A circulant graph is a graph of 
graph vertices in which the
th graph vertex
is adjacent to the 
th
and 
th
graph vertices for each 
in a list 
. The circulant graph 
gives the complete
graph 
and the graph 
gives the cyclic graph 
.
The circulant graph on 
vertices on an offset list 
is implemented in the Wolfram
Language as CirculantGraph[n,
l]. Precomputed properties are available using GraphData[
"Circulant", 
n, l
].
With the exception of the degenerate case of the path graph 
, connected circulant graphs are biconnected, bridgeless,
cyclic, Hamiltonian,
LCF, regular, traceable, and vertex-transitive.
A graph 
is a circulant iff the automorphism
group of 
contains at least one permutation consisting of a
minimal cycle of length 
.
The numbers of circulant graphs on 
, 2, ... nodes (counting empty
graphs as circulant graphs) are 1, 2, 2, 4, 3, 8, 4, 12, ... (OEIS A049287),
the first few of which are illustrated above. Note that these numbers cannot be counted
simply by enumerating the number of nonempty subsets of 
since, for example, 
. There is an easy formula for prime orders,
and formulas are known for squarefree and prime-squared orders.
The numbers of connected circulant graphs on 
, 2, ... nodes are 0, 1, 1, 2, 2, 5, 3, 8, ..., illustrated
above.
Classes of graphs that are circulant graphs include the Andrásfai graphs, antiprism graphs, cocktail
party graphs 
,
complete graphs, complete
bipartite graphs 
,
crown graphs 
, empty graphs, KC
graphs 
for 
(i.e., 
and 
relatively prime) and where 
denotes a Cartesian
product, rook graphs 
for 
, Möbius ladders,
Paley graphs of prime order, prism
graphs 
,
and torus grid graphs 
with 
corresponding to 
). Special cases are summarized in the table below.
Families of unit-distance connected circulant graphs include:
1. cycle graphs 
,
2. Cartesian products of prism graphs 
and 
, yielding torus grid graphs 
.
and Degree 4 or 5" [Chinese]. Dianzi Keji Daxue Xuebao 25,
272-276, 1996.