Keller Graph

Author

Eric W. Weisstein
August 2, 2023

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

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

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

Construction

Table[KellerGraph[n],{n,3}]

Table[GraphData[{"Keller",n}],{n,4}]

ToEntity/@%

KellerCond[v_]:=Module[{t=Transpose[v],diff},
Length[diff=Pick[t,UnsameQ@@@t]]>1&&!FreeQ[Mod[Subtract@@@diff,4],2]
]

KellerGraph0[n_,opts___?OptionQ]:=Module[{v=Tuples[Range[0,3],{n}]},
System`Graph[v,UndirectedEdge@@@Select[Subsets[v,{2}],KellerCond[#]&],opts]
]

KellerGraph0[2,VertexLabels->"Name"]

LabeledKellerGraph[2]//ShowGraph

(cg=ChangeVertices[KellerGraph[2],Flatten[Table[{x,y},{x,0,3},{y,0,3}],1]])//ShowGraph

RecognizeGraph/@KellerGraph/@Range[4]

KellerGraph[6] takes too long to compute using the above simple code; ran overnight without returning

4^Range[6]

v=ToCommonEdges[cg,Graph["ClebschGraph"]]

i=GraphData["ClebschGraph","AllImages"]

Length[i]

l={{"ClebschGraph",{0,ReplacePart[GraphData["ClebschGraph","Embeddings"],4->v]}}}

GraphData["ClebschGraph","EmbeddingClasses"]

l={{"ClebschGraph",{0,ReplacePart[GraphData["ClebschGraph","EmbeddingClasses"],4->{"IntegerCoordinates"}]}}}

ShowGraph[KellerGraph[3]]/.{PointSize[_]->AbsolutePointSize[5],GrayLevel[0]->Opacity[.2]}

MinimumVertexColoring

Stan Wagon 2013/01/18

There is a trivial algorithm to get a 2^n vertex coloring of the graph Keller (n).Here is code for that:

KellerGraphColoring[n_]:=(verts=Tuples[Range[0,3],n];
col1=Select[verts,{}==Complement[Union[#],{0,1}]&];
Table[Position[col1,Floor[v/2]][[1,1]],{v,verts}])

Table[VertexColoringQ[GraphData[{"Keller",n}],KellerGraphColoring[n]],{n,5}]

Select[GraphData["Keller"],Head[GraphData[#,"MinimumVertexColoring"]]===Missing&]

l=Table[{{"Keller",n},{0,KellerGraphColoring[n]}},{n,4,5}]

Discussion

Stan Wagon, 2013/01/24

Wendy Myrvold has a quite simple construction of a Ham cycle in ALL Keller graphs.  And I just extended it to prove that all Kellers are Hamilton-connected (pretty easy, though I did use base cases of d=2,3,4,5 which I did by computation). 

I find that the vertex order of Keller graphs in GraphData does not match the order of the 4-tuples in the natural way (Tuples[{0,1,2,3}, d]). I looked at your file, and you seem to build it the same way I did, but yet the vertex order is different. My defn of Keller graphs, with a special case for d=2 (note the permutation at the VerexCoords option, which comes from an isomorphism) so that the vertex locations match the nice ones for the Clebsch case is:

KellerGraphSW[2] := (verts = Tuples[Range[0, 3], 2];
  System`Graph[Range[Length[verts]], 
    UndirectedEdge @@@ (Most /@ 
       Select[
        Flatten[
         Table[{i, j, 
           If[
            Count[verts[[i]] - verts[[j]], x_ /; x != 0] >= 2 &&
             Count[verts[[i]] - verts[[j]], x_ /; Mod[x, 4] == 2] >= 1, True, 
            False]}, {i, Length[verts] - 1}, {j, i + 1, Length[verts]}], 
         1], #[[-1]] &]), 
    System`VertexCoordinates -> (System`VertexCoordinates /. 
        AbsoluteOptions[GraphData[{"Keller", 2}], VertexCoordinates])[[{1, 2, 
        4, 15, 9, 12, 7, 14, 8, 3, 10, 5, 16, 11, 13, 6}]]]);

KellerGraphSW[n_] := (verts = Tuples[Range[0, 3], n];
  System`Graph[Range[Length[verts]], 
    UndirectedEdge @@@ (Most /@ 
       Select[
        Flatten[
         Table[{i, j, 
           If[
            Count[verts[[i]] - verts[[j]], x_ /; x != 0] >= 2 &&
             Count[verts[[i]] - verts[[j]], x_ /; Mod[x, 4] == 2] >= 1, True, 
            False]}, {i, Length[verts] - 1}, {j, i + 1, Length[verts]}], 
         1], #[[-1]] &])]);

RecognizeGraph/@Table[KellerGraph[n],{n,4}]

RecognizeGraph/@Table[KellerGraphSW[n],{n,4}]

{CombinatoricaGraphToV8Graph@KellerGraph[2],KellerGraphSW[2]}

{CombinatoricaGraphToV8Graph@KellerGraph[3],KellerGraphSW[3]}

With the defn above (and, yes, it is too slow and too big to try to get Keller_6 as a graph, though one can work with 6 and 7 in some cases where full graph is not needed), the following code gets a Ham cycle

KellerHamiltonianCycle[d_] := (verts = Tuples[Range[0, 3], d];
  ord = Partition[verts, 4^(d - 1)][[{1, 3, 2, 4}]];
  ord[[1]] = Flatten[Transpose[Partition[ord[[1]], 4^(d - 2)]], 1];
  ord[[2]] = Flatten[RotateRight[#, 1] & /@ Transpose[Partition[ord[[2]], 4^(d - 2)]], 1];
  ord[[3]] = Flatten[Transpose[Partition[ord[[3]], 4^(d - 2)]], 1];
  ord[[4]] = Flatten[RotateRight[#, 1] & /@ Transpose[Partition[ord[[4]], 4^(d - 2)]], 1];
  (cyV = Join[Riffle[ord[[1]], ord[[2]]], Riffle[ord[[3]], ord[[4]]]]);
  cy = Flatten[Position[verts, #] & /@ cyV];
  UndirectedEdge @@@ Partition[Append[cy, cy[[1]]], 2, 1])

There is really nothing to it, as you can see from an explicit listing of the tuples in order of a Ham cycle. The first entries alternate  0 2 0 2 0 2…. 0 2   1 3 1 3 1 3 1 3   etc
And the second entries are chosen in an easy order  to make it all work.

