratingFunction[seq,x]//FullSimplify

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

FindLinearRecurrence[seq]

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

Ceiling[Fibonacci[Range[20],2]/2]

EdgeCoverPolynomial

(poly=Factor@Table[GraphData[{"Pell",n},"EdgeCoverPolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

EdgeCutCount

A113225

== EdgeCoverNumber

seq=Table[GraphData[{"Pell",n},"EdgeCoverNumber"],{n,0,9}]

EdgeTransitive

Table[GraphData[{"Pell",n},"EdgeTransitive"],{n,0,9}]

EigenvectorCentralities

seq=Table[GraphData[{"Pell",n},"EigenvectorCentralities"],{n,0,9}]

EulerianCycleCount

seq=Table[GraphData[{"Pell",n},"EulerianCycleCount"],{n,0,9}]

FaceCount

seq=Table[GraphData[{"Pell",n},"FaceCount"],{n,0,9}]

FixingNumber

seq=Table[GraphData[{"Pell",n},"FixingNumber"],{n,0,9}]

FlowPolynomial

(poly=Factor@Table[GraphData[{"Pell",n},"FlowPolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

FractionalArboricity

seq=Table[GraphData[{"Pell",n},"FractionalArboricity"],{n,0,9}]

FractionalChromaticNumber

seq=Table[GraphData[{"Pell",n},"FractionalChromaticNumber"],{n,0,9}]

FractionalEdgeChromaticNumber

seq=Table[GraphData[{"Pell",n},"FractionalEdgeChromaticNumber"],{n,0,9}]

Genus

seq=Table[GraphData[{"Pell",n},"Genus"],{n,0,9}]

Geodetic

Table[GraphData[{"Pell",n},"Geodetic"],{n,0,9}]

Girth

seq=Table[GraphData[{"Pell",n},"Girth"],{n,0,9}]

GlobalClusteringCoefficient

seq=Table[GraphData[{"Pell",n},"GlobalClusteringCoefficient"],{n,0,9}]

Gonality

seq=Table[GraphData[{"Pell",n},"Gonality"],{n,0,9}]

Graceful

Table[GraphData[{"Pell",n},"Graceful"],{n,0,9}]

GracefulLabelingCount

seq=Table[GraphData[{"Pell",n},"GracefulLabelingCount"],{n,0,9}]

HadwigerNumber

Table[GraphData[{"Pell",n},"HadwigerNumber"],{n,0,9}]

HamiltonConnected

Table[GraphData[{"Pell",n},"HamiltonConnected"],{n,0,9}]

HamiltonDecomposable

Table[GraphData[{"Pell",n},"HamiltonDecomposable"],{n,0,9}]

HamiltonDecompositionCount

seq=Table[GraphData[{"Pell",n},"HamiltonDecompositionCount"],{n,0,9}]

Hamiltonian

Table[GraphData[{"Pell",n},"Hamiltonian"],{n,0,9}]

HamiltonianCycleCount

seq=Table[GraphData[{"Pell",n},"HamiltonianCycleCount"],{n,0,9}]

HamiltonianNumber

seq=Table[GraphData[{"Pell",n},"HamiltonianNumber"],{n,0,9}]

HamiltonianPathCount

seq=Table[GraphData[{"Pell",n},"HamiltonianPathCount"],{n,0,9}]

HamiltonianWalkCount

seq=Table[GraphData[{"Pell",n},"HamiltonianWalkCount"],{n,0,9}]

HamiltonLaceable

Table[GraphData[{"Pell",n},"HamiltonianWalkCount"],{n,0,9}]

HararyIndex

seq=Table[GraphData[{"Pell",n},"HararyIndex"],{n,0,9}]

FindSequenceFunction[DeleteMissing@seq,n+1]//Head//Timing

HexagonCount

A364651

seq=Table[GraphData[{"Pell",n},"HexagonCount"],{n,0,9}]

FindSequenceFunction[seq,n+1]//Head//Timing

Do[Print[{{"Pell",n},CycleCount[PellGraph[n],6]//Timing}],{n,0,10}]

seq={0,0,0,20,206,1282,6302,26942,104948,382444,1325444};

FindLinearRecurrence[seq]//Head//Timing

FindGeneratingFunction[seq,x]//Head//Timing

HosoyaIndex

seq=Table[GraphData[{"Pell",n},"HosoyaIndex"],{n,0,9}]

HStarConnected

Table[GraphData[{"Pell",n},"HStarConnected"],{n,0,9}]

Hypohamiltonian

Table[GraphData[{"Pell",n},"Hypohamiltonian"],{n,0,9}]

