Cocktail Party Graph

Author

Eric W. Weisstein
October 3, 2024

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

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

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

Figure

bug(452535): PlotLabel's are cut off in V14.1

GraphicsGrid[Partition[Show[StyleGraphs@CocktailPartyGraph[#],PlotLabel->Style[n==#,Directive[FontFamily->"Times",14,Black]]]&/@Range[6],UpTo[3]],ImageSize->400,Alignment->Top,Dividers->All]

Special cases

ToEntity/@CocktailPartyGraph/@Range[8]

Embeddings

<<MathWorld`Graphs`

{"CocktailParty", 1} == {"Empty", 2}

Names

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

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

{"CocktailParty", 2} == "SquareGraph"

Names

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

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

{"CocktailParty", 3} == "OctahedralGraph"

Names

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

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

{"CocktailParty", 4} == "SixteenCellGraph"

Names

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

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

{"CocktailParty", 5}

Names

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

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

Primary

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

All

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

Inexact

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

Embedding types

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

Default embedding

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

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

RecognizeGraph[g]

GraphPlot[RotateGraph[Graph[GraphData[{"CocktailParty",5},"Edges"],GraphLayout->"CircularEmbedding"],Pi/2]]

Complement[Subsets[Range[10,{2}]],Partition[Range[3,8],2,1,1]]

Subsets[Range[10,{2}]]

Partition[Range[3,8],2,1,1]

v=UnitCirclePoints[10];
g=Graph[Range[Length[v]],UndirectedEdge@@@Complement[Subsets[Range[10],{2}],Partition[Range[3,8],2],{{1,9},{2,10}}],VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

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

RecognizeGraph@g

Degenerate

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

Bilateral

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

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

Circular

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

Planar

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

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

Table[MinimalCrossingGraph[GraphData[{"CocktailParty",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[{"CocktailParty",5},"Circulant"]

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

Circulant construction

CirculantIndices[GraphData[{"CocktailParty",5}],All]

CirculantGraph1@@@%

Halin

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

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

LCF

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

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

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

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

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

LCF compute

Length[FindHamiltonianCycle[GraphData[{"CocktailParty",5}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"CocktailParty",5}],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFEmbedding[#],#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]}]

Length[gs]

UnitDistance

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

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

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"CocktailParty",5},"VertexList"],GraphData[{"CocktailParty",5},"EdgeList"]]]

GraphPlot@RotateGraph[,-.1]

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

GraphPlot@

GraphPlot@RotateGraph[,-.3]

UnitDistance by FindUnitDistanceEmbedding

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

Matchstick

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

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

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

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

{"CocktailParty", 6}

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

Primary

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

All

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

Inexact

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

Embedding types

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

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

Bilateral

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

Circular

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

Planar

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

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

Table[MinimalCrossingGraph[GraphData[{"CocktailParty",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[{"CocktailParty",6},"Circulant"]

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

Circulant construction

CirculantGraph1@@@CirculantIndices[GraphData[{"CocktailParty",6}]]

Halin

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

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

LCF

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

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

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

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

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

LCF compute

Length[FindHamiltonianCycle[GraphData[{"CocktailParty",6}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"CocktailParty",6}],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFEmbedding[#],#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]}]

Length[gs]

UnitDistance

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

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

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"CocktailParty",6},"VertexList"],GraphData[{"CocktailParty",6},"EdgeList"]]]

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

UnitDistance by FindUnitDistanceEmbedding

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

Matchstick

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

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

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

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

{"CocktailParty", 7}

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

Primary

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

All

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

Inexact

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

Embedding types

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

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

Bilateral

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

Circular

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

Planar

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

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

Table[MinimalCrossingGraph[GraphData[{"CocktailParty",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[{"CocktailParty",7},"Circulant"]

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

Circulant construction

CirculantGraph1@@@CirculantIndices[GraphData[{"CocktailParty",7}]]

Halin

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

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

LCF

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

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

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

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

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

LCF compute

Length[FindHamiltonianCycle[GraphData[{"CocktailParty",7}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"CocktailParty",7}],All])[[All,2]]]//Timing