Properties

Acyclic

Entity["Graph",{"Keller",n}]["Acyclic"]

Table[GraphData[{"Keller",n},"Acyclic"],{n,5}]

AdjacencyMatrixCount

Entity["Graph",{"Keller",n}]["AdjacencyMatrixCount"]

seq=Table[GraphData[{"Keller",n},"AdjacencyMatrixCount"],{n,4}]

(((4^n)!)/(|Aut|))

Arboricity

Entity["Graph",{"Keller",n}]["Arboricity"]

Table[GraphData[{"Keller",n},"Arboricity"],{n,5}]

ArcTransitive

Entity["Graph",{"Keller",n}]["ArcTransitive"]

Table[GraphData[{"Keller",n},"ArcTransitive"],{n,5}]

ArcTransitivity

Entity["Graph",{"Keller",n}]["ArcTransitivity"]

Table[GraphData[{"Keller",n},"ArcTransitivity"],{n,5}]

AutomorphismCount

Entity["Graph",{"Keller",n}]["AutomorphismCount"]

seq=Table[GraphData[{"Keller",n},"AutomorphismCount"],{n,5}]

FindSequenceFunction[Rest@seq,n-1]

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

(seq=Table[GroupOrder[GraphAutomorphismGroup[KellerGraph[n]]],{n,6}])//Timing

FindSequenceFunction[{24,1920,3072,98304,3932160,188743680},n]

FindSequenceFunction[Rest@{24,1920,3072,98304,3932160,188743680},n-1]

FindGeneratingFunction[{24,1920,3072,98304,3932160,188743680},x]

BalabanIndex

Entity["Graph",{"Keller",n}]["BalabanIndex"]

seq=Table[GraphData[{"Keller",n},"BalabanIndex"],{n,5}]

FindSequenceFunction[seq,n]

Bandwidth

Entity["Graph",{"Keller",n}]["Bandwidth"]

Table[GraphData[{"Keller",n},"Bandwidth"],{n,5}]

BridgeCount

Entity["Graph",{"Keller",n}]["BridgeCount"]

seq=Table[GraphData[{"Keller",n},"BridgeCount"],{n,5}]

BurningNumber

Entity["Graph",{"Keller",n}]["BurningNumber"]

seq=Table[GraphData[{"Keller",n},"BurningNumber"],{n,5}]

ChordCount

Entity["Graph",{"Keller",n}]["ChordCount"]

seq=Table[GraphData[{"Keller",n},"ChordCount"],{n,5}]

Chords[GraphData[{"Keller",2}]]//Length//Timing

Table[(2^(2n-1)) ((4^n)-(3^n)-n),{n,5}]

FindSequenceFunction[seq,n]

ChordlessCycleCount

Entity["Graph",{"Keller",n}]["ChordlessCycleCount"]

seq=Table[GraphData[{"Keller",n},"ChordlessCycleCount"],{n,5}]

FindSequenceFunction[seq,n]

ChromaticInvariant

Entity["Graph",{"Keller",n}]["ChromaticInvariant"]

seq=Table[GraphData[{"Keller",n},"ChromaticInvariant"],{n,5}]

ChromaticNumber

Entity["Graph",{"Keller",n}]["ChromaticNumber"]

seq=Table[GraphData[{"Keller",n},"ChromaticNumber"],{n,5}]

FindSequenceFunction[seq,n]

Table[2^n,{n,5}]

Circumference

Entity["Graph",{"Keller",n}]["Circumference"]

seq=Table[GraphData[{"Keller",n},"Circumference"],{n,5}]

Table[4^n,{n,5}]

Classes

Entity["Graph",{"Keller",n}]["Classes"]

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

CliqueCount

A295902

Entity["Graph",{"Keller",n}]["CliqueCount"]

seq=Table[GraphData[{"Keller",n},"CliqueCount"],{n,5}]

FindSequenceFunction[seq,n]

CliqueCoveringNumber

Entity["Graph",{"Keller",n}]["CliqueCoveringNumber"]

seq=Table[GraphData[{"Keller",n},"CliqueCoveringNumber"],{n,5}]

FindSequenceFunction[Rest@seq,n-3]//FullSimplify

gf=x FindGeneratingFunction[seq,x]

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

Table[3 (2^(n-3)),{n,5}]

CliquePolynomial

Entity["Graph",{"Keller",n}]["CliquePolynomial"]

(poly=Factor@Table[GraphData[{"Keller",n},"CliquePolynomial"],{n,k0=1,kmax=5}]/.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,20}]

FullSimplify[poly2-poly]

CliqueNumber

Entity["Graph",{"Keller",n}]["CliqueNumber"]

A202604

seq=Table[GraphData[{"Keller",n},"CliqueNumber"],{n,5}]

seq={1,2,5,12,28,60,124,256,512,1024,2048,4096,8192,16384,32768,65536,131072,262144,524288,1048576,2097152,4194304,8388608,16777216,33554432,67108864,134217728,268435456,536870912,1073741824,2147483648,4294967296,8589934592};

FindLinearRecurrence[Drop[seq,7]]

seq[[8]]

Join[{1,2,5,12,28,60,124},LinearRecurrence[{2},{256},14]]

gf=x FindGeneratingFunction[seq,x]

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

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

Table[Piecewise[{{1,n==1},{2,n==2},{5,n==3},{2^n-4,4<=n<=7}},2^n],{n,20}]

ComplementOddChordlessCycleCount

Entity["Graph",{"Keller",n}]["ComplementOddChordlessCycleCount"]

seq=Table[GraphData[{"Keller",n},"ComplementOddChordlessCycleCount"],{n,5}]

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

ConnectedComponentCount

Entity["Graph",{"Keller",n}]["ConnectedComponentCount"]

seq=Table[GraphData[{"Keller",n},"ConnectedComponentCount"],{n,5}]

ConnectedDominatingSetCount

Entity["Graph",{"Keller",n}]["ConnectedDominatingSetCount"]

seq=Table[GraphData[{"Keller",n},"ConnectedDominatingSetCount"],{n,5}]

FindSequenceFunction[DeleteCases[seq,_Missing],n]

ConnectedDominationNumber

Entity["Graph",{"Keller",n}]["ConnectedDominationNumber"]

seq=Table[GraphData[{"Keller",n},"ConnectedDominationNumber"],{n,5}]

FindSequenceFunction[seq,n]

ConnectedDominationPolynomial

Entity["Graph",{"Keller",n}]["ConnectedDominationPolynomial"]

