Ladder Graph

Author

Eric W. Weisstein
September 29, 2024

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

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

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

Figure

Graphs.m using Combinatorica (old)

Show[GraphicsArray[Block[{$DisplayFunction=Identity},Table[ShowGraph[LadderGraph[k],PlotLabel->TraditionalForm[n==k],VertexStyle->Disk[.1]],{k,6}]
]],ImageSize->500]

Graphs.m

GraphicsRow[Table[LadderGraph[n,VertexSize->{"Scaled",.07},System`VertexStyle->Red,System`EdgeStyle->Black],{n,6}]]

Special cases

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

Embeddings

<<MathWorld`Graphs`

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

Names

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

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

{"Ladder", 2} == "SquareGraph"

Names

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

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

{"Ladder", 3} == "DominoGraph"

Names

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

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

{"Ladder", 4}

Names

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

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

Primary

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

Graphs.m

Table[LadderGraph[4,GraphLayout->layout],{layout,{"GridEmbedding","CircularEmbedding"}}]//StyleGraphs

All

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

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData[{"Ladder",4}]]

Default embedding

GraphPlot[Graph[GraphData[{"Ladder",4},"VertexList"],GraphData[{"Ladder",4},"Edges"]]]

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

RecognizeGraph@g

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

RecognizeGraph@g

Degenerate

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

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

Configuration

GetGraphData[{"Ladder",4},"File"]

GraphData[{"Ladder",4},"Configuration"]

GraphData[{"Ladder",4},"Graph","Configuration"]//StyleGraphs

Incidence

GetGraphData[{"Ladder",4},"File"]

GraphData[{"Ladder",4},"Incidence"]

Select[GraphData[All],GraphData[#,"LeviGraph","Name"]==={"Ladder",4}&]

lines=GraphData[{"Ladder",4},"IncidenceLines"]

Perspective

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

HoneycombToroidal construction

GraphData[{"Ladder",4},"NotationRules","HoneycombToroidal"]

GeneralizedPetersen

GraphData[{"Ladder",4},"Graph","GeneralizedPetersen"]//StyleGraphs

IGraph

GraphData[{"Ladder",4},"Graph","IGraph"]//StyleGraphs

Bilateral

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

Grid[DeleteCases[Monitor[{g=#,BilateralGraphEmbeddingTypes[#],GraphSymmetryGroup[#]}&/@GraphData[{"Ladder",4},"Graph","All"],g],{_,_,TransformationGroup[{}]}],Dividers->All,Alignment->{{Center,{Left}}}]

Circular

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

Circular construction

GraphData[{"Ladder",4},"Hamiltonian"]

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

Length[hc=FindHamiltonianCycle[GraphData[{"Ladder",4}],All][[All,All,1]]]//Timing

(circ=CircularEmbeddingFromHamiltonianCycle[GraphData[{"Ladder",4}],#]&/@hc);//Timing

sym={#,GraphSymmetryGroup[#]}&/@circ;//Timing

Tally[sym[[All,-1]]]

Cases[sym,{gn_,TransformationGroup[XXX]}:>gn]

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph/@circ],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

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

quad

Planar

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

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

Planar construction

PlanarGraphEmbedding[GraphData[{"Ladder",4}],Method->#]&/@{"TutteLayout","DavidsonHarelLayout"}

Table[PlanarGraphEmbedding[GraphData[{"Ladder",4}]],10]

PlanarGraphEmbeddings

GraphData[{"Ladder",4},"PlanarEmbeddingCount"]

(plemb=PlanarGraphEmbeddings[GraphData[{"Ladder",4}]])//Timing

Length[plemb]

MinimalCrossing

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

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

MinimalLocalCrossing

GraphData[{"Ladder",4},{"LocalCrossingNumber","RectilinearLocalCrossingNumber"}]

GraphData[{"Ladder",4},"Graph","MinimalLocalCrossing"]//StyleGraphs

Circulant

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

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

Circulant construction

GraphData[{"Ladder",4},"NotationRules","Circulant"]

(ci=CirculantIndices[GraphData[{"Ladder",4}]])//Timing

CirculantGraph1@@@ci

Cases[GraphData[{"Ladder",4},"StandardNames"],{"Circulant",i_}:>i]

CirculantGraph1@@@%

Complement graph

compl=GraphData[{"Ladder",4},"ComplementGraph","Name"]

Line graph

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

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

RootGraph

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

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

RecognizeGraph[%]

Halin

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

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

Rigid

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

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

Flexible

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

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

LCF

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

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

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

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

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

gs={,,,,};

Matchstick

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

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

3D default

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

3D construction

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

RecognizeGraph[g]

3D All

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

3D UnitDistance

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

3D Polyhedron

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

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

{"Ladder", 5}

Names

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

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

Primary

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

Graphs.m

Table[LadderGraph[4,GraphLayout->layout],{layout,{"GridEmbedding","CircularEmbedding"}}]//StyleGraphs

All

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

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData[{"Ladder",5}]]

Default embedding

GraphPlot[Graph[GraphData[{"Ladder",5},"VertexList"],GraphData[{"Ladder",5},"Edges"]]]

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

RecognizeGraph@g

GraphReplot@

v={#/4,#2/3}&@@@Join[{{0,0},{4,0},{4,3},{0,3}},Sequence@@Table[{i,j},{i,3},{j,2}]];
g=Graph[Range@Length@v,UndirectedEdge@@@{e[1,2,3,4,1,5,6,4],e[2,7,8,3],e[10,9],{8,10},{7,9}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

v={#/4,#2/3}&@@@Join[{{0,0},{4,0},{4,3},{0,3}},Sequence@@Table[{i,j},{i,3},{j,2}]];
g=Graph[Range@Length@v,UndirectedEdge@@@{e[1,2,3,4,1,5,6,4],e[3,10,8,7,9,2],{9,10}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

gs=UnlabelGraph/@{,,}

Degenerate

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

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

Configuration

GetGraphData[{"Ladder",5},"File"]

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

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

Incidence

GetGraphData[{"Ladder",5},"File"]

GraphData[{"Ladder",5},"Incidence"]

Select[GraphData[All],GraphData[#,"LeviGraph","Name"]==={"Ladder",5}&]

lines=GraphData[{"Ladder",5},"IncidenceLines"]

Perspective

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

HoneycombToroidal construction

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

GeneralizedPetersen

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

IGraph

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

Bilateral

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

Grid[DeleteCases[Monitor[{g=#,BilateralGraphEmbeddingTypes[#],GraphSymmetryGroup[#]}&/@GraphData[{"Ladder",5},"Graph","All"],g],{_,_,TransformationGroup[{}]}],Dividers->All,Alignment->{{Center,{Left}}}]

