The Brinkmann graph (misspelled by Cancela et al. (2004) as "Brinkman") is a weakly regular quartic graph on 21 vertices and 42 edges. It was first mentioned in Brinkmann (1992) and first published as a note in Brinkmann and Meringer (1997). It is implemented in the Wolfram Language as GraphData["BrinkmannGraph"].
Grünbaum conjectured that for every integer 
, 
, there exists an 
-regular, 
-chromatic graph of girth at least
. This result is trivial for 
and 
, but only a small number of other such graphs are known,
including the Brinkmann graph, illustrated above, Chvátal
graph, and 25-Grünbaum graph.
The plots above show the adjacency, incidence, and distance matrices of the graph.