Hypotraceable

Table[GraphData[{"Pell",n},"Hypotraceable"],{n,0,9}]

Identity

Table[GraphData[{"Pell",n},"Identity"],{n,0,9}]

IdiosyncraticPolynomial

(poly=Factor@Table[GraphData[{"Pell",n},"IdiosyncraticPolynomial"],{n,k0=0,9}]/.f_Function:>f[x,y])//Column

IndependenceNumber

A113225

seq=Table[GraphData[{"Pell",n},"IndependenceNumber"],{n,0,9}]

IndependencePolynomial

(poly=Table[GraphData[{"Pell",n},"IndependencePolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

IndependenceRatio

seq=Table[GraphData[{"Pell",n},"IndependenceRatio"],{n,0,9}]

FindSequenceFunction[seq,n+1]//Head//Timing

Table[((1+(((-1))^n)+2 Fibonacci[1+n,2])/(4 Fibonacci[1+n,2])),{n,0,9}]//Expand

IndependentEdgeSetCount

seq=Table[GraphData[{"Pell",n},"IndependentEdgeSetCount"],{n,0,9}]

IndependentVertexSetCount

seq=Table[GraphData[{"Pell",n},"IndependentVertexSetCount"],{n,0,9}]

IntersectionNumber

seq=Table[GraphData[{"Pell",n},"IntersectionNumber"],{n,0,9}]

IrredundanceNumber

seq=Table[GraphData[{"Pell",n},"IrredundanceNumber"],{n,0,9}]

IrredundancePolynomial

(poly=Table[GraphData[{"Pell",n},"IrredundancePolynomial"],{n,0,9}]/.f_Function:>f[x])//Column

IrredundantSetCount

seq=Table[GraphData[{"Pell",n},"IrredundantSetCount"],{n,0,9}]

IsolatedPointCount

seq=Table[GraphData[{"Pell",n},"IsolatedPointCount"],{n,0,9}]

KatzCentralities

Table[GraphData[{"Pell",n},"KatzCentralities"],{n,0,9}]/.f_Function:>f[x,y]//Column

KirchhoffIndex

seq=Table[GraphData[{"Pell",n},"KirchhoffIndex"],{n,0,9}]

KirchhoffSumIndex

seq=Table[GraphData[{"Pell",n},"KirchhoffSumIndex"],{n,0,9}]

LaplacianPolynomial

(poly=Factor@Table[GraphData[{"Pell",n},"LaplacianPolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

poly=ReadProperty["LaplacianPolynomial",GraphNameToFileName[{"Pell",9}]<>".gz"]

Exponent[poly,x]

$MaxRootDegree=Exponent[poly,x]

spec=CharacteristicPolynomialToSpectrum[poly,x];//Timing

N[spec]

LaplacianSpectralRadius

seq=Table[GraphData[{"Pell",n},"LaplacianSpectralRadius"],{n,0,9}]

poly=ReadProperty["LaplacianPolynomial",GraphNameToFileName[{"Pell",9}]<>".gz"]

Exponent[poly,x]

$MaxRootDegree=Exponent[poly,x]

spec=CharacteristicPolynomialToSpectrum[poly,x];//Timing

N[spec]

Laplacian spectral radius:

laplacianspectralradius=spec[[-1]]

ByteCount[laplacianspectralradius]

Algebraic connectivity:

algebraiconnectivity=spec[[2]]

ByteCount[algebraiconnectivity]

Laplacian spectral ratio:

?ToAlgebraicRoot

ToAlgebraicRoot[(laplacianspectralradius/algebraiconnectivity),"MaxDegree"->1024]//Timing

LaplacianSpectralRatio

seq=Table[GraphData[{"Pell",n},"LaplacianSpectralRadius"],{n,0,9}]

poly=ReadProperty["LaplacianPolynomial",GraphNameToFileName[{"Pell",9}]<>".gz"]

Exponent[poly,x]

$MaxRootDegree=Exponent[poly,x]

spec=CharacteristicPolynomialToSpectrum[poly,x];//Timing

N[spec]

Laplacian spectral radius:

laplacianspectralradius=spec[[-1]]

ByteCount[laplacianspectralradius]

Algebraic connectivity:

algebraiconnectivity=spec[[2]]

ByteCount[algebraiconnectivity]

Laplacian spectral ratio:

?ToAlgebraicRoot

ToAlgebraicRoot[(laplacianspectralradius/algebraiconnectivity),"MaxDegree"->1024]//Timing

LeafCount

seq=Table[GraphData[{"Pell",n},"LeafCount"],{n,0,9}]

Likelihood

seq=Table[GraphData[{"Pell",n},"Likelihood"],{n,0,9}]

