Ladder Rung Graph

Author

Eric W. Weisstein
September 13, 2017

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

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

©2017 Wolfram Research, Inc. except for portions noted otherwise

Graphic

Graphs.m

GraphicsGrid[Partition[Table[Show[LadderRungGraph[n,VertexSize->{"Scaled",.05},System`VertexStyle->Red,System`EdgeStyle->Black],PlotLabel->Style[HoldForm[n]==n,FontFamily->"Times"]],{n,6}],3],Dividers->All,ImageSize->500]

Old

Special cases

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

Relation to Kneser

Must run in V7; buggy in V8

V/@Table[KneserGraph[2n,n],{n,6}]/2

FullSimplify[FunctionExpand@FindSequenceFunction[s,n],n∈Integers&&n>0]

Table[Binomial[2n,n],{n,6}]/2

FindGeneratingFunction[s,x]

RecognizeGraph[GraphUnion[#,Graph["P2"]]]&/@{1,3,10}

Properties

Acyclic

seq=Table[GraphData[{"LadderRung",n},"Acyclic"],{n,20}]

AdjacencyMatrixCount

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

FullSimplify[(((2n)!)/((2^n)n!))]

Table[(2n-1)!!,{n,10}]

(2Range[0,10]-1)!!

Arboricity

seq=Table[GraphData[{"LadderRung",n},"Arboricity"],{n,20}]

ArcTransitive

seq=Table[GraphData[{"LadderRung",n},"ArcTransitive"],{n,20}]

ArcTransitivity

seq=Table[GraphData[{"LadderRung",n},"ArcTransitivity"],{n,20}]

AutomorphismCount

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

FindSequenceFunction[seq,n]//FunctionExpand

Table[(2^n)n!,{n,10}]

Table[(2n)!!,{n,10}]

(2Range[0, 20])!!

FindLinearRecurrence[%]

BalabanIndex

seq=Table[GraphData[{"LadderRung",n},"BalabanIndex"],{n,20}]//FullSimplify

Bandwidth

seq=Table[GraphData[{"LadderRung",n},"Bandwidth"],{n,20}]

BridgeCount

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

BurningNumber

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

CharacteristicPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

Chordal

seq=Table[GraphData[{"LadderRung",n},"Chordal"],{n,20}]

ChordCount

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

Chordless

seq=Table[GraphData[{"LadderRung",n},"Chordless"],{n,20}]

ChordlessCycleCount

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

Chords

seq=Table[GraphData[{"LadderRung",n},"Chords"],{n,20}]

ChromaticallyUnique

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

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

ChromaticInvariant

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

ChromaticNumber

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

ChromaticPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

Circumference

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

Classes

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

CliqueCount

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

FindSequenceFunction[seq,n]

CliqueCoveringNumber

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

CliqueNumber

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

CliquePolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

CochromaticGraphNames

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

Select[GraphData["LadderRung"],GraphData[#,"ChromaticallyUnique"]=!=False&]

ComplementOddChordlessCycleCount

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

ConnectedComponentCount

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

ConnectedDominatingSetCount

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

ConnectedDominationNumber

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

ConnectedDominationPolynomial

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

ConnectedInducedSubgraphCount

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

ConnectedInducedSubgraphPolynomial

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

CrossingNumber

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

CycleCount

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

CyclePolynomial

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

Cyclic

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

CyclomaticNumber

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

DetourIndex

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

DetourMatrix

MatrixForm/@Table[GraphData[{"LadderRung",n},"DetourMatrix"],{n,6}]

DetourPolynomial

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

Diameter

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

DistanceMatrix

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

DistancePolynomial

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

DistinguishingNumber

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

DomaticNumber

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

DominatingSetCount

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

FindSequenceFunction[seq,n]

DominationNumber

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

DominationPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

EdgeChromaticNumber

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

FindSequenceFunction[seq,n]-2

gf=x FindGeneratingFunction[seq,x]

EdgeConnectivity

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

EdgeCount

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

EdgeCoverCount

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

EdgeCoverNumber

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

EdgeCoverPolynomial

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

EulerianCycleCount

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

FaceCount

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

FlowPolynomial

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

FractionalChromaticNumber

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

FractionalEdgeChromaticNumber

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

Girth

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

Gonality

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

HamiltonConnected

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

HamiltonDecompositionCount

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

Hamiltonian

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

HamiltonianCycleCount

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

HamiltonianNumber

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

HamiltonianPathCount

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

HamiltonianWalkCount

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

HararyIndex

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

HexagonCount

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

IdiosyncraticPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

IndependenceNumber

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

IndependencePolynomial

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

IndependenceRatio

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

IndependentEdgeSetCount

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

FindSequenceFunction[seq,n]

