Euclidean Distance Graph

Author

Eric W. Weisstein
July 18, 2025

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

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

©2025 Wolfram Research, Inc. except for portions noted otherwise

Examples

"Antelope"

Table[
IsomorphicGraphQ@@{AntelopeGraph[m,n],EuclideanDistanceGraph[GridGraph1[{m,n}],5]},
{m,15},{n,m}]//Column

"Camel"

Table[
IsomorphicGraphQ@@{CamelGraph[m,n],EuclideanDistanceGraph[GridGraph1[{m,n}],(Sqrt[10])]},
{m,15},{n,m}]//Column

"Fiveleaper"

Table[
IsomorphicGraphQ@@{FiveleaperGraph[m,n],EuclideanDistanceGraph[GridGraph1[{m,n}],5]},
{m,15},{n,m}]//Column

"Giraffe"

Table[
IsomorphicGraphQ@@{GiraffeGraph[m,n],EuclideanDistanceGraph[GridGraph1[{m,n}],(Sqrt[17])]},
{m,15},{n,m}]//Column

"GrabarchukGraph"

EuclideanDistanceGraph[GridGraph1[{4,4,4}],3]//RecognizeGraph

"Knight"

Table[
IsomorphicGraphQ@@{KnightGraph[m,n],EuclideanDistanceGraph[GridGraph1[{m,n}],(Sqrt[5])]},
{m,15},{n,m}]//Column

"Zebra"

Table[
IsomorphicGraphQ@@{ZebraGraph[m,n],EuclideanDistanceGraph[GridGraph1[{m,n}],(Sqrt[13])]},
{m,15},{n,m}]//Column

Grids

n×n

Grid[
DeleteCases[
Flatten[Table[{k,n,RecognizeGraph@EuclideanDistanceGraph[GridGraph1[{n,n}],k]},{n,20},{k,2,10}],1],
{_,_,{}|{"Empty",_}|"SingletonGraph"}],
Dividers->All,Alignment->{{{Decimal},Left}}]

n×n×n

Grid[
DeleteCases[
Flatten[Table[{k,n,RecognizeGraph@EuclideanDistanceGraph[GridGraph1[{n,n,n}],k]},{n,10},{k,2,10}],1],
{_,_,{}|{"Empty",_}|"SingletonGraph"}],
Dividers->All,Alignment->{{{Decimal},Left}}]