Tally[(lcf=ReadProperty["LCFNotations",GraphNameToFileName[{"CocktailParty",7},".m.gz"]])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFEmbedding[#],#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]}]

Length[gs]

UnitDistance

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

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

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"CocktailParty",7},"VertexList"],GraphData[{"CocktailParty",7},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

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

UnitDistance by FindUnitDistanceEmbedding

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

Matchstick

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

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

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

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

Definitions

k-partite graph (Subscript[K, (Underscript[(Underscript[2,…,2, ⏟]), n])])

RecognizeGraph/@Table[CompleteGraph[Table[2,n]],{n,10}]

Polytope construction

mag[v_]:=N[Sqrt[v.v]]

edgefind[poly_]:=Block[{a,b},b=First@Union[Map[mag[First@poly-#]&,Rest@poly],SameTest->(Chop[#1-#2]==0&)];
Flatten[Table[a=Drop[poly,i-1];
{First@a,#}&/@Select[Rest@a,Abs[mag[First@a-#]-b]<.001&],{i,Length[poly]}],1]]

xyCross[n_]:=Module[{a,r,k},(*section 13.4 of Regular Polytopes*)a=Table[Flatten[Table[Through[{Cos,Sin}[(2 r-1) Pi k/n]],{r,Floor[n/2]}]],{k,2 n}];
If[OddQ[n],Table[Join[a[[k]],{(-1)^k/Sqrt[2]}],{k,2 n}],(*Else*)a]]

PolytopeGraph[poly_,opts___]:=Module[
{
a=Map[#[[{1,2}]]&,poly],
b=Map[#[[{1,2}]]&,edgefind[poly],{2}],
e
},
e=edgefind[poly]/.Thread[poly->Range[Length[poly]]];
System`Graph[Range[Length[a]],e,VertexCoordinates->a]
]

Table[PolytopeGraph[xyCross[n]],{n,2,6}]

RecognizeGraph/@%

Length/@Table[xyCross[n],{n,5}]

Properties

Acyclic

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

AdjacencyMatrixCount

A001147

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

FindSequenceFunction[seq,n]//FunctionExpand//FullSimplify

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

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

FindSequenceFunction[seq,n]

Arboricity

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

ArcTransitive

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

ArcTransitivity

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

AutomorphismCount

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

FindSequenceFunction[seq,n]//FunctionExpand

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

BalabanIndex

seq=Table[GraphData[{"CocktailParty",n},"BalabanIndex"],{n,20}]

FindSequenceFunction[seq,n]

Bandwidth

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

FindSequenceFunction[seq,n]

BridgeCount

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

BurningNumber

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

CharacteristicPolynomial

(poly=Factor[Table[GraphData[{"CocktailParty",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]

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

FullSimplify[poly2-poly]

Chordal

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

ChordCount

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

Table[GraphData[{"CocktailParty",n},"EdgeCount"],{n,20}]

FindSequenceFunction[seq,n]

Chordless

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

ChordlessCycleCount

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

FindSequenceFunction[DeleteCases[seq,_Missing],n]

Binomial[n,2]

ChordlessCycleSignature

seq=Table[GraphData[{"CocktailParty",n},"ChordlessCycleSignature"],{n,20}]

Rest@DeleteMissing@seq/.{{4,x_}}:>x

FindSequenceFunction[%,n-1]

Chords

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

ChromaticallyUnique

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

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

ChromaticInvariant

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

FindLinearRecurrence[Drop[DeleteCases[seq,_Missing],2]]

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

gf=(x^3)FindSequenceFunction[DeleteCases[seq,_Missing],x]

ChromaticNumber

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

FindSequenceFunction[seq,n]

ChromaticPolynomial

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

CoefficientList[Take[poly,5],x]

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

Factor[Table[System`ChromaticPolynomial[CocktailPartyGraph[n],x]/Pochhammer[-x,n],{n,4}]]

CoefficientList[%,x]

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

FullSimplify[poly2-poly]

Circumference

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

FindSequenceFunction[seq,n]

Classes

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

CliqueCount

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

FindSequenceFunction[seq,n]

CliqueCoveringNumber

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

CliqueNumber

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

CliquePolynomial

(poly=Factor[Table[GraphData[{"CocktailParty",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]