LinearArboricity

seq=Table[GraphData[{"Pell",n},"LinearArboricity"],{n,0,9}]

LocalClusteringCoefficients

Tally/@Table[GraphData[{"Pell",n},"LocalClusteringCoefficients"],{n,0,9}]//Column

LongestCycleCount

seq=Table[GraphData[{"Pell",n},"LongestCycleCount"],{n,0,9}]

LongestPathCount

seq=Table[GraphData[{"Pell",n},"LongestPathCount"],{n,0,9}]

LongestPathLength

seq=Table[GraphData[{"Pell",n},"LongestPathLength"],{n,0,9}]

LovaszNumber

A113225

seq=Table[GraphData[{"Pell",n},"LovaszNumber"],{n,0,9}]

LowerCliqueNumber

seq=Table[GraphData[{"Pell",n},"LowerCliqueNumber"],{n,0,9}]

LowerIndependenceNumber

seq=Table[GraphData[{"Pell",n},"LowerIndependenceNumber"],{n,0,9}]

LowerMatchingNumber

seq=Table[GraphData[{"Pell",n},"LowerMatchingNumber"],{n,0,9}]

MatchingGeneratingPolynomial

(poly=Table[GraphData[{"Pell",n},"MatchingGeneratingPolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

MatchingNumber

A105635

seq=Table[GraphData[{"Pell",n},"MatchingNumber"],{n,0,9}]

seq=Table[(Fibonacci[n+1,2]-Fibonacci[n+1,0])/2,{n,0,20}]

Table[Fibonacci[n+1,0],{n,10}]

gf=FindGeneratingFunction[seq,x]

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

Table[((-(((1-(Sqrt[2])))^n) (-2+(Sqrt[2]))+(((1+(Sqrt[2])))^n) (2+(Sqrt[2])))/4),{n,0,10}]//Expand

Table[(2Fibonacci[n+1,2]-(((-1))^n)-1)/4,{n,0,10}]

Floor[Fibonacci[Range[20],2]/2]

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

FindLinearRecurrence[seq]

LinearRecurrence[{2,2,-2,-1},{0,1,2,6,14},20]

MatchingPolynomial

(poly=Table[GraphData[{"Pell",n},"MatchingPolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

MaximalCliqueCount

A364553

seq=Table[GraphData[{"Pell",n},"MaximalCliqueCount"],{n,0,9}]

MaximalCliquePolynomial

(poly=Table[GraphData[{"Pell",n},"MaximalCliquePolynomial"],{n,0,9}]/.f_Function:>f[x])

MaximalIndependencePolynomial

(poly=Table[GraphData[{"Pell",n},"MaximalIndependencePolynomial"],{n,0,9}]/.f_Function:>f[x])//Column

MaximalIndependentEdgeSetCount

seq=Table[GraphData[{"Pell",n},"MaximalIndependentEdgeSetCount"],{n,0,9}]

MaximalIndependentVertexSetCount

seq=Table[GraphData[{"Pell",n},"MaximalIndependentVertexSetCount"],{n,0,9}]

MaximalIrredundancePolynomial

(poly=Table[GraphData[{"Pell",n},"MaximalIrredundancePolynomial"],{n,0,9}]/.f_Function:>f[x])//Column

MaximalIrredundantSetCount

seq=Table[GraphData[{"Pell",n},"MaximalIrredundantSetCount"],{n,0,9}]

MaximalMatchingGeneratingPolynomial

(poly=Table[GraphData[{"Pell",n},"MaximalMatchingGeneratingPolynomial"],{n,0,9}]/.f_Function:>f[x])//Column

MaximumCliqueCount

A364553

seq=Table[GraphData[{"Pell",n},"MaximumCliqueCount"],{n,0,9}]

MaximumIndependentEdgeSetCount

seq=Table[GraphData[{"Pell",n},"MaximumIndependentEdgeSetCount"],{n,0,9}]

MaximumIndependentVertexSetCount

Inferred for n == 9:

seq=Table[GraphData[{"Pell",n},"MaximumIndependentVertexSetCount"],{n,0,9}]

MaximumIrredundantSetCount

seq=Table[GraphData[{"Pell",n},"MaximumIrredundantSetCount"],{n,0,9}]

MaximumLeafNumber

seq=Table[GraphData[{"Pell",n},"MaximumLeafNumber"],{n,0,9}]

MaximumVertexDegree

seq=Table[GraphData[{"Pell",n},"MaximumVertexDegree"],{n,0,9}]

Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

Table[2n-1,{n,0,9}]

MeanClusteringCoefficient

seq=Table[GraphData[{"Pell",n},"MeanClusteringCoefficient"],{n,0,9}]