(poly=DeleteCases[Factor@Table[GraphData[{"Keller",n},"ConnectedDominationPolynomial"][x],{n,k0=1,kmax=5}],_Missing[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

ConnectedInducedSubgraphCount

Entity["Graph",{"Keller",n}]["ConnectedInducedSubgraphCount"]

seq=Table[GraphData[{"Keller",n},"ConnectedInducedSubgraphCount"],{n,5}]

FindSequenceFunction[DeleteCases[seq,_Missing],n]

ConnectedInducedSubgraphPolynomial

Entity["Graph",{"Keller",n}]["ConnectedInducedSubgraphPolynomial"]

(poly=DeleteCases[Factor@Table[GraphData[{"Keller",n},"ConnectedInducedSubgraphPolynomial"][x],{n,k0=1,kmax=5}],_Missing[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

CrossingNumber

Entity["Graph",{"Keller",n}]["CrossingNumber"]

seq=Table[GraphData[{"Keller",n},"CrossingNumber"],{n,5}]

CycleCount

Entity["Graph",{"Keller",n}]["CycleCount"]

seq=Table[GraphData[{"Keller",n},"CycleCount"],{n,5}]

Cyclic

Entity["Graph",{"Keller",n}]["Cyclic"]

Table[GraphData[{"Keller",n},"Cyclic"],{n,5}]

CyclomaticNumber

Entity["Graph",{"Keller",n}]["CyclomaticNumber"]

seq=Table[GraphData[{"Keller",n},"CyclomaticNumber"],{n,5}]

FindSequenceFunction[seq,n]

Table[(2^(2n-1)) ((4^n)-(3^n)-n)-(4^n)+1,{n,5}]

FullSimplify[(2^(2n-1)) ((4^n)-(3^n)-n)-(4^n)+1]

Table[(2^(2n-1)) (-(3^n)-n)+((2^(2n-1))-1)(4^n)+1,{n,5}]

FullSimplify[((2^(2n-1))-1)(4^n)]

DetourIndex

Entity["Graph",{"Keller",n}]["DetourIndex"]

seq=Table[GraphData[{"Keller",n},"DetourIndex"],{n,5}]

Diameter

Entity["Graph",{"Keller",n}]["Diameter"]

seq=Table[GraphData[{"Keller",n},"Diameter"],{n,5}]

DistinguishingNumber

Entity["Graph",{"Keller",n}]["DistinguishingNumber"]

seq=Table[GraphData[{"Keller",n},"DistinguishingNumber"],{n,5}]

DominatingSetCount

Entity["Graph",{"Keller",n}]["DominatingSetCount"]

seq=Table[GraphData[{"Keller",n},"DominatingSetCount"],{n,5}]

DominationNumber

Entity["Graph",{"Keller",n}]["DominationNumber"]

seq=Table[GraphData[{"Keller",n},"DominationNumber"],{n,5}]

EdgeChromaticNumber

Entity["Graph",{"Keller",n}]["EdgeChromaticNumber"]

Jarnicki et al. 2017: edge chromatic number of (Subscript[K, n]) equals its degree

A284850

seq=Table[GraphData[{"Keller",n},"EdgeChromaticNumber"],{n,5}]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

Table[((4^n)-(3^n)-n),{n,30}]

EdgeConnectivity

Entity["Graph",{"Keller",n}]["EdgeConnectivity"]

A284850

Same as EdgeChromaticNumber == Degree

Table[GraphData[{"Keller",n},"EdgeConnectivity"],{n,5}]

Table[((4^n)-(3^n)-n),{n,5}]

EdgeCount

Entity["Graph",{"Keller",n}]["EdgeCount"]

A284838

seq=Table[GraphData[{"Keller",n},"EdgeCount"],{n,5}]

seq=Table[2^(2 n-1) (4^n-3^n-n),{n,15}];

coef=FindLinearRecurrence[seq]

LinearRecurrence[{36,-432,1984,-3072},{0,40,1088,21888},20]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[-((8 x (-5+44 x))/((((1-4 x))^2) (1-28 x+192 (x^2)))),{x,0,20}],x]

EdgeCoverCount

Entity["Graph",{"Keller",n}]["EdgeCoverCount"]

seq=Table[GraphData[{"Keller",n},"EdgeCoverCount"],{n,5}]

EdgeCoverNumber

Entity["Graph",{"Keller",n}]["EdgeCoverNumber"]

seq=Table[GraphData[{"Keller",n},"EdgeCoverNumber"],{n,5}]

FindSequenceFunction[seq,n]//Factor

Table[((256 (-1+n))/(82-31 n+3 (n^2))),{n,5}]

gf=x FindGeneratingFunction[seq,x]

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

EulerianCycleCount

Entity["Graph",{"Keller",n}]["EulerianCycleCount"]

seq=Table[GraphData[{"Keller",n},"EulerianCycleCount"],{n,5}]

FaceCount

Entity["Graph",{"Keller",n}]["FaceCount"]

seq=Table[GraphData[{"Keller",n},"FaceCount"],{n,5}]

FibonacciCube

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

seq=Table[GraphData[{"Keller",n},"FibonacciCube"],{n,5}]

FractionalChromaticNumber

Entity["Graph",{"Keller",n}]["FractionalChromaticNumber"]

seq=Table[GraphData[{"Keller",n},"FractionalChromaticNumber"],{n,5}]

Table[Piecewise[{{1,n==1},{16/5,n==2}},2^n],{n,5}]

FractionalEdgeChromaticNumber

Entity["Graph",{"Keller",n}]["FractionalEdgeChromaticNumber"]

seq=Table[GraphData[{"Keller",n},"FractionalEdgeChromaticNumber"],{n,5}]

A284850

Table[((4^n)-(3^n)-n),{n,5}]

Girth

Entity["Graph",{"Keller",n}]["Girth"]

seq=Table[GraphData[{"Keller",n},"Girth"],{n,5}]

Gonality

Entity["Graph",{"Keller",n}]["Gonality"]

seq=Table[GraphData[{"Keller",n},"Gonality"],{n,5}]

HamiltonConnected

Entity["Graph",{"Keller",n}]["HamiltonConnected"]

seq=Table[GraphData[{"Keller",n},"HamiltonConnected"],{n,5}]

HamiltonDecompositionCount

Entity["Graph",{"Keller",n}]["HamiltonDecompositionCount"]

seq=Table[GraphData[{"Keller",n},"HamiltonDecompositionCount"],{n,5}]

Hamiltonian

Entity["Graph",{"Keller",n}]["Hamiltonian"]