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

FullSimplify[poly2-poly]

CochromaticGraphNames

Table[GraphData[{"CocktailParty",n},"CochromaticGraphNames"],{n,20}]

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

ComplementOddChordlessCycleCount

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

ConnectedComponentCount

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

ConnectedDominatingSetCount

A289254

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

FindSequenceFunction[DeleteCases[seq,_Missing],n]

seq=Table[(4^n)-3n-1,{n,40}]

FindLinearRecurrence[seq]

LinearRecurrence[{6,-9,4},{0,9,54},20]

gf=x FindGeneratingFunction[seq,x]

ToString[gf,InputForm]

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

ConnectedDominationNumber

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

ConnectedDominationPolynomial

(poly=Factor@Table[GraphData[{"CocktailParty",n},"ConnectedDominationPolynomial"],{n,k0=2,kmax=13}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

poly2=Table[-1+(((1+x))^(2 n))-n x (2+x),{n,k0,kmax-k0+2}]

Length/@{poly,poly2}

FullSimplify[Take[poly2-poly,12]]

ConnectedInducedSubgraphCount

A290721(n) for n > 1

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

FindSequenceFunction[DeleteCases[seq,_Missing],n]

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

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

seq=Table[(4^n)-n-1,{n,40}]

FindLinearRecurrence[seq]

LinearRecurrence[{6,9,-4},{2,13,60},20]

gf=x FindGeneratingFunction[seq,x]

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

ConnectedInducedSubgraphPolynomial

(poly=DeleteCases[Factor@Table[GraphData[{"CocktailParty",n},"ConnectedInducedSubgraphPolynomial"][x],{n,k0=1,20}],_Missing[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

p=.

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]

Length[poly]

poly2=Table[-1-n (x^2)+(((1+x))^(2 n)),{n,k0,Length[poly]+k0-1}]

FullSimplify[poly2-poly]

CrossingNumber

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

CycleCount

A167987

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

seq=Table[Sum[(((-1))^j)Binomial[n,j]Binomial[2(n-j),k-2j](2^j)(k-j-1)!/2,{k,3,2n},{j,0,Floor[k/2]}],{n,50}];

FindLinearRecurrence[DeleteMissing[seq]]

FindLinearRecurrence[DeleteMissing[Rest@seq]]

FindSequenceFunction[DeleteMissing[seq],n]

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

Table[Sum[(((-1))^j)Binomial[n,j]Binomial[2(n-j),k-2j](2^j)(k-j-1)!/2,{k,3,2n},{j,0,Floor[k/2]}],{n,10}]

Sum[(((-1))^j)Binomial[n,j]Binomial[2(n-j),k-2j](2^j)(k-j-1)!/2,{j,0,Floor[k/2]}]//FullSimplify

Sum[(1/2) Binomial[2 n,k] Gamma[k] HypergeometricPFQ[{(1/2)-(k/2),-(k/2)},{1-k,(1/2)-n},-2],{k,3,2n}]

Table[Sum[(1/2) Binomial[2 n,k] Gamma[k] HypergeometricPFQ[{(1/2)-(k/2),-(k/2)},{1-k,(1/2)-n},-2],{k,3,2n}],{n,10}]

CyclePolynomial

A167986

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

CoefficientList[Take[poly,4],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]

Derived from the sum for counts:

poly2=Table[Sum[(x^k)(((-1))^j)Binomial[n,j]Binomial[2(n-j),k-2j](2^j)(k-j-1)!/2,{k,3,2n},{j,0,Floor[k/2]}],{n,k0,kmax}];

FullSimplify[Take[poly2-poly,10]]

Sum[(x^k)(((-1))^j)Binomial[n,j]Binomial[2(n-j),k-2j](2^j)(k-j-1)!/2,{k,3,2n},{j,0,Floor[k/2]}]

FullSimplify[Sum[(x^k)Binomial[2(n-j),k-2j](2^j)(k-j-1)!/2,{k,3,2n}]]

Sum[(-1)^j*Binomial[n,j]*Binomial[2*(n-j),k-2*j]*2^j*(k-j-1)!/2,{j,0,Floor[k/2]}]//FullSimplify