MeanDistance

seq=Table[GraphData[{"Pell",n},"MeanDistance"],{n,0,9}]

FindSequenceFunction[seq,n+1]//Head//Timing

Median

Dovslic, T.  "Metallic Cubes.'" 26 Jul 2023.  https://arxiv.org/abs/2307.14054
Munarini [11] proved that Pell graphs are median graphs
Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

seq=Table[GraphData[{"Pell",n},"Median"],{n,0,9}]

Print[MedianGraphQ[#]//Timing]&/@pellgraphs

MetricDimension

seq=Table[GraphData[{"Pell",n},"MetricDimension"],{n,0,9}]

MinimalConnectedDominatingSetCount

seq=Table[GraphData[{"Pell",n},"MinimalConnectedDominatingSetCount"],{n,0,9}]

MinimalConnectedDominationPolynomial

(seq=Table[GraphData[{"Pell",n},"MinimalConnectedDominationPolynomial"],{n,0,9}]/.f_Function:>f[x])//Column

MinimalDominatingSetCount

seq=Table[GraphData[{"Pell",n},"MinimalDominatingSetCount"],{n,0,9}]

MinimalDominatingSetSignature

seq=Table[GraphData[{"Pell",n},"MinimalDominatingSetSignature"],{n,0,9}]//Column

MinimalDominationPolynomial

(poly=Table[GraphData[{"Pell",n},"MaximalMatchingGeneratingPolynomial"],{n,0,9}]/.f_Function:>f[x])//Column

MinimalEdgeCoverCount

seq=Table[GraphData[{"Pell",n},"MinimalEdgeCoverCount"],{n,0,9}]

MinimalEdgeCoverPolynomial

(poly=Table[GraphData[{"Pell",n},"MinimalEdgeCoverPolynomial"],{n,0,9}]/.f_Function:>f[x])//Column

MinimalEdgeCutCount

seq=Table[GraphData[{"Pell",n},"MinimalEdgeCutCount"],{n,0,9}]

MinimalEdgeCutPolynomial

(poly=Table[GraphData[{"Pell",n},"MinimalEdgeCutPolynomial"],{n,0,9}]/.f_Function:>f[x])//Column

MinimalTotalDominatingSetCount

seq=Table[GraphData[{"Pell",n},"MinimalTotalDominatingSetCount"],{n,0,9}]

MinimalTotalDominationPolynomial

(poly=Table[GraphData[{"Pell",n},"MinimalTotalDominationPolynomial"],{n,0,9}]/.f_Function:>f[x])//Column

MinimalVertexCoverCount

seq=Table[GraphData[{"Pell",n},"MinimalVertexCoverCount"],{n,0,9}]

MinimalVertexCoverPolynomial

(poly=Table[GraphData[{"Pell",n},"MinimalVertexCoverPolynomial"],{n,0,9}]/.f_Function:>f[x])//Column

MinimalVertexCutCount

seq=Table[GraphData[{"Pell",n},"MinimalVertexCutCount"],{n,0,9}]

MinimalVertexCutPolynomial

(poly=Table[GraphData[{"Pell",n},"MinimalVertexCutPolynomial"],{n,0,9}]/.f_Function:>f[x])//Column

MinimumCliqueCoveringCount

seq=Table[GraphData[{"Pell",n},"MinimumCliqueCoveringCount"],{n,0,9}]

MinimumConnectedDominatingSetCount

seq=Table[GraphData[{"Pell",n},"MinimumConnectedDominatingSetCount"],{n,0,9}]

FindSequenceFunction[DeleteMissing@seq,n]

MinimumCoveringsByMaximalCliquesCount

seq=Table[GraphData[{"Pell",n},"MinimumCoveringsByMaximalCliquesCount"],{n,0,9}]

MinimumCyclicEdgeCutCount

seq=Table[GraphData[{"Pell",n},"MinimumCyclicEdgeCutCount"],{n,0,9}]

MinimumDistinguishingLabelingCount

seq=Table[GraphData[{"Pell",n},"MinimumDistinguishingLabelingCount"],{n,0,9}]

FindSequenceFunction[seq,n+1]

MinimumDominatingSetCount

seq=Table[GraphData[{"Pell",n},"MinimumDominatingSetCount"],{n,0,9}]

MinimumEdgeCoverCount

seq=Table[GraphData[{"Pell",n},"MinimumEdgeCoverCount"],{n,0,9}]

MinimumEdgeCutCount

seq=Table[GraphData[{"Pell",n},"MinimumEdgeCutCount"],{n,0,9}]