Circular

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

Circular construction

GraphData[{"Ladder",5},"Hamiltonian"]

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

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

(circ=CircularEmbeddingFromHamiltonianCycle[GraphData[{"Ladder",5}],#]&/@hc);//Timing

sym={#,GraphSymmetryGroup[#]}&/@circ;//Timing

Tally[sym[[All,-1]]]

Cases[sym,{gn_,TransformationGroup[XXX]}:>gn]

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph/@circ],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

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

quad

Planar

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

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

Planar construction

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

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

PlanarGraphEmbeddings

GraphData[{"Ladder",5},"PlanarEmbeddingCount"]

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

Length[plemb]

MinimalCrossing

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

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

MinimalLocalCrossing

GraphData[{"Ladder",5},{"LocalCrossingNumber","RectilinearLocalCrossingNumber"}]

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

Circulant

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

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

Circulant construction

GraphData[{"Ladder",5},"NotationRules","Circulant"]

(ci=CirculantIndices[GraphData[{"Ladder",5}]])//Timing

CirculantGraph1@@@ci

Cases[GraphData[{"Ladder",5},"StandardNames"],{"Circulant",i_}:>i]

CirculantGraph1@@@%

Complement graph

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

Line graph

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

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

RootGraph

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

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

RecognizeGraph[%]

Halin

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

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

Rigid

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

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

Flexible

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

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

LCF

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

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

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

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

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

gs={,};

Matchstick

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

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

3D default

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

3D construction

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

RecognizeGraph[g]

3D All

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

3D UnitDistance

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

3D Polyhedron

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

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

{"Ladder", 6}

Names

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

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

Primary

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

Graphs.m

Table[LadderGraph[4,GraphLayout->layout],{layout,{"GridEmbedding","CircularEmbedding"}}]//StyleGraphs

All

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

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData[{"Ladder",6}]]

Default embedding

GraphPlot[Graph[GraphData[{"Ladder",6},"VertexList"],GraphData[{"Ladder",6},"Edges"]]]

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

RecognizeGraph@g

GraphPlot[Graph[GraphData[{"Ladder",6},"VertexList"],GraphData[{"Ladder",6},"Edges"]]]

GraphReplot@

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

RecognizeGraph@g

gs=UnlabelGraph/@{,}

RecognizeGraph/@gs

Degenerate

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

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

Configuration

GetGraphData[{"Ladder",6},"File"]

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

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

Incidence

GetGraphData[{"Ladder",6},"File"]

GraphData[{"Ladder",6},"Incidence"]

Select[GraphData[All],GraphData[#,"LeviGraph","Name"]==={"Ladder",6}&]

lines=GraphData[{"Ladder",6},"IncidenceLines"]

Perspective

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

HoneycombToroidal construction

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

GeneralizedPetersen

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

IGraph

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

Bilateral

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

Grid[DeleteCases[Monitor[{g=#,BilateralGraphEmbeddingTypes[#],GraphSymmetryGroup[#]}&/@GraphData[{"Ladder",6},"Graph","All"],g],{_,_,TransformationGroup[{}]}],Dividers->All,Alignment->{{Center,{Left}}}]

Circular

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

Circular construction

GraphData[{"Ladder",6},"Hamiltonian"]

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

Length[hc=FindHamiltonianCycle[GraphData[{"Ladder",6}],All][[All,All,1]]]//Timing

(circ=CircularEmbeddingFromHamiltonianCycle[GraphData[{"Ladder",6}],#]&/@hc);//Timing

sym={#,GraphSymmetryGroup[#]}&/@circ;//Timing

Tally[sym[[All,-1]]]

Cases[sym,{gn_,TransformationGroup[XXX]}:>gn]

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph/@circ],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

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

quad

Planar

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

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

Planar construction

PlanarGraphEmbedding[GraphData[{"Ladder",6}],Method->#]&/@{"TutteLayout","DavidsonHarelLayout"}

Table[PlanarGraphEmbedding[GraphData[{"Ladder",6}]],10]

PlanarGraphEmbeddings

GraphData[{"Ladder",6},"PlanarEmbeddingCount"]

(plemb=PlanarGraphEmbeddings[GraphData[{"Ladder",6}]])//Timing

Length[plemb]

MinimalCrossing

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

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

MinimalLocalCrossing

GraphData[{"Ladder",6},{"LocalCrossingNumber","RectilinearLocalCrossingNumber"}]

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

Circulant

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

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

Circulant construction

GraphData[{"Ladder",6},"NotationRules","Circulant"]

(ci=CirculantIndices[GraphData[{"Ladder",6}]])//Timing

CirculantGraph1@@@ci

Cases[GraphData[{"Ladder",6},"StandardNames"],{"Circulant",i_}:>i]

CirculantGraph1@@@%

Complement graph

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

Line graph

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

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

RootGraph

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

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

RecognizeGraph[%]

Halin

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

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

Rigid

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

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

Flexible

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

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

LCF

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

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

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

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

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

gs={,,,,,,};

Matchstick

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

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

3D default

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

3D construction

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

RecognizeGraph[g]

3D All

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

3D UnitDistance

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

3D Polyhedron

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

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

Properties

Acyclic

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

AdjacencyMatrixCount

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

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

Table[(2n)!/4,{n,20}]

Arboricity

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

ArcTransitive

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

ArcTransitivity

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

AutomorphismCount

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

BalabanIndex

Table[GraphData[{"Ladder",n},"BalabanIndex"],{n,10}]//FullSimplify

Bandwidth

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

BridgeCount

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

GraphData[{"Ladder",1}]

BurningNumber

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

CharacteristicPolynomial

(poly=Factor@Table[GraphData[{"Ladder",n},"CharacteristicPolynomial"][x],{n,k0=1,10}])//Column

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindPolynomialRecurrence[poly,6]

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^(k0-1))FindGeneratingFunction[poly,z]]

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

ChordCount

seq=Table[GraphData[{"Ladder",n},"ChordCount"],{n,20}]

GraphData[{"Ladder",5},"Graph","Labeled"]

GraphData[{"Ladder",5},"Chords"]

ChordlessCycleCount

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

ChromaticInvariant

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

ChromaticallyUnique

seq=Table[GraphData[{"Ladder",n},"ChromaticallyUnique"],{n,20}]

GraphData[{"ChromaticallyUnique","Ladder"}]

ChromaticNumber

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

ChromaticPolynomial

(poly=Factor@Table[GraphData[{"Ladder",n},"ChromaticPolynomial"][x],{n,k0=1,10}])//Column