IndependentVertexSetCount

A000244

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

FindSequenceFunction[seq,n]

FindLinearRecurrence[seq]

seq=LinearRecurrence[{3},{1},20]

FindGeneratingFunction[seq,x]

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

IntersectionNumber

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

IrredundantSetCount

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

FindSequenceFunction[DeleteMissing@seq,n]

KirchhoffIndex

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

KirchhoffSumIndex

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

LaplacianMatrix

MatrixForm/@Table[GraphData[{"LadderRung",n},"LaplacianMatrix"],{n,5}]

LaplacianPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

LeafCount

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

LongestCycleCount

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

LovaszNumber

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

MatchingGeneratingPolynomial

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

MatchingNumber

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

MatchingPolynomial

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

MaximalCliqueCount

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

FindSequenceFunction[seq,n]

MaximalIndependentEdgeSetCount

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

MaximalIndependentVertexSetCount

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

FindSequenceFunction[DeleteMissing@seq,n]

MaximalIrredundantSetCount

seq=Table[GraphData[{"LadderRung",n},"MaximalIrredundantSetCount"],{n,20}]

FindSequenceFunction[DeleteMissing@seq,n]

MaximumCliqueCount

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

FindSequenceFunction[seq,n]

MaximumIndependentEdgeSetCount

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

MaximumIndependentVertexSetCount

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

FindSequenceFunction[seq,n]

MaximumLeafNumber

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

MaximumVertexDegree

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

MeanDistance

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

MinimalConnectedDominatingSetCount

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

Table[Piecewise[{{2,n==1}},0],{n,20}]

MinimalDominatingSetCount

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

FindSequenceFunction[DeleteMissing@seq,n]

MinimalDominatingSetSignature

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

FindSequenceFunction[DeleteMissing@seq,n]

MinimalEdgeCoverCount

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

MinimalEdgeCoverSignature

seq=Table[GraphData[{"LadderRung",n},"MinimalEdgeCoverSignature"],{n,20}]//Column

MinimumCliqueCoveringCount

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

MinimumConnectedDominatingSetCount

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

MinimumDistinguishingLabelingCount

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

FindSequenceFunction[DeleteMissing@seq,n]

MinimumDominatingSetCount

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

FindSequenceFunction[seq,n]

MinimumEdgeCoverCount

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

MinimumVertexDegree

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

MolecularTopologicalIndex

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

Nonhamiltonian

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

NormalizedLaplacianMatrix

MatrixForm/@Table[GraphData[{"LadderRung",n},"NormalizedLaplacianMatrix"],{n,5}]

OddChordlessCycleCount

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

PathCount

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

PathPolynomial

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

PentagonCount

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

Perfect

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

ProjectivePlaneCrossingNumber

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

Radius

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

RankPolynomial

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

RectilinearCrossingNumber

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

ReliabilityPolynomial

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

Skewness

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

SigmaPolynomial

alternate rows of A274106

Take[(poly=Factor[Table[GraphData[{"LadderRung",n},"SigmaPolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x]]),10]//Column

CoefficientList[Take[poly,4],x]

SpanningTreeCount

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

SquareCount

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

StabilityIndex

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

FindSequenceFunction[seq,n]

TopologicalIndex

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

ToroidalCrossingNumber

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

Traceable

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

TriangleCount

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

TuttePolynomial

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

Untraceable

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

VertexConnectivity

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

VertexCount

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

FindSequenceFunction[seq,n]

VertexCoverNumber

vertex cover number == vertex count - independence number

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

VertexCoverPolynomial

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

VertexDegrees

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

VertexTransitive

Table[GraphData[{"LadderRung",n},"VertexTransitive"],{n,20}]

WienerIndex

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

WienerSumIndex

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

GraphData

GraphClassString

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

GraphDataString

Print[GraphDataString[LadderRungGraph[#],
"Name"->ToString[#]<>"-ladder rung graph",
"StandardName"->{"LadderRung",#},
"Classes"->{"LadderRung"},
TimeConstraint->60
]
]&/@Range[20]