The Doob graph 
,
also called the Egawa graph, is the graph given by the graph
Cartesian product of 
copies of the Shrikhande graph with a Hamming
graph 
(i.e., with the graph Cartesian product
of 
copies of the tetrahedral
graph 
).
therefore has 
nodes.
Doob graphs are distance-regular and integral with the same parameters as 
(Brouwer et al. 1989, p. 262). However,
they are not distance-transitive.
Doob graphs for small 
are implemented in the Wolfram Language
as GraphData[
"Doob", 
m, n
].
Special cases are summarized in the following table, where 
denotes the Shrikhande graph.
-
by the Parameters." J. Combin. Th., Ser. A 31,
108-125, 1981.Godsil, C. D. "Eigenpolytopes of Distance Regular
Graphs." Canad. J. Math. 59, 739-755, 1998.