The graph Cartesian product 
of a complete graph 
and a cycle
graph 
has been termed a "KC graph" by Knuth (2024, p. 22), who restricts
their parameters to 
and 
. The KC graphs are regular
with degree 
and have vertex count and edge
count
 |  |  |
(1)
|
 |  |  |
(2)
|
Many KC graphs are circulant graphs. In particular, for any 
(i.e., relatively
prime so as not to contain a common divisor), 
is isomorphic to the circulant
graph 
,
where the indices are a subset of integer multiples of 
and 
less than or equal to 
.
Other special cases are summarized in the table below.
| graph | special case |
 | cycle graph  |
 | prism
graph  |
 | rook graph  |
 | generalized
quadrangle  |
for  ,  |  torus grid graph |
is ungraceful
for 
odd (Knuth 2024, p. 22).
The scramble number of the KC graph 
with 
and 
is 
(Echavarria et al. 2021).
