Triangular Honeycomb King Graph

Author

Eric W. Weisstein
December 10, 2024

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

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

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

Figure

3

g=TriangularHoneycombKingGraph[3]

RecognizeGraph@g

EdgeLengths[g]//Tally

v=GraphEmbedding[g];

edges=List@@@EdgeList[g];

g3=Graphics[{
FaceForm[LightGray],EdgeForm[Black],Polygon[CirclePoints[#,{1/Sqrt[3],Pi/2},6]]&/@v,
Thick,Line[v,edges],
FaceForm[White],Disk[v,.1]
},
PlotLabel->"3\[Hyphen]triangular honeycomb king graph",
LabelStyle->Directive[Black,FontFamily->"Times",14,Italic]]

4

g=TriangularHoneycombKingGraph[4]

RecognizeGraph@g

EdgeLengths[g]//Tally

v=GraphEmbedding[g];

edges=List@@@EdgeList[g];

g4=Graphics[{
FaceForm[LightGray],EdgeForm[Black],Polygon[CirclePoints[#,{1/Sqrt[3],Pi/2},6]]&/@v,
Thick,Line[v,edges],
FaceForm[White],Disk[v,.1]
},
PlotLabel->"4\[Hyphen]triangular honeycomb king graph",
LabelStyle->Directive[Black,FontFamily->"Times",14,Italic]]

Combined

GraphicsRow[{g3,g4},ImageSize->600,Alignment->Top]

Classes

GraphClassesThatAre["TriangularHoneycombKing"]

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

StringReplace[StringRiffle[GraphData[#,"Description"]&/@%,", "]," graph"->""]

Properties

ChromaticNumber

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

Wagon (2014)

CliqueCoveringNumber

A301752

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

seq==Table[(Sqrt[3] (16+3 n (3+n))-9 Cos[n Pi/6]+2 Sqrt[3] Cos[2 n Pi/3]+9 Cos[5 n Pi/6]+9 Sin[n Pi/6]-9 Sin[5 n Pi/6])/(18 Sqrt[3]),{n,0,19}]

Wagon (2014)

DominationNumber

A251419

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

seq=BFileSequence[251419]

gf=FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[(x^22-x^21-x^19+2*x^18-x^17-x^14+2*x^13-2*x^11+2*x^10-2*x^9+x^8+x^7-2*x^6+x^5-x^3+x^2-x)/(x^9-2*x^8+x^7-x^2+2*x-1),{x,0,n}],n>13&&n∈Integers]

Table[Floor[(((n^2)+7n-23)/14)],{n,14,38}]

Wagon (2014)

EdgeChromaticNumber

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

Wagon (2014)

IndependenceNumber

A239438

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

seq==Table[Piecewise[{{3,n==3},{6,n==5}},Ceiling[((n(n+1))/6)]],{n,20}]

Wagon (2014)

6. A. V. Geramita, D. Gregory, L. Roberts, Monomial ideals and points in projective space, J. Pure Appl. Algebra 40 (1986) 33–62. http://dx.doi.org/10.1016/0022-4049(86)90029-0.

GraphData

GraphDataClassString

GraphDataClassString["TriangularHoneycombKing"]