Table[GraphData[{"Keller",n},"Hamiltonian"],{n,5}]

HamiltonianCycleCount

Entity["Graph",{"Keller",n}]["HamiltonianCycleCount"]

seq=Table[GraphData[{"Keller",n},"HamiltonianCycleCount"],{n,5}]

HamiltonianNumber

Entity["Graph",{"Keller",n}]["HamiltonianNumber"]

seq=Table[GraphData[{"Keller",n},"HamiltonianNumber"],{n,5}]

Table[Piecewise[{{Missing["NotApplicable"],n==1}},4^n],{n,5}]

HamiltonianPathCount

Entity["Graph",{"Keller",n}]["HamiltonianPathCount"]

seq=Table[GraphData[{"Keller",n},"HamiltonianPathCount"],{n,5}]

HamiltonianWalkCount

Entity["Graph",{"Keller",n}]["HamiltonianWalkCount"]

seq=Table[GraphData[{"Keller",n},"HamiltonianWalkCount"],{n,5}]

HararyIndex

Entity["Graph",{"Keller",n}]["HararyIndex"]

A296189

seq=Table[GraphData[{"Keller",n},"HararyIndex"],{n,5}]

FindSequenceFunction[seq,n]

x FindGeneratingFunction[seq,x]

Table[(2^(2n-1)) ((4^n)-(3^n)-n),{n,15}]

HexagonCount

Entity["Graph",{"Keller",n}]["HexagonCount"]

A300849

seq=Table[GraphData[{"Keller",n},"HexagonCount"],{n,5}]

Do[Print[{{"Keller",n},CycleCount[KellerGraph[n],6]//Timing}],{n,6}]

Length[bfile=Import["https://oeis.org/A300849/b300849.txt","Table"][[All,2]]]

(PARI)
G(n, k, bset)={
my(r=vector(n), m=sum(i=0, k-1, 1<<(2*i)), u=vector(1<<(2*k), i, i==1));
for(n=1, n,
  my(v=vector(#u));
  for(b=1, #bset,
    my(d=bitxor(bset[b], bset[b]<<2), t1=bitand(d, m), t2=bitand((d>>1), m-t1));
    for(i=0, #u-1,
      my(k=bitor(bitor(i, t1), bitor(t2<<1, bitand(i>>1, t2))));
      v[k+1] += u[i+1] ));
  u = v;
  r[n] = 4^n*u[#u]/(2*k) );
r
}
\\ this takes a few seconds per term
seq(n)={my(q2=G(n, 2, [0..3])*4, q3=G(n, 3, [0..15])*6, q4=G(n, 4, [0..63])*8, q6=G(n, 6, [0..1023])*12, diamonds=G(n, 6, apply(t->bitor(t, bitand(t, 12)<<6), [0..63]))*12);
vector(n, n, (q6[n] - (6*q4[n]*q2[n] + 3*q3[n]^2)/4^n + 6*q4[n] + 9*diamonds[n] + 7*q2[n]^3/16^n - 12*q2[n]^2/4^n - 4*q3[n] + 4*q2[n])/12)
} \\ Andrew Howroyd, Mar 16 2018

seq={0,320,103493760,1989020096512,18004320077137920,119822580205402103808,679908187040469153808384,3502748255987493030839058432,16926129866817207966343976976384,78226597001366370548567920133275648,350205926622184430366093984866429304832,1532001517834279405829297957506135552425984,6589377576999932329601746449243034070800138240,27990447048040710479802148885641964782216623947776,117804289180944144163631320509669322048991333915295744,492413106077838014832560453002564757791822773573564497920,2047747788835807072030716465666963713188560683319932535963648,8483316914615688414450693784822513074136071385397669517292732416,35044227181036770816513705747824212758932746329134210684114490621952,144457848998673154725774576352422970707356773699688327409009532631777280,594528529797894697550105490727259397243042604371247526033403999855700869120,2443910369469337978844623546245628348818345187157720004603022987029328942858240,10037119268600850501028541363718182386791026515833687418235952264009754824797061120,41194723176251595217509342609553626297788168457649441916056945392037080240240514301952,168987959554720841427073001607855799218638007588799552440977193324627551779585464740085760,692956976203008671232804461455812489921967615204599857648667677639101797885354064284984803328,2840756961276946720987155954065692207422344613745381489945510227988233415175495665197284887363584,11643133811630994524687198409065138731003954229835271688412644024726835537699116079864039540923564032,47712998825928935504790264903231958750110463680735311283368197038612140719803494355566040488613934792704,195502271677254226354417873380528910966917470751002738325083129339538937623998548306368302738487096196988928,800991873783102285628073809829281249012253689948193462253331702321968161510419809034295498782439606043270971392,3281521999741081747617325447469640148771003583254068033564234156348772314177905925045017284320136270165604233641984,13443139730363367316676067017688082914136977790473628505328513580799263238202651533345207072815548288807591267281141760,55069323722214654831578132523520154386830057117066302437706212204464850753819721986639720583175963789274816913846636118016,225583069785510784551916272846397945376118769489535573602633900078080993950985407001721284940136504301423070655124436936556544,924046993556661724337120667898368109791036081186894470349951347484510640552250611948075380391858787128751498867686162321480089600,3785076942377693223061243493278511290557466949762097205244602744154654216276099307627826606189579918197788850685028682507115716149248,15504229538447662074382239942100731200318123529237218799803521292119019549824057650694882699824405578838518714806943259524116289899462656,63507027296829951326610046753552011581658923416875876887133616994011196737523577731159186425243447484268111823738862105329578665354232594432,260130015855899837943670891988284113233628354552454520183532018002752522690516065739223126731262309855715359352659386382581656398964880845045760,1065508618033207050911587186630663435557051918080861313009600697525991539318103319973922801436977105268264077001328233175626740735336900191101911040,4364372676531754506398806347992772122882143658613426248479985374864637707599603358822327968801442772697732422312032418968993473904894658190195724124160,17876622170677514104150598143863225240715451197998525846320865938409255564280602939215376336216124769555252303950517611193508873318054761709548132085268480,73223110398296028086110454837825279913283218302812727558124647524733727701105437119375257315481659055358522884937703351561645062818406236091919351983732948992,299923291710744570530762864152937123435946072429961126107375087801008266026222581812689736871354445269735162346595573377076082804797013647272667567388902608076800,1228490200489878228976446551206706568911316299407014921825435122937873643246850795475681002701007275525671691657573252166043781525578462026109819189451442249739010048,5031909370800081609572133226116314312648639644977674225824567104313390264463110568337404727839237115155867302998464198503116653042307911311243172601704766506500551081984,20610742284349736086734512059499311632742719293424498209899696259555340286194205222894351535613500500031979880877591206403006217192831641300777800303124544624257113975160832,84421727889653302262002941669187140324738819646910358110483425805106312752258821436513068915579672991562093878451684480728576924750856284588636210975181641489752616430336999424,345791789094814451519840303482945466901442765990361530360777437476596483902302536005883213578279412083740456792320637526180052210677611376468183810016750167455860836963228799991808,1416364371309170462351774333457091245372308686283507608820002133710500877421532407751313726978144366076968002819465856160781792451560680446429700069946367051695151271447399817283633152,5801432161043813429893013807155169010393013475624833156906963110815086805383737995837081358399321306641357094811517148697288121634795600334088436802751210496664458884660985753354039721984,23762677486248712852609569729209907652291607595937083651744124887112470617555387668098818047724272454623021771357559161998667641818207013687701687425719953569764462633432483773369286320455680,97331961865058794586542857700361920095257355207586998825115011735080373274554646614073043180262044859299410996347951490197320370698108422069954338518000647122008271116901089327926611638753427456,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,};

