The Hamming graph 
, sometimes also denoted 
, is the graph Cartesian product of 
copies of the complete graph 
. 
therefore has 
vertices.
has chromatic number 
(S. Wagon, pers. comm., Feb. 16, 2013) and graph diameter 
. Hammin graphs are distance-regular and geomtric (Koolen et al. 2023).
Special cases are summarized in the following table.
The Doob graph 
is the graph given by the graph Cartesian product of 
copies of the Shrikhande graph with a Hamming graph 
. 
is cospectral with the Doob graph 
and shares the same regularity parameters.
Some order-3 LCF notations of 
are illustrated above.
Because the graph Cartesian product of unit-distance graphs are themselves unit-distance, the Hamming graphs 
and 
are unit-distance. A (degenerate) unit-distance embedding of 
is shown above (E. Gerbracht, pers. comm., May 2008).
." 
." 15 Nov 2023.