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).