Table[(1/2) Binomial[2 n,k] Gamma[k] HypergeometricPFQ[{(1/2)-(k/2),-(k/2)},{1-k,(1/2)-n},-2],{n,2,7},{k,3,2n}]

Cyclic

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

CyclomaticNumber

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

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

DetourIndex

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

FindSequenceFunction[Rest@seq,n-1]

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

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

DetourMatrix

MatrixForm/@Table[GraphData[{"CocktailParty",n},"DetourMatrix"],{n,20}]

MatrixForm/@Table[SparseArray[{i_,j_}:>Abs[i-j],n],{n,20}]

DetourPolynomial

(poly=Factor[Table[GraphData[{"CocktailParty",n},"DetourPolynomial"],{n,k0=2,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

Diameter

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

DistanceMatrix

MatrixForm/@Table[GraphData[{"CocktailParty",n},"DistanceMatrix"],{n,20}]

MatrixForm/@Table[SparseArray[{i_,j_}:>Abs[i-j],n],{n,20}]

DistancePolynomial

(poly=Factor[Table[GraphData[{"CocktailParty",n},"DistancePolynomial"],{n,k0=2,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

DistinguishingNumber

Possibly 4 for n>3

seq=Table[GraphData[{"CocktailParty",n},"DistinguishingNumber"],{n,3,20}]

DomaticNumber

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

DominatingSetCount

A289255

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

FindSequenceFunction[seq,n]

seq=Table[(4^n)-2 n-1,{n,40}]

FindLinearRecurrence[seq]

LinearRecurrence[{6,-9,4},{1,11,57},20]

gf=x FindGeneratingFunction[seq,x]

ToString[gf,InputForm]

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

DominationNumber

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

DominationPolynomial

(poly=Factor[Table[GraphData[{"CocktailParty",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]

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

FullSimplify[poly2-poly]

EdgeChromaticNumber

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

FindSequenceFunction[seq,n]

EdgeConnectivity

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

EdgeCount

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

FindSequenceFunction[DeleteCases[seq,_Missing],n]

4Binomial[n,2]

EdgeCoverCount

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

EdgeCoverNumber

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

EdgeCoverPolynomial

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

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

EulerianCycleCount

seq=Table[GraphData[{"CocktailParty",n},"EulerianCycleCount"],{n,3,20}]

FaceCount

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

FlowPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

FractionalChromaticNumber

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

FractionalEdgeChromaticNumber

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

Girth

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

Gonality

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

HamiltonConnected

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

HamiltonDecompositionCount

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

Hamiltonian

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

HamiltonianCycleCount

A129348(n)/2

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

Table[(-1+2 n)! Hypergeometric1F1[-n,1-2 n,-2],{n,20}]/2

FindLinearRecurrence[seq]

FindSequenceFunction[DeleteCases[seq,_Missing],n]

gf=x FindGeneratingFunction[seq,x]

HamiltonianNumber

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

HamiltonianPathCount

A007060(n)/2

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

Table[(2 n)! Hypergeometric1F1[-n,-2 n,-2],{n,20}]/2

FindLinearRecurrence[seq]

FindSequenceFunction[DeleteCases[seq,_Missing],n]

gf=x FindGeneratingFunction[seq,x]

HamiltonianWalkCount

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

HararyIndex

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

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

HexagonCount

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

FindSequenceFunction[seq,n]//Factor

IdiosyncraticPolynomial

(poly=Factor[Table[GraphData[{"CocktailParty",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]

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

FullSimplify[poly2-poly]

IndependenceNumber

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

IndependencePolynomial

l=Table[GraphData[{"CocktailParty",n},"#"]->{blah,Factor[(1+2n x+n x^2)]},{n,20}]

IndependenceRatio

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

FindSequenceFunction[seq,n]

IndependentEdgeSetCount

A293073

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

FindSequenceFunction[DeleteCases[seq,_Missing],n]