Take[seq,100]==bfile

coef=FindLinearRecurrence[seq]

Length[coef]

(eqn=Prepend[Table[a[k]==seq[[k]],{k,Length[coef]}],a[n]==Total[Table[a[n-k] coef[[k]],{k,Length[coef]}]]])//TraditionalForm

RSolve[eqn,a[n],n]

FullSimplify[%]

InputForm[%373[[1,1,1,2]]]

Take[seq,Length[coef]]

LinearRecurrence[coef,Take[seq,Length[coef]],Length[seq]]==seq

IndependenceNumber

A258935

Entity["Graph",{"Keller",n}]["IndependenceNumber"]

seq=Table[GraphData[{"Keller",n},"IndependenceNumber"],{n,5}]

Table[Piecewise[{{4,n==1},{5,n==2}},2^n],{n,10}]

gf=x FindGeneratingFunction[%,x]

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

IndependenceRatio

Entity["Graph",{"Keller",n}]["IndependenceRatio"]

seq=Table[GraphData[{"Keller",n},"IndependenceRatio"],{n,5}]

Table[((Piecewise[{{4,n==1},{5,n==2}},2^n])/(4^n)),{n,10}]

FullSimplify[((Piecewise[{{4,n==1},{5,n==2}},2^n])/(4^n))]

IndependentEdgeSetCount

Entity["Graph",{"Keller",n}]["IndependentEdgeSetCount"]

seq=Table[GraphData[{"Keller",n},"IndependentEdgeSetCount"],{n,5}]

IndependentVertexSetCount

A284854

Entity["Graph",{"Keller",n}]["IndependentVertexSetCount"]

seq=Table[GraphData[{"Keller",n},"IndependentVertexSetCount"],{n,5}]

FindSequenceFunction[DeleteCases[seq,_Missing],n]

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

FindLinearRecurrence[seq]

IntersectionNumber

Entity["Graph",{"Keller",n}]["IntersectionNumber"]

seq=Table[GraphData[{"Keller",n},"IntersectionNumber"],{n,5}]

KirchhoffIndex

Entity["Graph",{"Keller",n}]["KirchhoffIndex"]

seq=Table[GraphData[{"Keller",n},"KirchhoffIndex"],{n,5}]

FindSequenceFunction[seq,n]

KirchhoffSumIndex

Entity["Graph",{"Keller",n}]["KirchhoffSumIndex"]

seq=Table[GraphData[{"Keller",n},"KirchhoffSumIndex"],{n,5}]

LeafCount

Entity["Graph",{"Keller",n}]["LeafCount"]

seq=Table[GraphData[{"Keller",n},"LeafCount"],{n,5}]

LongestCycleCount

Entity["Graph",{"Keller",n}]["LongestCycleCount"]

Same as HamiltonianCycleCount

seq=Table[GraphData[{"Keller",n},"LongestCycleCount"],{n,5}]

LovaszNumber

Entity["Graph",{"Keller",n}]["LovaszNumber"]

Fung (2011)

seq=Table[GraphData[{"Keller",n},"LovaszNumber"],{n,5}]

Table[Piecewise[{{4,n==1},{6,n==2},{28/3,n==3}},2^n],{n,5}]

LucasCube

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

seq=Table[GraphData[{"Keller",n},"LucasCube"],{n,5}]

MatchingNumber

Entity["Graph",{"Keller",n}]["MatchingNumber"]

seq=Table[GraphData[{"Keller",n},"MatchingNumber"],{n,5}]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

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

MaximalCliqueCount

A296101

Entity["Graph",{"Keller",n}]["MaximalCliqueCount"]

seq=Table[GraphData[{"Keller",n},"MaximalCliqueCount"],{n,5}]

MaximalCliquePolynomial

Entity["Graph",{"Keller",n}]["MaximalCliquePolynomial"]

(poly=Factor@Table[GraphData[{"Keller",n},"MaximalCliquePolynomial"],{n,k0=1,kmax=5}]/.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,20}]

FullSimplify[poly2-poly]

MaximalIndependentEdgeSetCount

Entity["Graph",{"Keller",n}]["MaximalIndependentEdgeSetCount"]

seq=Table[GraphData[{"Keller",n},"MaximalIndependentEdgeSetCount"],{n,5}]

MaximalIndependentVertexSetCount

Entity["Graph",{"Keller",n}]["MaximalIndependentVertexSetCount"]

A290607

seq=Table[GraphData[{"Keller",n},"MaximalIndependentVertexSetCount"],{n,5}]

MaximumCliqueCount

A296100

Entity["Graph",{"Keller",n}]["MaximumCliqueCount"]

seq=Table[GraphData[{"Keller",n},"MaximumCliqueCount"],{n,5}]

MaximumIndependentEdgeSetCount

Entity["Graph",{"Keller",n}]["MaximumIndependentEdgeSetCount"]

seq=Table[GraphData[{"Keller",n},"MaximumIndependentEdgeSetCount"],{n,5}]

MaximumIndependentVertexSetCount

Entity["Graph",{"Keller",n}]["MaximumIndependentVertexSetCount"]

Probably 4^n

seq=Table[GraphData[{"Keller",n},"MaximumIndependentVertexSetCount"],{n,5}]

