Complete Tree

Author

Eric W. Weisstein
July 28, 2021

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

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

©2021 Wolfram Research, Inc. except for portions noted otherwise

Graphics

GraphData

GraphData[{"CompleteTree",{3,3}},"Graph","All"]//StyleGraphs

Built-in

CompleteKaryTree[3,3]//StyleGraphs

Combinatorica

ShowGraph[CompleteNaryTree[2^2-1,2]]

With[{k=2,n=3},ShowGraph[CompleteNaryTree[((-1+(k^n))/(-1+k)),k]]]

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

Embeddings

DeleteCases[GraphData["CompleteTree"],{"Star",_}]

Select[GraphData["CompleteTree"],FreeQ[GraphData[#,"EmbeddingClasses"],"UnitDistance"]&]

2, 3

GraphData[{"CompleteTree",{2,3}},"Graph","All"]//StyleGraphs

g=CompleteNaryTree[2,3,GraphLayout->"UnitDistanceEmbedding","Angle"->{180}Degree]

g=CompleteNaryTree[2,3,GraphLayout->"UnitDistanceEmbedding","Angle"->{90}Degree]

RecognizeGraph[g]

GraphEmbedding[g]//FullSimplify

UnitDistanceGraphEmbeddingQ[g]

2, 4

GraphData[{"CompleteTree",{2,4}},"Graph","All"]//StyleGraphs

g=CompleteNaryTree[2,4,GraphLayout->"UnitDistanceEmbedding","Angle"->180Degree]

RecognizeGraph[g]

GraphEmbedding[g]//FullSimplify

UnitDistanceGraphEmbeddingQ[g]

2, 5

GraphData[{"CompleteTree",{2,5}},"Graph","All"]//StyleGraphs

CompleteKaryTree[5,2]

<<MathWorld`Graphs`

CompleteNaryTree[2,5]

g=CompleteNaryTree[2,5,GraphLayout->"UnitDistanceEmbedding","Angle"->{180,90,53.}Degree]

RecognizeGraph[g]

UnitDistanceGraphEmbeddingQ[g]

3, 3

GraphData[{"CompleteTree",{3,3}},"Graph","All"]//StyleGraphs

g=CompleteNaryTree[3,3,GraphLayout->"UnitDistanceEmbedding","Angle"->60Degree]

RecognizeGraph[g]

GraphEmbedding[g]//FullSimplify

UnitDistanceGraphEmbeddingQ[g]

3, 4

GraphData[{"CompleteTree",{3,4}},"Graph","All"]//StyleGraphs

g=CompleteNaryTree[3,4,GraphLayout->"UnitDistanceEmbedding","Angle"->{80.,30}Degree]

RecognizeGraph[g]

GraphEmbedding[g]//FullSimplify

UnitDistanceGraphEmbeddingQ[g]

3, 5

GraphData[{"CompleteTree",{3,5}},"Graph","All"]//StyleGraphs

g=CompleteNaryTree[3,5,GraphLayout->"UnitDistanceEmbedding","Angle"->{80,30,15}Degree]

RecognizeGraph[g]

GraphEmbedding[g]//FullSimplify

UnitDistanceGraphEmbeddingQ[g]

4, 3

GraphData[{"CompleteTree",{4,3}},"Graph","All"]//StyleGraphs

g=CompleteNaryTree[4,3,GraphLayout->"UnitDistanceEmbedding","Angle"->45Degree]

RecognizeGraph[g]

GraphEmbedding[g]//FullSimplify

UnitDistanceGraphEmbeddingQ[g]

4, 4

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

g=CompleteNaryTree[4,4,GraphLayout->"UnitDistanceEmbedding","Angle"->{45,15}Degree]

RecognizeGraph[g]

UnitDistanceGraphEmbeddingQ[g]