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindPolynomialRecurrence[poly,6]

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]

Expand[(3+(-3+x) x)]

Table[x(x-1)(((x^2-3x+3))^(n-1)),{n,10}]

FullSimplify[%-poly]

Circumference

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

Classes

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

CliqueCount

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

FindSequenceFunction[seq,n]

CliqueCoveringNumber

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

CliqueNumber

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

CliquePolynomial

(poly=Factor@Table[GraphData[{"Ladder",n},"CliquePolynomial"][x],{n,k0=1,10}])//Column

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindPolynomialRecurrence[poly,6]

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]

Table[n x(3x+2)-2x^2+1,{n,10}]

FullSimplify[%-poly]

CochromaticGraphNames

Table[GraphData[{"Ladder",n},"CochromaticGraphNames"],{n,20}]//Column

ComplementGraphName

Table[GraphData[{"Ladder",n},"ComplementGraphName"],{n,20}]

ComplementOddChordlessCycleCount

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

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

ConnectedComponentCount

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

ConnectedDominatingSetCount

A291706

seq=Table[GraphData[{"Ladder",n},"ConnectedDominatingSetCount"],{n,20}]

gf=x FindGeneratingFunction[DeleteCases[seq,_Missing],x]

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

seq=Table[Piecewise[{{3^n,n==1||n==2}},4LucasL[n-1,2]],{n,30}]

FindLinearRecurrence[Drop[seq,2]]

Join[{3,9},LinearRecurrence[{2,1},{24,56},20]]

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

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

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

ConnectedDominationNumber

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

FindSequenceFunction[seq,n]

Table[Piecewise[{{2,n==3}},n],{n,20}]

%==seq

ConnectedDominationPolynomial

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

CoefficientList[Take[poly,5],x]

Table[Function[x,Evaluate@Factor@LinearRecurrence[{x (1+x),x^3},{x^2 (1+x)^2 (1+4 x+x^2),x^4 (1+x)^2 (7+6 x+x^2)},{n-2}][[1]]][x],{n,3,10}]//Column

Take[poly,2]

Take[poly,{3,4}]

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

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

LinearRecurrence[{x (1+x),(x^3)},{(((((1+x))^2) (1+(x^2)))/(x^2)),(((1+x))^2) (-1+2 x+(x^2))},3]

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[Piecewise[{{x (2+x),n==1},{(x^2) (((2+x))^2),n==2}},(1/((x^4) (Sqrt[1+x (6+x)])))(2^(-1-n)) (((1+x))^2) ((((x (1+x-(Sqrt[1+x (6+x)]))))^n) (-1-(Sqrt[1+x (6+x)])-x (5+x+(x^2)+2 (Sqrt[1+x (6+x)])-x (Sqrt[1+x (6+x)])))+(((x (1+x+(Sqrt[1+x (6+x)]))))^n) (1-(Sqrt[1+x (6+x)])+x (5-2 (Sqrt[1+x (6+x)])+x (1+x+(Sqrt[1+x (6+x)])))))],{n,5}]//Expand//Column

FullSimplify[poly2-poly]

4LucasL[n-1,2]

Take[poly,4]//Expand//Column

Table[LucasL[n-1,(((x+1))^2)],{n,4}]//Expand//Column

CoefficientList[Table[(1/((x^4) (Sqrt[1+x (6+x)])))(2^(-1-n)) ((((x (1+x-(Sqrt[1+x (6+x)]))))^n) (-1-(Sqrt[1+x (6+x)])-x (5+x+(x^2)+2 (Sqrt[1+x (6+x)])-x (Sqrt[1+x (6+x)])))+(((x (1+x+(Sqrt[1+x (6+x)]))))^n) (1-(Sqrt[1+x (6+x)])+x (5-2 (Sqrt[1+x (6+x)])+x (1+x+(Sqrt[1+x (6+x)]))))),{n,3,7}],x]//Expand

ConnectedInducedSubgraphCount

A059020

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

FindLinearRecurrence[DeleteCases[seq,_Missing]]

LinearRecurrence[{4,-4,0,1},{3,13,40,108},20]

FindSequenceFunction[DeleteCases[seq,_Missing],n]//FullSimplify

gf=x FindGeneratingFunction[DeleteCases[seq,_Missing],x]

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

Table[(7-5 (Sqrt[2])) (((1-(Sqrt[2])))^n)+(((1+(Sqrt[2])))^n) (7+5 (Sqrt[2])),{n,10}]//Expand

Table[LucasL[n+3,2],{n,10}]

Table[(LucasL[n+3,2]-8n-14)/4,{n,10}]

ConnectedInducedSubgraphPolynomial

Probably a Lucas polynomial

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

Take[poly,4]//Column

Table[(LucasL[n+3,x+1]-8n-14)/4,{n,4}]

CoefficientList[Take[poly,5],x]

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[-1+XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

CrossingNumber

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

CycleCount

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

FindSequenceFunction[seq,n]

CyclePolynomial

(poly=Factor@Table[GraphData[{"Ladder",n},"CyclePolynomial"][x],{n,k0=1,10}])//Column

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindPolynomialRecurrence[poly,6]

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]

Table[((((x^(2 n))-n ((x^2)-1)-1)(x^4) )/((((x^2)-1))^2)),{n,10}]//Factor

FullSimplify[%-poly]

Cyclic

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

CyclomaticNumber

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

DetourIndex

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

FindSequenceFunction[seq,n]//FullSimplify

gf=x FindGeneratingFunction[seq,x]

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

Factor[(14+n (-13+8 n))]

Table[(2 n (14-13 n+8 (n^2))-15-(((-1))^n))/4,{n,20}]==seq

DetourPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindPolynomialRecurrence[poly,6]

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]

Table[((((x^(2 n))-n ((x^2)-1)-1)(x^4) )/((((x^2)-1))^2)),{n,10}]//Factor

FullSimplify[%-poly]

Diameter

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

DistanceMatrix

MatrixForm/@(seq=Table[GraphData[{"Ladder",n},"DistanceMatrix"],{n,5}])

DistancePolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindPolynomialRecurrence[poly,6]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n]

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]

Table[((((x^(2 n))-n ((x^2)-1)-1)(x^4) )/((((x^2)-1))^2)),{n,10}]//Factor

FullSimplify[%-poly]

DistinguishingNumber

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

DomaticNumber

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

Table[LadderGraph[n,VertexLabels->"Name"],{n,6}]

Monitor[Table[DomaticNumber[g=LadderGraph[n]],{n,6}],n]//Timing

DominatingSetCount

A218348

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

FindLinearRecurrence[seq]

seq=LinearRecurrence[{3,2,2,-1,-1},{3,11,41,149,547},20]