Table[Piecewise[{{1,n==1}},4^n],{n,5}]

MaximumLeafNumber

Entity["Graph",{"Keller",n}]["MaximumLeafNumber"]

seq=Table[GraphData[{"Keller",n},"MaximumLeafNumber"],{n,5}]

MaximumVertexDegree

Entity["Graph",{"Keller",n}]["MaximumVertexDegree"]

seq=Table[GraphData[{"Keller",n},"MaximumVertexDegree"],{n,5}]

Table[4^n-3^n-n,{n,5}]

MeanDistance

Entity["Graph",{"Keller",n}]["MeanDistance"]

seq=Table[GraphData[{"Keller",n},"MeanDistance"],{n,5}]

seq=Table[4^n GraphData[{"Keller",n},"MeanDistance"],{n,5}]

FindSequenceFunction[Rest@seq,n-1]

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

Table[Mean[Flatten[Normal[GraphDistanceMatrix[KellerGraph[n]]]]],{n,5}]//Timing

{∞,(25/16),(23/16),(339/256),(635/512)}Table[(4^n),{n,5}]

d=Table[Sort[Tally[Flatten[GraphDistanceMatrix[KellerGraph[n]]]]],{n,2,6}];//Timing

d//ColumnForm

A000012, A284850, A301571

Transpose[d[[All,All,2]]/Table[4^n,{n,2,6}]]

sums=Dot@@@(Transpose/@d)

sums/16^Range[2,Length[d]+1]

distance-0:

FindSequenceFunction[{16,64,256,1024,4096},n-1]

==VertexCount

distance-1:

FindSequenceFunction[{80,2176,43776,794624,13766656},n-1]

Table[2^(2 n-1) (4^n-3^n-n),{n,6}]

==2*EdgeCount

distance-2:

d[[All,3,2]]

FindSequenceFunction[d[[All,3,2]]/Table[4^n,{n,2,6}],n-1]

FindGeneratingFunction[{10,29,84,247,734},x]

Table[3^n+n-1,{n,10}]

Table[(2^(-1+2 n)) (-(3^n)+(4^n)-n)+2(-1+(3^n)+n),{n,2,5}]

Table[2 2^(2 n-1) (4^n-3^n-n),{n,2,6}]

Table[(4^n)(-1+(3^n)+n),{n,2,6}]

FullSimplify[2 2^(2 n-1) (4^n-3^n-n)+2 4^n(-1+(3^n)+n)]

Table[(4^n) (-2+(3^n)+(4^n)+n),{n,10}]

Table[(((4^n) (-2+(3^n)+(4^n)+n))/(16^n)),{n,2,6}]

FullSimplify[(((4^n) (-2+(3^n)+(4^n)+n))/(16^n))]

Table[((3^n)+n-2)/4^n+1,{n,6}]

Table[Mean[Flatten[Normal[GraphDistanceMatrix[KellerGraph[n]]]]],{n,5}]//Timing

MinimalDominatingSetCount

Entity["Graph",{"Keller",n}]["MinimalDominatingSetCount"]

seq=Table[GraphData[{"Keller",n},"MinimalDominatingSetCount"],{n,5}]

MinimalDominatingSetSignature

Entity["Graph",{"Keller",n}]["MinimalDominatingSetSignature"]

(seq=Table[GraphData[{"Keller",n},"MinimalDominatingSetSignature"],{n,5}])//Column

MinimumCliqueCoveringCount

Entity["Graph",{"Keller",n}]["MinimumCliqueCoveringCount"]

seq=Table[GraphData[{"Keller",n},"MinimumCliqueCoveringCount"],{n,5}]

MinimumConnectedDominatingSetCount

Entity["Graph",{"Keller",n}]["MinimumConnectedDominatingSetCount"]

seq=Table[GraphData[{"Keller",n},"MinimumConnectedDominatingSetCount"],{n,5}]

FindSequenceFunction[DeleteCases[seq,_Missing],n]

MinimumDistinguishingLabelingCount

Entity["Graph",{"Keller",n}]["MinimumDistinguishingLabelingCount"]

seq=Table[GraphData[{"Keller",n},"MinimumDistinguishingLabelingCount"],{n,5}]

MinimumDominatingSetCount

Entity["Graph",{"Keller",n}]["MinimumDominatingSetCount"]

seq=Table[GraphData[{"Keller",n},"MinimumDominatingSetCount"],{n,5}]

MinimumEdgeCoverCount

Entity["Graph",{"Keller",n}]["MinimumEdgeCoverCount"]

seq=Table[GraphData[{"Keller",n},"MinimumEdgeCoverCount"],{n,5}]

MinimumVertexDegree

Entity["Graph",{"Keller",n}]["MinimumVertexDegree"]

seq=Table[GraphData[{"Keller",n},"MinimumVertexDegree"],{n,5}]

Table[4^n-3^n-n,{n,5}]

MolecularTopologicalIndex

Entity["Graph",{"Keller",n}]["MolecularTopologicalIndex"]

seq=Table[GraphData[{"Keller",n},"MolecularTopologicalIndex"],{n,5}]

FindSequenceFunction[Rest@seq,n-1]

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

Nonhamiltonian

Entity["Graph",{"Keller",n}]["Nonhamiltonian"]

Table[GraphData[{"Keller",n},"Nonhamiltonian"],{n,5}]

OddChordlessCycleCount

Entity["Graph",{"Keller",n}]["OddChordlessCycleCount"]

seq=Table[GraphData[{"Keller",n},"OddChordlessCycleCount"],{n,5}]

PathCount

Entity["Graph",{"Keller",n}]["PathCount"]

seq=Table[GraphData[{"Keller",n},"PathCount"],{n,5}]

PathPolynomial

Entity["Graph",{"Keller",n}]["PathPolynomial"]

(poly=Factor[Table[GraphData[{"Keller",n},"PathPolynomial"],{n,k0=1,kmax=5}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

poly2=Table[XXX,{n,20}]

FullSimplify[poly2-poly]

Pell

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

seq=Table[GraphData[{"Keller",n},"Pell"],{n,5}]

PentagonCount

Entity["Graph",{"Keller",n}]["PentagonCount"]

A300848

seq=Table[GraphData[{"Keller",n},"PentagonCount"],{n,5}]

Do[Print[{{"Keller",n},CycleCount[KellerGraph[n],5]//Timing}],{n,6}]

seq=Import["https://oeis.org/A300848/b300848.txt","Table"][[All,2]];