MinimumLeafNumber

seq=Table[GraphData[{"Pell",n},"MinimumLeafNumber"],{n,0,9}]

MinimumPathCoveringCount

seq=Table[GraphData[{"Pell",n},"MinimumPathCoveringCount"],{n,0,9}]

MinimumTotalDominatingSetCount

seq=Table[GraphData[{"Pell",n},"MinimumTotalDominatingSetCount"],{n,0,9}]

MinimumVertexCoverCount

seq=Table[GraphData[{"Pell",n},"MinimumVertexCoverCount"],{n,0,9}]

MinimumVertexCutCount

seq=Table[GraphData[{"Pell",n},"MinimumVertexCutCount"],{n,0,9}]

MinimumVertexDegree

seq=Table[GraphData[{"Pell",n},"MinimumVertexDegree"],{n,0,9}]

Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

Table[Ceiling[n/2],{n,0,9}]==seq

MolecularTopologicalIndex

seq=Table[GraphData[{"Pell",n},"MolecularTopologicalIndex"],{n,0,9}]

FindSequenceFunction[DeleteMissing@seq,n+1]//Head//Timing

Nonhamiltonian

Table[GraphData[{"Pell",n},"Nonhamiltonian"],{n,0,9}]

NoPerfectMatching

Table[GraphData[{"Pell",n},"NoPerfectMatching"],{n,0,9}]

OddChordlessCycleCount

seq=Table[GraphData[{"Pell",n},"OddChordlessCycleCount"],{n,0,9}]

OddChordlessCyclePolynomial

poly=Table[GraphData[{"Pell",n},"OddChordlessCyclePolynomial"],{n,0,9}]/.f_Function:>f[x]

Outerplanar

Table[GraphData[{"Pell",n},"Outerplanar"],{n,0,9}]

PageRankCentralities

(seq=Table[GraphData[{"Pell",n},"PageRankCentralities"],{n,0,9}]/.f_Function:>f[x,y])//Column

PathCount

seq=Table[GraphData[{"Pell",n},"PathCount"],{n,0,9}]

PathCoveringNumber

seq=Table[GraphData[{"Pell",n},"PathCoveringNumber"],{n,0,9}]

PathPolynomial

(poly=Table[GraphData[{"Pell",n},"PathPolynomial"],{n,k0=0,kmax=9}]/.f_Function:>f[x])//Column

CoefficientList[Take[poly,7],x]

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

PentagonCount

seq=Table[GraphData[{"Pell",n},"PentagonCount"],{n,0,9}]

Perfect

Table[GraphData[{"Pell",n},"Perfect"],{n,0,9}]

PerfectMatching

Table[GraphData[{"Pell",n},"PerfectMatching"],{n,0,9}]

Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

Periphery

seq=Table[GraphData[{"Pell",n},"Periphery"],{n,0,9}]

Planar

Table[GraphData[{"Pell",n},"Planar"],{n,0,9}]

ProjectivePlaneCrossingNumber

seq=Table[GraphData[{"Pell",n},"ProjectivePlaneCrossingNumber"],{n,0,9}]

Pseudoarboricity

seq=Table[GraphData[{"Pell",n},"Pseudoarboricity"],{n,0,9}]

Ptolemaic

Inferred for n > 6:

seq=Table[GraphData[{"Pell",n},"Ptolemaic"],{n,0,9}]

QChromaticPolynomial

(poly=Factor[Table[GraphData[{"Pell",n},"QChromaticPolynomial"],{n,k0=0,kmax=9}]/.f_Function:>f[x]])//Column

QuadraticallyEmbeddable

Inferred for n > 7:

seq=Table[GraphData[{"Pell",n},"QuadraticallyEmbeddable"],{n,0,9}]

QuadraticEmbeddingConstant

seq=Table[GraphData[{"Pell",n},"QuadraticEmbeddingConstant"],{n,0,9}]

RadialityCentralities

seq=Table[GraphData[{"Pell",n},"RadialityCentralities"],{n,0,9}]

RadioLabelingCount

seq=Table[GraphData[{"Pell",n},"RadioLabelingCount"],{n,0,9}]

RadioNumber

seq=Table[GraphData[{"Pell",n},"RadioNumber"],{n,0,9}]

Radius

Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

seq=Table[GraphData[{"Pell",n},"Radius"],{n,0,9}]

Rank

A005409

seq=Table[GraphData[{"Pell",n},"Rank"],{n,0,9}]

Join[{1},Fibonacci[Range[2,20],2]-1]

seq=Fibonacci[Range[2,20],2]-1

FindLinearRecurrence[seq]

Join[{1},LinearRecurrence[{3,-1,-1},{1,4,11},20]]