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

FindSequenceFunction[Rest@seq,n-1]

gf=x FindGeneratingFunction[seq,x]

gf=((x (3+2 x+2 (x^2)-2 (x^3)-(x^4)))/(1-3 x-2 (x^2)-2 (x^3)+(x^4)+(x^5)));

soln=Simplify[SeriesCoefficient[gf,{x,0,n}],n>2]

Table[soln,{n,5}]//FullSimplify

(List@@soln/.n->0)//N//Chop

FullSimplify[(List@@soln/.n->0)[[1]]]

Map[RootReduce,(-1+((Root[1+#1-2 ((#1)^2)-2 ((#1)^3)-3 ((#1)^4)+((#1)^5)&,1])^4))/((Root[1+#1-2 ((#1)^2)-2 ((#1)^3)-3 ((#1)^4)+((#1)^5)&,1]-Root[1+#1-2 ((#1)^2)-2 ((#1)^3)-3 ((#1)^4)+((#1)^5)&,2]) (Root[1+#1-2 ((#1)^2)-2 ((#1)^3)-3 ((#1)^4)+((#1)^5)&,1]-Root[1+#1-2 ((#1)^2)-2 ((#1)^3)-3 ((#1)^4)+((#1)^5)&,3]) (Root[1+#1-2 ((#1)^2)-2 ((#1)^3)-3 ((#1)^4)+((#1)^5)&,1]-Root[1+#1-2 ((#1)^2)-2 ((#1)^3)-3 ((#1)^4)+((#1)^5)&,4]) (Root[1+#1-2 ((#1)^2)-2 ((#1)^3)-3 ((#1)^4)+((#1)^5)&,1]-Root[1+#1-2 ((#1)^2)-2 ((#1)^3)-3 ((#1)^4)+((#1)^5)&,5])),{2}]

RootReduce[(Root[-269056-474112 ((#1)^2)+41984 ((#1)^4)+261632 ((#1)^6)-63616 ((#1)^8)+113424 ((#1)^10)+18176 ((#1)^12)-10048 ((#1)^14)+1168 ((#1)^16)-56 ((#1)^18)+((#1)^20)&,1] Root[-269056-474112 ((#1)^2)+41984 ((#1)^4)+261632 ((#1)^6)-63616 ((#1)^8)+113424 ((#1)^10)+18176 ((#1)^12)-10048 ((#1)^14)+1168 ((#1)^16)-56 ((#1)^18)+((#1)^20)&,3] Root[-269056-474112 ((#1)^2)+41984 ((#1)^4)+261632 ((#1)^6)-63616 ((#1)^8)+113424 ((#1)^10)+18176 ((#1)^12)-10048 ((#1)^14)+1168 ((#1)^16)-56 ((#1)^18)+((#1)^20)&,11] Root[-269056-474112 ((#1)^2)+41984 ((#1)^4)+261632 ((#1)^6)-63616 ((#1)^8)+113424 ((#1)^10)+18176 ((#1)^12)-10048 ((#1)^14)+1168 ((#1)^16)-56 ((#1)^18)+((#1)^20)&,12])]

Adam, 2017-07-12:
Here it is.

It is the first time I used RootApproximant to derive something
I didn't know (the Root representation of the denominator --
it seems to be too hard for RootReduce).

In[1]:= e=(Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 1]^
            n (-1 + Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &,
               1]^4))/((Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 1] -
              Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 2]) (Root[
               1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 1] -
              Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 3]) (Root[
               1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 1] -
              Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 4]) (Root[
               1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 1] -
              Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 5])) + (Root[
             1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 2]^
            n (-1 + Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &,
               2]^4))/((-Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 1] +
               Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 2]) (Root[
               1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 2] -
              Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 3]) (Root[
               1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 2] -
              Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 4]) (Root[
               1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 2] -
              Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 5])) + (Root[
             1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 3]^
            n (-1 + Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &,
               3]^4))/((-Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 1] +
               Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &,
               3]) (-Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 2] +
              Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 3]) (Root[
               1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 3] -
              Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 4]) (Root[
               1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 3] -
              Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 5])) + (Root[
             1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 4]^
            n (-1 + Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &,
               4]^4))/((-Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 1] +
               Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &,
               4]) (-Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 2] +
              Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &,
               4]) (-Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 3] +
              Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 4]) (Root[
               1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 4] -
              Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 5])) + (Root[
             1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 5]^
            n (-1 + Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &,
               5]^4))/((-Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 1] +
               Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &,
               5]) (-Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 2] +
              Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &,
               5]) (-Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 3] +
              Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &,
               5]) (-Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 4] +
              Root[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, 5]));

In[2]:= ee=RootSum[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &,
        (-167+525 #-73 #^2+819 #^3-218 #^4)#^n&]/2102;

In[3]:= Table[N[e-ee, 1000], {n, 21}]//Quiet//Chop

Out[3]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

In[4]:= Table[RootReduce[ee], {n, 21}]

Out[4]= {3, 11, 41, 149, 547, 2007, 7361, 27001, 99043, 363299, 1332617,
>    4888173, 17930307, 65770159, 241251521, 884934705, 3246028995,
>    11906758971, 43675182633, 160204937605, 587647732323}

Table[((RootSum[1+#-2 ((#)^2)-2 ((#)^3)-3 ((#)^4)+((#)^5)&,(-167+525 #-73 ((#)^2)+819 ((#)^3)-218 ((#)^4)) ((#)^n)&])/2102),{n,20}]

%-seq

DominationNumber

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

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

%==seq

DominationPolynomial

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

CoefficientList[Take[poly,5],x]

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}]

RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n]

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]]

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

Eccentricities

seq=Table[GraphData[{"Ladder",n},"Eccentricities"],{n,5}]

EdgeChromaticNumber

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

EdgeConnectivity

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

EdgeCount

seq=Table[GraphData[{"Ladder",n},"M"],{n,20}]

FindSequenceFunction[seq,n]

EdgeCoverCount

A286911

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

FindLinearRecurrence[seq]

LinearRecurrence[{6,3,-2},{1,7,43},20]

FindSequenceFunction[seq,n]

FullSimplify[%]

Table[((Root[2-3 #1-6 ((#1)^2)+((#1)^3)&,3])^n) Root[-4+45 #1-150 ((#1)^2)+150 ((#1)^3)&,1]+((Root[2-3 #1-6 ((#1)^2)+((#1)^3)&,1])^n) Root[-4+45 #1-150 ((#1)^2)+150 ((#1)^3)&,2]+((Root[2-3 #1-6 ((#1)^2)+((#1)^3)&,2])^n) Root[-4+45 #1-150 ((#1)^2)+150 ((#1)^3)&,3],{n,0,1}]//FullSimplify