FindGeneratingFunction[seq,x,TimeConstraint->60,ValidationLength->20]//Head//Timing

FindGeneratingFunction[seq,x,TimeConstraint->120,ValidationLength->20]//Head//Timing

coef=FindLinearRecurrence[seq]

Length[coef]

(eqn=Prepend[Table[a[k]==seq[[k]],{k,Length[coef]}],a[n]==Total[Table[a[n-k] coef[[k]],{k,Length[coef]}]]])//TraditionalForm

RSolve[eqn,a[n],n]

FullSimplify[Factor[%]]

Table[(2^(-1+2 n)) (1323 (5 (3^(1+2 n))+5 (3^(1+3 n))-5 (3^(1+n)) (4^n)+5 (3^n) (4^(1+2 n))-5 (6^(1+2 n))-5 (7^n)+5 (2^(1+4 n)) (9^n)+5 (16^n)-5 (21^n)-5 (2^(1+2 n)) (27^n)+5 (28^n)-(61^n)-5 (64^n)+5 (81^n)-5 (192^n)+(256^n))+5 n (-9 (7^n) (225+140 (3^n)+216 n)+49 (4 (27^(1+n))+(3^(3+2 n)) (7-3 (2^(1+2 n))+6 n)+27 (-1-(64^n)-3 n+3 (n^2)+(n^3)+(2^(1+4 n)) (2+n)-(2^(1+2 n)) n (3+n))+(3^n) (67+27 (4^(1+2 n))+129 n+92 (n^2)-27 (2^(1+2 n)) (5+3 n)))))/6615,{n,100}]==seq

Table[(1/6615)(2^(-1+2 n)) (1323 (5 (3^(1+2 n))+5 (3^(1+3 n))-5 (3^(1+n)) (4^n)+5 (3^n) (4^(1+2 n))-5 (6^(1+2 n))-5 (7^n)+5 (2^(1+4 n)) (9^n)+5 (16^n)-5 (21^n)-5 (2^(1+2 n)) (27^n)+5 (28^n)-(61^n)-5 (64^n)+5 (81^n)-5 (192^n)+(256^n))+5 n (-9 (7^n) (225+140 (3^n)+216 n)+49 (4 (27^(1+n))+(3^(3+2 n)) (7-3 (2^(1+2 n))+6 n)+27 (-1-(64^n)-3 n+3 (n^2)+(n^3)+(2^(1+4 n)) (2+n)-(2^(1+2 n)) n (3+n))+(3^n) (67+27 (4^(1+2 n))+129 n+92 (n^2)-27 (2^(1+2 n)) (5+3 n))))),{n,20}]

LinearRecurrence[coef,Take[seq,Length[coef]],100]==seq

Perfect

Entity["Graph",{"Keller",n}]["Perfect"]

Table[GraphData[{"Keller",n},"Perfect"],{n,5}]

ProjectivePlaneCrossingNumber

Entity["Graph",{"Keller",n}]["ProjectivePlaneCrossingNumber"]

seq=Table[GraphData[{"Keller",n},"ProjectivePlaneCrossingNumber"],{n,5}]

Radius

Entity["Graph",{"Keller",n}]["Radius"]

seq=Table[GraphData[{"Keller",n},"Radius"],{n,5}]

RectilinearCrossingNumber

Entity["Graph",{"Keller",n}]["RectilinearCrossingNumber"]

seq=Table[GraphData[{"Keller",n},"RectilinearCrossingNumber"],{n,5}]

SpanningTreeCount

Entity["Graph",{"Keller",n}]["SpanningTreeCount"]

seq=Table[GraphData[{"Keller",n},"SpanningTreeCount"],{n,3}]

Log2/@seq

FactorInteger/@seq

FindSequenceFunction[seq,n]

SpectralRadius

Entity["Graph",{"Keller",n}]["SpectralRadius"]

seq=Table[GraphData[{"Keller",n},"SpectralRadius"],{n,5}]

Table[-(3^n)+(4^n)-n,{n,5}]

Spectrum

Entity["Graph",{"Keller",n}]["Spectrum"]

Table[GraphData[{"Keller",n},"Spectrum"],{n,3}]

SquareCount

Entity["Graph",{"Keller",n}]["SquareCount"]

A300842

seq=Table[GraphData[{"Keller",n},"SquareCount"],{n,5}]

Do[Print[{{"Keller",n},CycleCount[KellerGraph[n],4]//Timing,MaxMemoryUsed[]/10^9. GB}],{n,5}]

seq=Import["https://oeis.org/A300842/b300842.txt","Table"][[All,2]];

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

FindGeneratingFunction[seq,x,ValidationLength->10]//Head//Timing

FindGeneratingFunction[seq,x,ValidationLength->20]//Head//Timing

gf=Factor[x FindGeneratingFunction[seq,x,TimeConstraint->60,ValidationLength->20]]

Rest[CoefficientList[Series[gf,{x,0,100}],x]]==seq

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

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

Table[2^(2n-3) (21(-2^(4n+1)-3^n+4^n+3^n 4^(n+1)-3^n 4^(2 n+1)+6^(2n+1)-2 9^n+21^n-4 27^n+64^n)+(-63-14 3^n+21 4^(n+1)+7 3^(n+2) 4^(n+1)-21 4^(2n+1)+72 7^n-28 9^(n+1)) n+7 (3+9 2^(2n+1)-28 3^n) n^2-84 n^3)/21,{n,20}]

CoefficientList[Series[-((8 x (5+13494 x-7659024 (x^2)+1619423712 (x^3)-136239729408 (x^4)-6166229419008 (x^5)+2751756296921088 (x^6)-323809706190127104 (x^7)+22263967229708795904 (x^8)-1004774859520610533376 (x^9)+30166466190897816010752 (x^10)-557827773596112789700608 (x^11)+3836796315532570883260416 (x^12)+98004685735481941350678528 (x^13)-3439510519824231292370681856 (x^14)+52560369429605424136068268032 (x^15)-472805698361893718926628487168 (x^16)+2514590036498863236378789937152 (x^17)-6697147772007341308881068359680 (x^18)+1543470882179190947000434556928 (x^19)+24429568859168763958151501316096 (x^20)))/((((-1+4 x))^4) (((-1+12 x))^3) (((-1+16 x))^3) (((-1+28 x))^2) (((-1+36 x))^2) (((-1+48 x))^2) (((-1+64 x))^2) (-1+84 x) (-1+108 x) (-1+144 x) (-1+192 x) (-1+256 x))),{x,0,20}],x]

coef=FindLinearRecurrence[seq]