FindSequenceFunction[Rest@DeleteCases[seq,_Missing],n-1]

seq=Table[DifferenceRoot[Function[{\[FormalY],\[FormalN]},{4 (1+\[FormalN]) (2+\[FormalN]) \[FormalY][\[FormalN]]+(4+10 \[FormalN]+4 (\[FormalN]^2)) \[FormalY][1+\[FormalN]]+(-9-4 \[FormalN]) \[FormalY][2+\[FormalN]]+\[FormalY][3+\[FormalN]]==0,\[FormalY][1]==1,\[FormalY][2]==7,\[FormalY][3]==51}]][n],{n,100}];

Head/@Table[FindLinearRecurrence[Drop[seq,k]],{k,0,4}]//Union

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

RecurrenceTable[{4(1+n)(2+n)a[n]+(4+10n+4n^2)a[1+n]+(-9-4n)a[2+n]+a[3+n]==0,a[1]==1,a[2]==7,a[3]==51},a,{n,20}]

IndependentVertexSetCount

A016777

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

FindSequenceFunction[DeleteCases[seq,_Missing],n]

3Range[0,20]+1

Table[3n+1,{n,0,20}]

FindLinearRecurrence[seq]

LinearRecurrence[{2,-1},{1,4},20]

seq=Table[3n+1,{n,0,20}]

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

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

IntersectionNumber

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

IrredundantSetCount

A084849

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

FindLinearRecurrence[DeleteMissing@seq]

LinearRecurrence[{3,-3,1},{4,11,22},{0,20}]

FindSequenceFunction[DeleteMissing@seq,n]

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

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

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

KirchhoffIndex

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

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

KirchhoffSumIndex

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

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

LaplacianMatrix

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

LaplacianPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

LeafCount

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

LongestCycleCount

Same as |HC|

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

LovaszNumber

Possibly 2?

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

MatchingGeneratingPolynomial

(poly=Factor[Table[GraphData[{"CocktailParty",n},"MatchingGeneratingPolynomial"],{n,k0=2,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

MatchingNumber

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

MatchingPolynomial

(poly=Factor[Table[GraphData[{"CocktailParty",n},"MatchingPolynomial"],{n,k0=2,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

MaximalCliqueCount

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

FindSequenceFunction[seq,n]

MaximalIndependentEdgeSetCount

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

FindSequenceFunction[DeleteMissing@seq,n]

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

MaximalIndependentVertexSetCount

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

MaximalIrredundantSetCount

A000384

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

PolygonalNumber[6,Range[20]]

FindLinearRecurrence[DeleteMissing@seq]

LinearRecurrence[{3,-3,1},{4,11,22},{0,20}]

FindSequenceFunction[DeleteMissing@seq,n]

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

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

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

MaximumCliqueCount

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

FindSequenceFunction[seq,n]

MaximumIndependentEdgeSetCount

A053871

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

Series[1/(Exp[x]Sqrt[1-2x]),{x,0,5}]

Range[0,20]!CoefficientList[Series[1/(Exp[x]Sqrt[1-2x]),{x,0,20}],x]

FullSimplify[SeriesCoefficient[1/(Exp[x]Sqrt[1-2x]),{x,0,n}],n>0]

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

FindSequenceFunction[seq,n]

FindSequenceFunction[Rest@seq,n-1]

gf=x FindGeneratingFunction[seq,x]

MaximumIndependentVertexSetCount

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

MaximumLeafNumber

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

MaximumVertexDegree

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

MeanDistance

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

MinimalDominatingSetCount

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

FindSequenceFunction[DeleteMissing@seq,n]//Factor

MinimalDominatingSetSignature

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

MinimumCliqueCoveringCount

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

MinimumConnectedDominatingSetCount

A046092

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

FindSequenceFunction[DeleteCases[seq,_Missing],n]

4Binomial[n,2]

MinimumDistinguishingLabelingCount

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

MinimumDominatingSetCount

A000384

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

FindSequenceFunction[seq,n]

MinimumEdgeCoverCount

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

MinimumVertexDegree

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

MolecularTopologicalIndex

A181773

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

FindLinearRecurrence[Rest@seq]

FindSequenceFunction[Rest@seq,n-1]

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

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

Nonhamiltonian

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

NormalizedLaplacianMatrix

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

OddChordlessCycleCount

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

PathCount

A286038

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

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

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

(1/2) * (-2*n - 1 + Sum_{j=0..n} Sum_{k=2*j..2*n} (-1)^j*2^j*(k-j)! * binomial(n,j) * binomial(2*n-2*j,k-2*j) ).

