The graph Cartesian product 
of a complete graph 
and a path
graph 
has been termed a "KP graph" by Knuth (Knuth 2020; Knuth 2024, pp. 20-21),
who restricts their parameters to 
and 
. The KP graphs have vertex
count, edge count, and (except for 
and 
)
graph automorphism count
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
KP graphs are regular only for 
(corresponding to complete
graphs) and 
(corresponding to 
rook graphs).
Special cases are summarized in the table below.
| graph | special case |
 | rook graph  |
 | path
graph  |
 | ladder graph  |
 | stacked
prism graph  |
 | complete graph  |
A KP graph 
is graceful iff there exists
an 
matrix of distinct integers 
in the range 
for which the differences
 |
(4)
|
are themselves distinct (Knuth 2020).
(i.e., the 
rook graphs) is ungraceful
for all 
(Knuth 2024, p. 22).
