For a connected graph 
of graph diameter 
, the graph distance-
graph 
for 
, ..., 
is a graph with the same vertex
set and having edge set consisting of the pairs of
vertices that lie a graph distance 
apart. It is therefore the case that 
(Brouwer et al. 1989, p. 437).
Graph distance graphs differ from Euclidean distance graphs in that the former are constructed based on graph distance between vertices, while the latter are constructed based on Euclidean distance.
Special classes of graphs isomorphic to their own graph 
-distance graph are summarized in the table below.
| condition | distance- graph is isomorphic to itself |
 |  -antiprism graph |
 odd |  -caveman graph |
 odd | prism
graph  |
 odd |  -sunlet graph |
 | torus
grid graph  |
 odd | hypercube
graph  |
Notable graphs that are isomorphic to their own distance-k graph are summarized in the following table.
 | graph and distance- graphs are isomorphic |
| 2 | generalized quadrangle  , icosahedral
graph, Paley graphs, Peisert
graphs |
| 3 | Franklin graph, Hoffman graph, Pasch graph, rhombic dodecahedral graph, Wells graph |
| 4 | dodecahedral graph, Greenfield graph |
| 5 | rhombic triacontahedral graph, Schläfli double six graph, subdivided cubical graph, truncated octahedral graph |
| 7 | Foster
graph  |
| 9 | Foster
graphs  ,
 ,
 ,
 ,
subdivided dodecahedral
graph |
| 11 | Foster
graphs  ,
 ,
 ,
 ,
 ,
 ,
 ,
 |
| 13 | Foster
graphs  ,
 ,
 ,
 |
| 17 | Foster
graphs  |