FindSequenceFunction[Prepend[seq,1],n]

f1=2-3 #1-6 ((#1)^2)+((#1)^3)&;
f2=-4+45 #1-150 ((#1)^2)+150 ((#1)^3)&;
pos={2,3,1};

Table[Sum[((Root[f1,k])^n)Root[f2,pos[[k]]],{k,Length[pos]}],{n,10}]//N//Chop

an=ToNumberField[Root[f2,pos[[#]]],Root[f1,#]]&/@Range[Length[pos]];

Equal@@Last/@an

RootSum[f1,Evaluate[((#)^(n+Range[Length[pos]]-1)).an[[1,2]]]&]

Table[-RootSum[2-3 #-6 ((#)^2)+((#)^3)&,-14 ((#)^n)-5 ((#)^(n+1))+((#)^(n+2))&]/30,{n,20}]

gf=x FindGeneratingFunction[seq,x]

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

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

EdgeCoverNumber

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

EdgeCoverPolynomial

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

CoefficientList[Take[poly,5],x]

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}]

RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n]

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]]

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

EulerianCycleCount

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

FaceCount

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

FibonacciCube

Entity["Graph",{"Ladder",n}]["FibonacciCube"]

seq=Table[GraphData[{"Ladder",n},"FibonacciCube"],{n,20}]

FlowPolynomial

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

Table[Function[x,Piecewise[{{0,n==1}},(x-2)^(n-2)(x-1)]][x],{n,20}]==poly

FractionalChromaticNumber

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

FractionalEdgeChromaticNumber

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

Girth

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

Gonality

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

HamiltonConnected

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

HamiltonDecompositionCount

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

Hamiltonian

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

HamiltonianCycleCount

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

HamiltonianNumber

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

HamiltonianPathCount

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

FindSequenceFunction[Rest@seq,n-1]//Factor

HamiltonianWalkCount

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

HararyIndex

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

FindSequenceFunction[seq,n]//FullSimplify

Table[(4 n+2) HarmonicNumber[n]-5n,{n,20}]==seq

HexagonCount

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

IndependencePolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindPolynomialRecurrence[poly,6]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n]

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]]

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

IndependenceNumber

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

IndependenceRatio

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

IndependentEdgeSetCount

A030186

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

FindLinearRecurrence[seq]

FindSequenceFunction[DeleteCases[seq,_Missing],n]//FullSimplify

f1=1-#1-3 ((#1)^2)+((#1)^3)&;
f2=-1+18 #1-74 ((#1)^2)+74 ((#1)^3)&;
pos={2,1,3};

Table[Sum[((Root[f1,k])^n)Root[f2,pos[[k]]],{k,Length[pos]}],{n,0,10}]//N//Chop

an=ToNumberField[Root[f2,pos[[#]]],Root[f1,#]]&/@Range[Length[pos]];

Equal@@Last/@an

RootSum[f1,Evaluate[((#)^(n+Range[Length[pos]]-1)).an[[1,2]]]&]

Table[RootSum[1-#-3#^2+#^3&,9#^n-10#^(n+1)+7#^(n+2)&]/74,{n,0,20}]

seq=LinearRecurrence[{3,1,-1},{2,7,22},{0,20}]

gf=x FindGeneratingFunction[DeleteCases[seq,_Missing],x]

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

IndependentVertexSetCount

A001333

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

FindLinearRecurrence[seq]

LinearRecurrence[{2,1},{3,7},20]

FindSequenceFunction[seq,n]//FullSimplify

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

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

IntersectionNumber

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

IrredundantSetCount

A290513

seq=Table[GraphData[{"Ladder",n},"IrredundantSetCount"],{n,20}]

seq={3,11,26,79,224,640,1828,5225,14928,42654,121873,348232,995003,2843014,8123337,23210809,66320216,189496620,541448364,1547079580,4420468031,12630596045,36089381477,103118131368,294639269914,841872308017,2405480380385,6873175192304};

coef=FindLinearRecurrence[DeleteMissing@seq]

Take[seq,Length[coef]]

LinearRecurrence[{2,1,3,5,-2,-6,-5,-4,-4,-3,-2,-2},{3,11,26,79,224,640,1828,5225,14928,42654,121873,348232},20]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

gf=x*(3+5*x+x^2+7*x^3-8*x^4-14*x^5-3*x^6-5*x^7-9*x^8-3*x^9-2*x^10-2*x^11)/(1-2*x-x^2-3*x^3-5*x^4+2*x^5+6*x^6+5*x^7+4*x^8+4*x^9+3*x^10+2*x^11+2*x^12);

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

CoefficientList[Series[((3+5 x+(x^2)+7 (x^3)-8 (x^4)-14 (x^5)-3 (x^6)-5 (x^7)-9 (x^8)-3 (x^9)-2 (x^10)-2 (x^11))/(1-2 x-(x^2)-3 (x^3)-5 (x^4)+2 (x^5)+6 (x^6)+5 (x^7)+4 (x^8)+4 (x^9)+3 (x^10)+2 (x^11)+2 (x^12))),{x,0,20}],x]

res=Refine[SeriesCoefficient[gf,{x,0,n}],n>0]

terms=List@@res;

terms[[1]]

N[terms/.n->0]//Chop

approx=RootApproximant[N[exact=terms/.n->0,300],12]

pos=approx[[All,-2]]

N[approx-exact,200]

f1=2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&;
f2=70756-38472136 #1+680287468 ((#1)^2)+505635636356 ((#1)^3)-34754559163944 ((#1)^4)-4137390838253281 ((#1)^5)+5383132958966821 ((#1)^6)-49379984740594571 ((#1)^7)+20309601766042971 ((#1)^8)-331498821583963071 ((#1)^9)+125141094330601095 ((#1)^10)-828526502634464593 ((#1)^11)+828526502634464593 ((#1)^12)&;

Table[Sum[((Root[f1,k])^n)Root[f2,pos[[k]]],{k,Length[pos]}],{n,10}]//N//Chop

an=ToNumberField[Root[f2,pos[[#]]],Root[f1,#]]&/@Range[Length[pos]]

Equal@@Last/@an

RootSum[f1,Evaluate[((#)^n) #^Range[0,Length[pos]-1].an[[1,2]]]&]