Table[(1/2)(-2n-1+Sum[(-1)^j*2^j*(k-j)!*Binomial[n,j]Binomial[2*n-2*j,k-2*j],{j,0,n},{k,2j,2n}]),{n,20}]

Sum[(-1)^j*2^j*(k-j)!*Binomial[n,j]Binomial[2*n-2*j,k-2*j],{k,2j,2n}]

Sum[(((-2))^j) Binomial[n,j] j! HypergeometricU[1+j,2-j+2 n,1],{j,0,n}]//FullSimplify

%78//FunctionExpand

Table[(Sum[(((-2))^k) Binomial[n,k] k! HypergeometricU[k+1,2n+2-k,1],{k,0,n}]-2n-1)/2,{n,20}]//FunctionExpand

PathPolynomial

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

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

PentagonCount

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

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]//Factor

Perfect

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

ProjectivePlaneCrossingNumber

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

Radius

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

RankPolynomial

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

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

RectilinearCrossingNumber

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

ReliabilityPolynomial

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

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

SigmaPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

SpanningTreeCount

A193130

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

FindLinearRecurrence[Rest@seq]

FindSequenceFunction[Rest@seq,n-1]

Table[n^(n-2)*(n-1)^n*4^(n-1),{n,20}]

SquareCount

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

FindSequenceFunction[seq,n]//Factor

StabilityIndex

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

FullSimplify[FindSequenceFunction[Rest@seq,n-1],n∈Integers]

gf=x FindGeneratingFunction[seq,x]

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

TopologicalIndex

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

FindSequenceFunction[Rest@seq,n-1]

FindGeneratingFunction[seq,x]

ToroidalCrossingNumber

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

Traceable

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

TriangleCount

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

FindSequenceFunction[seq,n]

TuttePolynomial

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

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

Untraceable

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

VertexConnectivity

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

VertexCount

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

FindSequenceFunction[seq,n]

VertexCoverNumber

vertex cover number == vertex count - independence number

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

VertexCoverPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

VertexDegrees

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

VertexTransitive

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

WienerIndex

A001105

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

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

WienerSumIndex

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

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

Embeddings

Graphics[{Circle[.5{1,1},.5],ShowLabeledGraph[g=CocktailPartyGraph[3],VertexStyle->Disk[.02]][[1]]}]

Graphics[{Circle[.5{1,1},.54],ShowLabeledGraph[g=CocktailPartyGraph[4],VertexStyle->Disk[.02]][[1]]}]

Graphics[{Circle[.5{1,1},.54],ShowLabeledGraph[g=CocktailPartyGraph[5],VertexStyle->Disk[.02]][[1]]}]

Other Embeddings

CocktailPartyGraph[n_Integer?Positive]:=ChangeVertices[GraphComplement[LadderRungGraph[n]],
Flatten[Transpose[{{(n-1)/2+Cos[π-# π/(n-1)],.1-.8Sin[π-# π/(n-1)]}&/@Range[0,n-1],{(n-1)/2+Cos[π-# π/(n-1)],1-.1+.8Sin[π-# π/(n-1)]}&/@Range[0,n-1]}],1]]

(g=CocktailPartyGraph[6])//ShowLabeledGraph

CocktailPartyGraph2[n_Integer?Positive]:=Module[{v,i},
v=Table[Through[{Cos,Sin}[2π (i+1/2)/(2n)]],{i,0,2n-1}];
ChangeVertices[GraphComplement[LadderRungGraph[n]],Join[v[[1;;2n;;2]],Reverse@v[[2;;2n;;2]]]
]]

ShowLabeledGraph[CocktailPartyGraph2[5],VertexStyle->Disk[.02]]

GraphData

GraphClassString

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

GraphDataString

Table[ClearSystemCache[];
Print[GraphDataString[CocktailPartyGraph[n],{{"CocktailParty",n},ToString[n]<>"-cocktail party graph","CocktailPartyGraph"},{300,10^9}]],{n,20}]