Length[coef]

(eqn=Prepend[Table[a[k]==seq[[k]],{k,Length[coef]}],a[n]==Total[Table[a[n-k] coef[[k]],{k,Length[coef]}]]])//TraditionalForm

RSolve[eqn,a[n],n]

Take[seq,23]

LinearRecurrence[coef,Take[seq,23],100]==seq

StabilityIndex

Entity["Graph",{"Keller",n}]["StabilityIndex"]

seq=Table[GraphData[{"Keller",n},"StabilityIndex"],{n,5}]

FindSequenceFunction[seq,n]

FindGeneratingFunction[seq,x]

TopologicalIndex

Entity["Graph",{"Keller",n}]["TopologicalIndex"]

seq=Table[GraphData[{"Keller",n},"TopologicalIndex"],{n,5}]

FindSequenceFunction[seq,n]

FindGeneratingFunction[seq,x]

ToroidalCrossingNumber

Entity["Graph",{"Keller",n}]["ToroidalCrossingNumber"]

seq=Table[GraphData[{"Keller",n},"ToroidalCrossingNumber"],{n,5}]

Traceable

Entity["Graph",{"Keller",n}]["Traceable"]

Table[GraphData[{"Keller",n},"Traceable"],{n,5}]

TriangleCount

Entity["Graph",{"Keller",n}]["TriangleCount"]

A300818

seq=Table[GraphData[{"Keller",n},"TriangleCount"],{n,5}]

(seq=Table[CycleCount[KellerGraph[n],3],{n,4}])//Timing

Do[Print[{{"Keller",n},CycleCount[KellerGraph[n],3]//Timing,MaxMemoryUsed[]/10^9. GB}],{n,5}]

seq=Import["https://oeis.org/A300818/b300818.txt","Table"][[All,2]];

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

FindGeneratingFunction[seq,x,TimeConstraint->30]//Head//Timing

gf=Factor[x FindGeneratingFunction[seq,x,ValidationLength->20]]

CoefficientList[Series[-((64 (x^2) (87-8648 x+319808 (x^2)-4495488 (x^3)-11911936 (x^4)+941117440 (x^5)-8636104704 (x^6)+25777668096 (x^7)))/((((-1+4 x))^3) (((-1+12 x))^2) (((-1+16 x))^2) (-1+28 x) (-1+36 x) (-1+48 x) (-1+64 x))),{x,0,20}],x]

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

FullSimplify[%,n>0]

Expand[%]//FullSimplify

Table[(2^(2 n-1)) ((3^(2 n+1))-(3^(n+1)) (4^n)-(7^n)+(16^n)+ (4 (3^n)-3 (4^n)+3 n-3)n)/3,{n,100}]==seq

Table[(2^(2 n-1)) ((3^(2 n+1))-(3^(n+1)) (4^n)-(7^n)+(16^n)+ (4 (3^n)-3 (4^n)+3 n-3)n)/3,{n,20}]

coef=FindLinearRecurrence[seq]

Length[coef]

(eqn=Prepend[Table[a[k]==seq[[k]],{k,Length[coef]}],a[n]==Total[Table[a[n-k] coef[[k]],{k,Length[coef]}]]])//TraditionalForm

RSolve[eqn,a[n],n]

Take[seq,11]

LinearRecurrence[coef,Take[seq,11],100]==seq

LinearRecurrence[{244,-25104,1434176,-50448128,1145183232,-17085018112,167078576128,-1046232170496,3986606260224,-8340356726784,7305739370496},{0,0,5568,805120,77137920,6307700736,475634876416,34155559059456,2374835654098944,161464681410592768,10803965184146669568},20]

Untraceable

Entity["Graph",{"Keller",n}]["Untraceable"]

Table[GraphData[{"Keller",n},"Untraceable"],{n,5}]

VertexConnectivity

Entity["Graph",{"Keller",n}]["VertexConnectivity"]

A284850

Same as EdgeChromaticNumber

Table[GraphData[{"Keller",n},"VertexConnectivity"],{n,5}]

Table[((4^n)-(3^n)-n),{n,5}]

VertexCount

Entity["Graph",{"Keller",n}]["VertexCount"]

seq=Table[GraphData[{"Keller",n},"VertexCount"],{n,5}]

FindSequenceFunction[seq,n]

VertexCoverNumber

Entity["Graph",{"Keller",n}]["VertexCoverNumber"]

Related to but not the same as IndependenceNumber

seq=Table[GraphData[{"Keller",n},"VertexCoverNumber"],{n,5}]

Table[4^n-Piecewise[{{4,n==1},{5,n==2}},2^n],{n,5}]

Table[Piecewise[{{0,n==1},{11,n==2}},4^n-2^n],{n,5}]

VertexDegrees

Entity["Graph",{"Keller",n}]["VertexDegrees"]

seq=Table[GraphData[{"Keller",n},"VertexDegrees"],{n,5}]

seq[[All,1]]

Table[4^n-3^n-n,{n,5}]

WienerIndex

Entity["Graph",{"Keller",n}]["WienerIndex"]

A292056

seq=Table[GraphData[{"Keller",n},"WienerIndex"],{n,5}]

Do[Print[{{"Keller",n},WienerIndex[KellerGraph[n]]//Timing}],{n,10}]

seq={∞,200,2944,43392,650240,9889792,152174592};

seq=Table[(4^n)((4^n)+(3^n)+n-2)/2,{n,20}]

coef=FindLinearRecurrence[Rest@seq]

LinearRecurrence[{36,-432,1984,-3072},{12,200,2944,43392},20]

gf=x FindGeneratingFunction[Rest@seq,x]

CoefficientList[Series[-((8 (-25+532 x-2976 (x^2)+4608 (x^3)))/((((-1+4 x))^2) (1-28 x+192 (x^2)))),{x,0,20}],x]

WienerSumIndex

Entity["Graph",{"Keller",n}]["WienerSumIndex"]

seq=Table[GraphData[{"Keller",n},"WienerSumIndex"],{n,5}]

FindSequenceFunction[seq,n]

GraphData

GraphDataString

file=OpenWrite["~/keller.m"]

Do[WriteString[file,GraphDataString[KellerGraph[n],
"StandardName"->{"Keller",n},
"Name"->ToString[n]<>"\[Hyphen]Keller graph",
"NotationRules"->{"Keller"->n},
TimeConstraint->10,
Debug->True
]],{n,3,6}]

Close[file]