Table[(1/2485579507903393779)RootSum[2+2 #1+3 ((#1)^2)+4 ((#1)^3)+4 ((#1)^4)+5 ((#1)^5)+6 ((#1)^6)+2 ((#1)^7)-5 ((#1)^8)-3 ((#1)^9)-((#1)^10)-2 ((#1)^11)+((#1)^12)&,-105577159431355949 ((#1)^n)-389671420034091247 ((#1)^(1+n))+29241021604101932 ((#1)^(2+n))+551171239538727862 ((#1)^(3+n))-835076245333578054 ((#1)^(4+n))-823168743731791895 ((#1)^(5+n))+1288188291747539683 ((#1)^(6+n))+114497498217658607 ((#1)^(7+n))-529545500369064866 ((#1)^(8+n))+763912087212707104 ((#1)^(9+n))-593709228550713556 ((#1)^(10+n))+130400048784652699 ((#1)^(11+n))&],{n,20}]

Table[RootSum[2+2 #+3 #^2+4 #^3+4 #^4+5 #^5+6 #^6+2 #^7-5 #^8-3 #^9-#^10-2 #^11+#^12&,-105577159431355949 #^n-389671420034091247 #^(n+1)+29241021604101932 #^(n+2)+551171239538727862 #^(n+3)-835076245333578054 #^(n+4)-823168743731791895 #^(n+5)+1288188291747539683 #^(n+6)+114497498217658607 #^(n+7)-529545500369064866 #^(n+8)+763912087212707104 #^(n+9)-593709228550713556 #^(n+10)+130400048784652699 #^(n+11)&]/2485579507903393779,{n,20}]

KirchhoffIndex

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

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

KirchhoffSumIndex

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

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

LaplacianPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindPolynomialRecurrence[poly,6]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n]

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]]

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

LeafCount

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

LineGraphName

Table[GraphData[{"Ladder",n},"LineGraphName"],{n,20}]

LongestCycleCount

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

LovaszNumber

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

LucasCube

Entity["Graph",{"Ladder",n}]["LucasCube"]

seq=Table[GraphData[{"Ladder",n},"LucasCube"],{n,20}]

MatchingNumber

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

MatchingPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindPolynomialRecurrence[poly,6]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n]

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]]

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

MaximalCliqueCount

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

FindSequenceFunction[seq,n]

MaximalIndependentEdgeSetCount

A286945

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

FindLinearRecurrence[seq]

LinearRecurrence[{2,0,0,1,1},{1,2,5,11,24},20]

FindSequenceFunction[seq,n]

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

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

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

(1/26) (6 Cos[((n π)/3)]+(1/(Sqrt[31]))4 I (((Root[-1-2 #1-((#1)^2)+((#1)^3)&,3])^n) Root[885391-3549 ((#1)^2)+333 ((#1)^4)+((#1)^6)&,1]+((Root[-1-2 #1-((#1)^2)+((#1)^3)&,1])^n) Root[885391-3549 ((#1)^2)+333 ((#1)^4)+((#1)^6)&,3]+((Root[-1-2 #1-((#1)^2)+((#1)^3)&,2])^n) Root[885391-3549 ((#1)^2)+333 ((#1)^4)+((#1)^6)&,5])-((10 Sin[((n π)/3)])/(Sqrt[3])))

RootReduce[Root[885391-3549 ((#1)^2)+333 ((#1)^4)+((#1)^6)&,1]/Root[-1-2 #1-((#1)^2)+((#1)^3)&,3]]

From Adam, 2017-07-12:

e=1/26 (6 Cos[(n π)/3] +
           1/Sqrt[31]
             4 I (Root[-1 - 2 #1 - #1^2 + #1^3 &, 3]^
                n Root[885391 - 3549 #1^2 + 333 #1^4 + #1^6 &, 1] +
               Root[-1 - 2 #1 - #1^2 + #1^3 &, 1]^
                n Root[885391 - 3549 #1^2 + 333 #1^4 + #1^6 &, 3] +
               Root[-1 - 2 #1 - #1^2 + #1^3 &, 2]^
                n Root[885391 - 3549 #1^2 + 333 #1^4 + #1^6 &, 5]) - (
           10 Sin[(n π)/3])/Sqrt[3]);

In[2]:= ee=(3*Cos[(n*Pi)/3])/13-5*Sin[(n*Pi)/3]/(13*Sqrt[3])+
        RootSum[-1 - 2 #1 - #1^2 + #1^3 &, (-6-72 #+80#^2)#^n&]/403;

In[3]:= Table[RootReduce[e-ee], {n, 1, 21}]

Out[3]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

Table[3Cos[n Pi/3]/13-5Sin[n Pi/3]/(13Sqrt[3])+RootSum[-1-2 #-#^2+#^3&,(-6-72 #+80#^2)#^n&]/403,{n,20}]

%-seq

MaximalIndependentVertexSetCount

A118658(n-1)

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

FindSequenceFunction[Rest@seq,n-1]//FullSimplify

Table[Fibonacci[n-1]+LucasL[n-1],{n,20}]

Table[Fibonacci[n-2]+LucasL[n-2],{n,0,20}]

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

2Fibonacci[Range[0,20]-1]

Table[LucasL[n]-Fibonacci[n],{n,0,20}]

Subtract@@@(Through[{LucasL,Fibonacci}[#]]&/@Range[0,20])

Through[{LucasL,Fibonacci}[#]]&/@Range[0,20]

Subtract@@@(Through[{LucasL,Fibonacci}[#]]&/@Range[0,20])

gf=x FindGeneratingFunction[%,x]//Factor

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

MaximumCliqueCount

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

FindSequenceFunction[seq,n]

MaximumIndependentEdgeSetCount

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

FindSequenceFunction[seq,n]

Table[Fibonacci[n+1],{n,20}]

MaximumIndependentVertexSetCount

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

MaximumLeafNumber

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

MaximumVertexDegree

seq=Table[GraphData[{"Ladder",n},"MaximumVertexDegree"],{n,20}]

MeanDistance

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

FindSequenceFunction[seq,n]//Factor

MinimalConnectedDominatingSetCount

seq=Table[GraphData[{"Ladder",n},"MinimalConnectedDominatingSetCount"],{n,20}]

Table[Piecewise[{{4,n==2},{3,n==3||n==4}},2],{n,20}]

MinimalDominatingSetCount

A290379

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

seq={2,6,7,18,39,75,155,310,638,1295,2624,5339,10853,22069,44836,91134,185259,376542,765331,1555567,3161843,6426646,13062506,26550391,53965428,109688223,222948193,453156469,921069708,1872133138,3805230243,7734373962,15720610559};

coef=FindLinearRecurrence[DeleteMissing@seq]

Take[seq,Length[coef]]