RankPolynomial

(poly=Factor@Table[GraphData[{"Pell",n},"RankPolynomial"],{n,k0=0,9}]/.f_Function:>f[x,y])//Column

RectilinearCrossingNumber

seq=Table[GraphData[{"Pell",n},"RectilinearCrossingNumber"],{n,0,9}]

Regular

Table[GraphData[{"Pell",n},"Regular"],{n,0,9}]

RegularParameters

seq=Table[GraphData[{"Pell",n},"RegularParameters"],{n,0,9}]

ReliabilityPolynomial

(poly=Factor@Table[GraphData[{"Pell",n},"ReliabilityPolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

ResistanceMatrix

MatrixForm/@Table[GraphData[{"Pell",n},"ResistanceMatrix"],{n,6}]

ScrambleNumber

seq=Table[GraphData[{"Pell",n},"ScrambleNumber"],{n,0,9}]

ShannonCapacity

A113225

seq=Table[GraphData[{"Pell",n},"ShannonCapacity"],{n,0,9}]

SigmaPolynomial

(poly=Factor@Table[GraphData[{"Pell",n},"SigmaPolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

Singlecross

Table[GraphData[{"Pell",n},"Singlecross"],{n,0,9}]

Skewness

seq=Table[GraphData[{"Pell",n},"Skewness"],{n,0,9}]

SpanningTreeCount

seq=Table[GraphData[{"Pell",n},"SpanningTreeCount"],{n,0,9}]

FindSequenceFunction[seq,n+1]

SpectralRadius

seq=Table[GraphData[{"Pell",n},"SpectralRadius"],{n,0,9}]

SquareCount

A364619

seq=Table[GraphData[{"Pell",n},"SquareCount"],{n,0,9}]

Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

FindGeneratingFunction[seq,x]//Head//Timing

Do[Print[{{"Pell",n},CycleCount[PellGraph[n],4]//Timing}],{n,0,10}]

seq={0,0,1,8,40,164,601,2048,6632,20680,62633};

FindLinearRecurrence[seq]//Head//Timing

FindSequenceFunction[seq,n]//Head//Timing

gf=FindGeneratingFunction[seq,x]

StringReplace[ToString[gf,InputForm]," "->""]

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

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

Table[(1/64) ((((1+(Sqrt[2])))^n) ((Sqrt[2])-2 (3+(Sqrt[2])) n+2 (2+(Sqrt[2])) (n^2))-(((1-(Sqrt[2])))^n) ((Sqrt[2])+2 n (3-(Sqrt[2])+(-2+(Sqrt[2])) n))),{n,0,10}]//Expand

(1/64) ((((1+(Sqrt[2])))^n) Expand[((Sqrt[2])-2 (3+(Sqrt[2])) n+2 (2+(Sqrt[2])) (n^2))]+(((1-(Sqrt[2])))^n) Expand[-((Sqrt[2])+2 n (3-(Sqrt[2])+(-2+(Sqrt[2])) n))])

Table[(1/64) (((((1+(Sqrt[2])))^n) +(((1-(Sqrt[2])))^n))(-6n+4n^2)+((((1+(Sqrt[2])))^n) -(((1-(Sqrt[2])))^n))((Sqrt[2])-2 (Sqrt[2]) n+2 (Sqrt[2]) (n^2))),{n,0,10}]//Expand

A001333

Table[(1/2)((((1+(Sqrt[2])))^n) +(((1-(Sqrt[2])))^n)),{n,10}]//Expand

Table[(-I)^n ChebyshevT[n,I],{n,10}]

A000129

Table[(1/(2(Sqrt[2])))((((1+(Sqrt[2])))^n) -(((1-(Sqrt[2])))^n)),{n,10}]//Expand

Table[Fibonacci[n,2],{n,10}]

Factor[((Sqrt[2])-2 (Sqrt[2]) n+2 (Sqrt[2]) (n^2))2(Sqrt[2])]

seq=Table[(n(2n-3)(-I)^n ChebyshevT[n,I]+(2n^2-2n+1)Fibonacci[n,2])/16,{n,0,28}]

StringLength[ToString[seq]]

coef=FindLinearRecurrence[seq]

Take[seq,Length@coef]

LinearRecurrence[{6,-9,-4,9,6,1},{0,0,1,8,40,164},20]

SquareFree

Table[GraphData[{"Pell",n},"SquareFree"],{n,0,9}]

StabilityIndex

seq=Table[GraphData[{"Pell",n},"StabilityIndex"],{n,0,9}]

StarArboricity

seq=Table[GraphData[{"Pell",n},"StarArboricity"],{n,0,9}]

