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, and the 
fiveleaper graph,
which is the Euclidean distance-5 graph on the vertex set
of the 
grid graph. Other examples 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.
