Let 
be a set of positive numbers, then the 
-Euclidean distance graph (or Euclidean distance-
graph) 
on a nonempty subset 
of Euclidean space is the graph
with vertex set 
and edge set 
, where 
is the Euclidean distance between vertices 
and 
.
Maehara (1992) additionally require the set 
to contain the distance 1.
Examples include the Grabarchuk graph, which is the Euclidean distance-3 graph on the vertex set
of the 
grid graph. A similar example is the sextic
Euclidean distance-
graph on 32 vertices obtained from the vertices of the
truncated octahedron with unit edge lengths
after adding the centers of the hexagonal faces (E. Pegg, Jr., pers. comm.,
Aug. 12, 2025; Pegg 2025).
The 
fiveleaper graph is the Euclidean distance-5 graph
on the vertex set of the 
grid graph.
Other examples of Euclidean distance graphs include 
-leaper graph when 
is not an integer, as summarized in the following table. Note that because the antelope graph is a 
-leaper graph with 
,
it is not a Euclidean distance graph.
 |  | Euclidean distance leaper graph |
 |  | fers graph |
 |  | knight graph |
 |  | camel graph |
 |  | giraffe graph |
 |  | flamingo graph |
 |  | alfil graph |
 |  | zebra graph |
 |  | lancer graph |
 |  | tripper graph |
 |  | commuter graph |
Euclidean distance graphs differ from graph distance graphs in that former are constructed based on Euclidean distance between vertices, while the latter are constructed based on graph distance.