StatusCentralities

seq=Table[GraphData[{"Pell",n},"StatusCentralities"],{n,0,9}]

Strength

seq=Table[GraphData[{"Pell",n},"Strength"],{n,0,9}]

StronglyPerfect

Table[GraphData[{"Pell",n},"StronglyPerfect"],{n,0,9}]

StronglyRegular

Table[GraphData[{"Pell",n},"StronglyRegular"],{n,0,9}]

Switchable

Table[GraphData[{"Pell",n},"Switchable"],{n,0,9}]

Symmetric

Table[GraphData[{"Pell",n},"Symmetric"],{n,0,9}]

SymmetricallyDistinctVertexPairCount

seq=Table[GraphData[{"Pell",n},"SymmetricallyDistinctVertexPairCount"],{n,0,9}]

FindGeneratingFunction[seq,x]//Head//Timing

SymmetricallyDistinctVertexPairSignature

seq=Table[GraphData[{"Pell",n},"SymmetricallyDistinctVertexPairSignature"],{n,0,9}]

Thickness

seq=Table[GraphData[{"Pell",n},"Thickness"],{n,0,9}]

TopologicalIndex

seq=Table[GraphData[{"Pell",n},"TopologicalIndex"],{n,0,9}]

FindSequenceFunction[seq,n+1]//FullSimplify

ToroidalCrossingNumber

seq=Table[GraphData[{"Pell",n},"ToroidalCrossingNumber"],{n,0,9}]

TotalDominatingSetCount

seq=Table[GraphData[{"Pell",n},"TotalDominatingSetCount"],{n,0,9}]

TotalDominationNumber

seq=Table[GraphData[{"Pell",n},"TotalDominationNumber"],{n,0,9}]

Do[Print[{gn={"Pell",n},Flatten[{set=Flatten@FindTotalDominatingSet[PellGraph[n],Method->"Mosek"],Length[set]}//Timing,1]}],{n,7}]

TotalDominationPolynomial

(poly=Table[GraphData[{"Pell",n},"TotalDominationPolynomial"],{n,0,9}]/.f_Function:>f[x])//Column

Toughness

seq=Table[GraphData[{"Pell",n},"Toughness"],{n,0,9}]

Traceable

Table[GraphData[{"Pell",n},"Traceable"],{n,0,9}]

TreeDepth

seq=Table[GraphData[{"Pell",n},"TreeDepth"],{n,0,9}]

TreeNumber

seq=Table[GraphData[{"Pell",n},"TreeNumber"],{n,0,9}]

Treewidth

seq=Table[GraphData[{"Pell",n},"Treewidth"],{n,0,9}]

Triameter

Inferred for n > 7:

seq=Table[GraphData[{"Pell",n},"Triameter"],{n,0,9}]

Do[Print[{{"Pell",n},GraphTriameter[PellGraph[n]]//Timing}],{n,0,9}]

TriangleCount

seq=Table[GraphData[{"Pell",n},"TriangleCount"],{n,0,9}]

TriangleFree

Table[GraphData[{"Pell",n},"TriangleFree"],{n,0,9}]

TuttePolynomial

(Table[GraphData[{"Pell",n},"TuttePolynomial"],{n,0,9}]/.f_Function:>f[x,y])//Column

Unicyclic

seq=Table[GraphData[{"Pell",n},"Unicyclic"],{n,0,9}]

UniquelyColorable

seq=Table[GraphData[{"Pell",n},"UniquelyColorable"],{n,0,9}]

UniquelyPancyclic

seq=Table[GraphData[{"Pell",n},"UniquelyPancyclic"],{n,0,9}]

UnitDistance

Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

Since (Subscript[Q, n]) is UD<, a subgraph of it is.

seq=Table[GraphData[{"Pell",n},"UnitDistance"],{n,0,9}]

UpperDominationNumber

seq=Table[GraphData[{"Pell",n},"UpperDominationNumber"],{n,0,9}]

UpperIrredundanceNumber

seq=Table[GraphData[{"Pell",n},"UpperIrredundanceNumber"],{n,0,9}]

VertexConnectivity

Table[GraphData[{"Pell",n},"VertexConnectivity"],{n,0,9}]

VertexCount

A000129

seq=Table[GraphData[{"Pell",n},"VertexCount"],{n,0,9}]

Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

FindSequenceFunction[seq,n+1]

VertexCoverCount

seq=Table[GraphData[{"Pell",n},"VertexCoverCount"],{n,0,9}]

VertexCoverNumber

A105635

seq=Table[GraphData[{"Pell",n},"VertexCoverNumber"],{n,0,9}]

VertexCoverPolynomial

(poly=Factor@Table[GraphData[{"Pell",n},"VertexCoverPolynomial"],{n,k0=0,9}]/.f_Function:>f[x])//Column

VertexCutCount

seq=Table[GraphData[{"Pell",n},"VertexCutCount"],{n,0,9}]

VertexDegreePolynomial

Munarini, E.  "Pell Graphs."  Disc. Math. 342, 2415–2428, 2019.