LinearRecurrence[{0,1,3,4,4,1,2,3,5,4,2},{2,6,7,18,39,75,155,310,638,1295,2624},20]

RSolve[Append[Table[a[k+2]==seq[[k]],{k,Length[coef]}],a[n]==Sum[coef[[k]]a[n-k],{k,Length[coef]}]],a[n],n]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[(((1+x) (2+4 x+(x^2)+5 (x^3)+(x^4)+3 (x^5)+5 (x^6)+3 (x^7)+2 (x^8)+2 (x^9)))/(1-(x^2)-3 (x^3)-4 (x^4)-4 (x^5)-(x^6)-2 (x^7)-3 (x^8)-5 (x^9)-4 (x^10)-2 (x^11))),{x,0,20}],x]

Series[gf=x*(1+x)*(2+4*x+x^2+5*x^3+x^4+3*x^5+5*x^6+3*x^7+2*x^8+2*x^9)/(1-x^2-3*x^3-4*x^4-4*x^5-x^6-2*x^7-3*x^8-5*x^9-4*x^10-2*x^11),{x,0,20}]

res=Refine[SeriesCoefficient[gf,{x,0,n}],n>0]

terms=List@@res;

terms[[1]]

N[terms/.n->0]//Chop

approx=RootApproximant[N[exact=terms/.n->0,200],12]

pos=approx[[All,-2]]

N[approx-exact,200]

Table[Sum[((Root[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,k])^n)Root[-3012352-30200832 #1-989507328 ((#1)^2)-3700047680 ((#1)^3)-6704334748 ((#1)^4)+21166036128 ((#1)^5)+17062711909 ((#1)^6)+220857547691 ((#1)^7)-554632114740 ((#1)^8)+136281756656 ((#1)^9)-2097121971853 ((#1)^10)+2097121971853 ((#1)^11)&,pos[[k]]],{k,11}],{n,10}]//N//Chop

seq

f1=-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&;
f2=-3012352-30200832 #1-989507328 ((#1)^2)-3700047680 ((#1)^3)-6704334748 ((#1)^4)+21166036128 ((#1)^5)+17062711909 ((#1)^6)+220857547691 ((#1)^7)-554632114740 ((#1)^8)+136281756656 ((#1)^9)-2097121971853 ((#1)^10)+2097121971853 ((#1)^11)&;

an=ToNumberField[Root[f2,pos[[#]]],Root[f1,#]]&/@Range[11]

Equal@@Last/@an

an[[1,2]]

RootSum[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,Evaluate[((#)^n) #^Range[0,10].{-(621827501801/2097121971853),(301456826961/2097121971853),-(280366986955/2097121971853),(1253389979482/2097121971853),-(843186094854/2097121971853),(87555893434/2097121971853),-(236346312907/2097121971853),(504072574383/2097121971853),-(231943645265/2097121971853),(618185916584/2097121971853),-(290649224768/2097121971853)}]&]

Table[-(1/2097121971853)RootSum[-2-4 #1-5 ((#1)^2)-3 ((#1)^3)-2 ((#1)^4)-((#1)^5)-4 ((#1)^6)-4 ((#1)^7)-3 ((#1)^8)-((#1)^9)+((#1)^11)&,621827501801 ((#1)^n)-301456826961 ((#1)^(1+n))+280366986955 ((#1)^(2+n))-1253389979482 ((#1)^(3+n))+843186094854 ((#1)^(4+n))-87555893434 ((#1)^(5+n))+236346312907 ((#1)^(6+n))-504072574383 ((#1)^(7+n))+231943645265 ((#1)^(8+n))-618185916584 ((#1)^(9+n))+290649224768 ((#1)^(10+n))&],{n,10}]

Table[-RootSum[-2-4 #-5 ((#)^2)-3 ((#)^3)-2 ((#)^4)-((#)^5)-4 ((#)^6)-4 ((#)^7)-3 ((#)^8)-((#)^9)+((#)^11)&,621827501801 ((#)^n)-301456826961 ((#)^(n+1))+280366986955 ((#)^(n+2))-1253389979482 ((#)^(n+3))+843186094854 ((#)^(n+4))-87555893434 ((#)^(n+5))+236346312907 ((#)^(n+6))-504072574383 ((#)^(n+7))+231943645265 ((#)^(n+8))-618185916584 ((#)^(n+9))+290649224768 ((#)^(n+10))&]/2097121971853,{n,10}]

MinimalDominatingSetSignature

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

MinimalEdgeCoverCount

A288029

seq=Table[GraphData[{"Ladder",n},"MinimalEdgeCoverCount"],{n,20}]

seq={1,2,6,17,45,120,324,873,2349,6322,17018,45809,123305,331904,893400,2404801,6473097,17423890,46900574,126244129,339816309,914696984,2462126012,6627401865,17839239445,48018585634,129253524146,347916817697,936501444241,2520817938240};

coef=FindLinearRecurrence[DeleteMissing@seq]

LinearRecurrence[{2,1,2,1,0,-1},{1,2,6,17,45,120},20]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

ToString[gf,InputForm]

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

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

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

Table[(2+I) (((-I))^n)+(2-I) (I^n),{n,10}]

Table[(3+(((-1))^n))(((-1))^(Floor[n/2])),{n,10}]

Table[(1/10) ((2+I) (((-I))^n)+(2-I) (I^n)+2 (((Root[1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,1])^n) Root[125-850 #1+2120 ((#1)^2)-2184 ((#1)^3)+728 ((#1)^4)&,1]+((Root[1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,2])^n) Root[125-850 #1+2120 ((#1)^2)-2184 ((#1)^3)+728 ((#1)^4)&,2]+((Root[1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,3])^n) Root[125-850 #1+2120 ((#1)^2)-2184 ((#1)^3)+728 ((#1)^4)&,3]+((Root[1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,4])^n) Root[125-850 #1+2120 ((#1)^2)-2184 ((#1)^3)+728 ((#1)^4)&,4])),{n,20}]//Round

RootSum[1-2 #^2-2 #^3+#^4&,#^n(47/91+33/182#-1/26#^2+4/91#^3)&]

2/182

FullSimplify[((2+I)(-I)^n+(2-I)I^n+ RootSum[1-2 ((#)^2)-2 ((#)^3)+((#)^4)&,94 ((#)^n)+33 ((#)^(n+1))-7 ((#)^(n+2))+8 ((#)^(n+3))&]/91)/10]

Table[(1/910) (RootSum[1-2 ((#)^2)-2 ((#)^3)+((#)^4)&,94 ((#)^n)+33 ((#)^(n+1))-7 ((#)^(n+2))+8 ((#)^(n+3))&]+182 (2 Cos[((n π)/2)]+Sin[((n π)/2)])),{n,20}]

MinimumCliqueCoveringCount

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

Table[Fibonacci[n+1],{n,20}]

MinimumConnectedDominatingSetCount

seq=Table[GraphData[{"Ladder",n},"MinimumConnectedDominatingSetCount"],{n,20}]

FindSequenceFunction[DeleteCases[seq,_Missing],n]

gf=x FindGeneratingFunction[DeleteCases[seq,_Missing],x]

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

InputForm[%]

MinimumDistinguishingLabelingCount

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

FindLinearRecurrence[DeleteCases[seq,_Missing]]

FindSequenceFunction[DeleteCases[seq,_Missing],n]

gf=x FindGeneratingFunction[DeleteCases[seq,_Missing],x]

MinimumDominatingSetCount

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

Table[Piecewise[{{6,n==2},{3,n==3},{17,n==6},{2(n+2),Mod[n,2]==0},{2,Mod[n,2]==1}}],{n,20}]

%==seq

MinimumEdgeCoverCount

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

Table[Fibonacci[n+1],{n,20}]

MinimumVertexDegree

seq=Table[GraphData[{"Ladder",n},"MinimumVertexDegree"],{n,20}]

MolecularTopologicalIndex

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

FindSequenceFunction[Rest@seq,n-1]//FullSimplify

Table[Piecewise[{{4,n==1}},2 n (2 (n^2)+n+7)-20],{n,20}]

%==seq

Nonhamiltonian

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

OddChordlessCycleCount

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

PathCount

A288516

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

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]//FullSimplify//Factor

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

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

Factor[(41+n (9+n))]

seq=Table[18((2^n)-1)-n((n^2)+9n+41)/3,{n,40}]

FindLinearRecurrence[seq]

LinearRecurrence[{6,-14,16,-9,2},{1,12,49,146,373},20]

CoefficientList[Series[(((-1-6 x+9 (x^2)-4 (x^3)))/((((-1+x))^4) (-1+2 x))),{x,0,20}],x]

PathPolynomial

(poly=Factor[Table[GraphData[{"Ladder",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}]

CoefficientList[%73,x]

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]

Factor[(1/((((-1+(x^2)))^3) (((-1+x+(x^2)))^2)))(x^2) (-2 (-1+(-2+x) x+(((x (1+x)))^n))+2 x (x (-4 x+(x^4)-(x^6)-(x^(2 n)) (((-1+x+(x^2)))^2) (-1-2 x+(x^4)))+(((x (1+x)))^n) (-2+(x^2) (4+5 x-5 (x^3)-4 (x^4)+2 (x^6)+(x^7))))-n (-1+x) (1+x) (-1+x+(x^2)) (3+x (3+x (-3+x (-4+x+(x^2)+(x^3)+2 (x^(2 n)) (-1+x+(x^2)))))))]

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

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

poly2=Table[(1/((((-1+(x^2)))^3) (((-1+x+(x^2)))^2)))(x^2) (-2 (-1+(-2+x) x+(((x (1+x)))^n))+2 x (x (-4 x+(x^4)-(x^6)-(x^(2 n)) (((-1+x+(x^2)))^2) (-1-2 x+(x^4)))+(((x (1+x)))^n) (-2+(x^2) (4+5 x-5 (x^3)-4 (x^4)+2 (x^6)+(x^7))))-n (-1+x) (1+x) (-1+x+(x^2)) (3+x (3+x (-3+x (-4+x+(x^2)+(x^3)+2 (x^(2 n)) (-1+x+(x^2))))))),{n,k0,kmax}]

FullSimplify[Take[poly2-poly,10]]

Pell

Entity["Graph",{"Ladder",n}]["Pell"]

seq=Table[GraphData[{"Ladder",n},"Pell"],{n,20}]

PentagonCount

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

Perfect

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

ProjectivePlaneCrossingNumber

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

Radius

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

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

RankPolynomial

(poly=Factor@Table[GraphData[{"Ladder",n},"RankPolynomial"][x,y],{n,k0=1,10}])//Column

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindPolynomialRecurrence[poly,6]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n]

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]

RectilinearCrossingNumber

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

ReliabilityPolynomial

(poly=Factor@Table[GraphData[{"Ladder",n},"ReliabilityPolynomial"][x,y],{n,k0=1,10}])//Column

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindPolynomialRecurrence[poly,6]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n]

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]

Skewness

seq=Table[GraphData[{"Ladder",n},"Skewness"],{n,20}]

SpanningTreeCount

A001353

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

FindSequenceFunction[seq,n]//FullSimplify

Table[GegenbauerC[n-1,1,2],{n,20}]

Table[ChebyshevU[-1+n,2],{n,20}]

Spectrum

Table[GraphData[{"Ladder",n},"Spectrum"],{n,10}]

SquareCount

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

FindSequenceFunction[seq,n]

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

FindGeneratingFunction[seq,x]

FindLinearRecurrence[Drop[seq,3]]

StabilityIndex

A138573

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

FindLinearRecurrence[seq]

FindSequenceFunction[Rest@seq,n-1]

gf=x FindGeneratingFunction[seq,x]

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

ToRadicals[%223]

FullSimplify[%]

Table[(ChebyshevT[n+1,GoldenRatio]-ChebyshevT[n+1,-1/GoldenRatio])/Sqrt[5],{n,20}]//FullSimplify

Table[Abs[Fibonacci[n,I+1]]^2,{n,20}]

TopologicalIndex

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

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

ToroidalCrossingNumber

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

Traceable

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

TriangleCount

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

TuttePolynomial

(poly=Factor@Table[GraphData[{"Ladder",n},"TuttePolynomial"][x,y],{n,k0=1,10}])//Column

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindPolynomialRecurrence[poly,6]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n]

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]

Untraceable

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

VertexConnectivity

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

VertexCount

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

FindSequenceFunction[seq,n]

VertexCoverNumber

Related to but not the same as IndependenceNumber

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

VertexCoverPolynomial

(poly=Factor@Table[GraphData[{"Ladder",n},"VertexCoverPolynomial"][x],{n,k0=1,10}])//Column

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindPolynomialRecurrence[poly,6]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n]

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]

VertexDegrees

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

WienerIndex

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

FindSequenceFunction[seq,n]//FullSimplify

WienerSumIndex

seq=Table[GraphData[{"Ladder",n},"WienerSumIndex"],{n,10}]

FindSequenceFunction[seq,n]

GraphData

GraphClassString

GraphClassString[Select[GraphData["Ladder"],GraphData[#,"V"]>2&]]