Total[(#2 x^#&@@@Tally[#])]&/@Table[GraphData[{"Pell",n},"VertexDegrees"],{n,0,9}]//Column

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

Table[Expand@SeriesCoefficient[((1+(x-(x^2))t)/(1-(x+(x^2))t-(x+(x^2)-(x^3))(t^2)-((x^2)-(x^3))(t^3))),{t,0,n}],{n,0,9}]//Column

VertexDegrees

Tally/@Table[GraphData[{"Pell",n},"VertexDegrees"],{n,0,9}]//Column

VertexTransitive

seq=Table[GraphData[{"Pell",n},"VertexTransitive"],{n,0,9}]

WeisfeilerLemanDimension

seq=Table[GraphData[{"Pell",n},"WeisfeilerLemanDimension"],{n,0,9}]

FindSequenceFunction[seq,n+1]

WienerIndex

seq=Table[GraphData[{"Pell",n},"WienerIndex"],{n,0,9}]

FindSequenceFunction[seq,n+1]//Head//Timing

FindGeneratingFunction[seq,x]//Head//Timing

WienerSumIndex

seq=Table[GraphData[{"Pell",n},"WienerSumIndex"],{n,0,9}]

FindSequenceFunction[DeleteMissing@seq,n+1]//Head//Timing

Algorithms

Pell strings

Munarini, E.  Pell Graphs.  Disc. Math. 342, 2415–2428, 2019.

https://re.public.polimi.it/retrieve/e0c31c11-517b-4599-e053-1705fe0aef77/11311-1088084_Munarini.pdf

PellEdgeQ[{e1_,e2_}]:=Module[{t=Sort/@Transpose[{e1,e2}]},
Or[
MatchQ[DeleteCases[t,{x_,x_}],{{0,1}}],
MatchQ[DeleteCases[Split[t],{x_,x_},{2}],{{}...,{{1,2},{1,2}},{}...}]
]
]

PellVertices[n_]:=Select[Flatten/@Tuples[{0,1,2},n],AllTrue[Length/@Cases[Split[#],{2..}],EvenQ]&]

PellGraph[n_]:=Module[{v=PellVertices[n],e},
e=Select[Subsets[Range@Length@v,{2}],PellEdgeQ[v[[#]]]&];
Graph[Range@Length@v,UndirectedEdge@@@e]
]

PellEdgeQ[{{0,0,0,0},{0,0,0,1}}]

PellEdgeQ[{{2,2,1,1},{2,2,2,2}}]

PellEdgeQ[{{2,2,1,1},{2,2,2,1}}]

<<MathWorld`Graphs`

pellgraphs=Table[PellGraph[n],{n,0,7}]

rules=DeleteCases[MapIndexed[RecognizeGraph[#]->(#2[[1]]-1)&,pellgraphs],HoldPattern[{}->_]]

VertexCount/@pellgraphs

Table[Fibonacci[n,2],{n,11}]

GraphData

GraphDataClassString

GraphDataClassString["Pell"]

GraphDataString

<<MathWorld`Graphs`

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

Do[WriteString[file,GraphDataString[
PellGraph[n],
"Skip"->{"ChromaticPolynomial","ConnectedInducedSubgraphPolynomial","ConnectedInducedSubgraphs","ConstructionNumber","CrossingNumber","CyclicEdgeConnectivity","Embeddings3D","HamiltonianWalks","LinklesslyEmbeddable","LovaszNumber","LowerMatchingNumber","MaximumIndependentVertexSets","MinimumCyclicEdgeCut","MinimumCyclicEdgeCutCount","MinimumCyclicEdgeCuts","MinimumDistinguishingLabelings","MinimumDominatingSet","MinimumVertexCut","MinimumTotalDominatingSet","MinorCount","Minors","ProjectivePlanar","QuadraticEmbeddingConstant","TuttePolynomial"},
"Embeddings"->None,
"StandardName"->{"Pell",n},
"Name"->ToString[n]<>"\[Hyphen]Pell graph",
"Classes"->{"Pell"},
"Notation"->Subscript["Π",n],
"Information"->"PellGraph",
"NotationRules"->{"Pell"->n},
TimeConstraint->10
]],{n,0,9}];

Close[file]