A circulant graph 
is a simple graph with vertices
labeled 0, 1, ..., 
,
where 
is a positive integer and 
is a list of distinct integer
offsets satisfying 
.
Each graph vertex 
is adjacent to 
and 
for every 
, with indices interpreted modulo 
by modular arithmetic.
The circulant graph 
is the complete graph 
, and for 
the circulant graph 
is the cycle graph 
. More generally, for a positive
integer 
and 
,
the cycle power graph 
is the circulant graph 
, where 
is the floor function.
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
].
All connected circulant graphs on at least three vertices 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
permutation cycle of length 
.
In other words, the cyclic symmetry of a circulant graph acts on all vertices in a single group orbit, while a bicirculant graph has a cyclic symmetry with exactly two equal group orbits.
The numbers of nonisomorphic 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. Simply counting all offset subsets
of 
,
including the empty subset, overcounts because distinct
offset lists may give isomorphic
graphs. For example, 
. There is an easy formula for prime graph orders, and formulas are known for squarefree graph orders and graph orders
that are squares of primes.
The numbers of connected circulant graphs on 
, 2, ... nodes
are 1, 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 
for odd 
, where 
is a perfect matching,
empty graphs, KC graphs 
for 
(i.e., 
and 
relatively prime) and
where 
denotes a graph Cartesian product, rook
graphs 
for 
,
Möbius ladders, Paley
graphs of prime graph
order, prism graphs 
for odd 
, and torus grid graphs 
for 
with 
, corresponding to 
. Special cases are summarized in the table below.
Families of unit-distance connected circulant graphs include:
1. cycle graphs 
for 
,
2. graph Cartesian products of prism graphs 
and 
,
yielding torus grid graphs 
for odd
.
Conder and Grande (2015, pp. 2 and 10) classified the connected circulant graphs of graph genus 1 and 2. Their proof
includes 
,
which has graph genus 2 but is omitted from the enumeration
in their Theorem 3. Further exact graph genera have
been computed by Metzger and Ulrigg (2026).
Exact graph crossing numbers for several infinite families of quartic circulant graphs are
summarized below (Gauci and Zerafa Xuereb 2024). All parameters are positive
integers, 
denotes the floor function, and 
denotes the remainder on
division of 
by 
.
| graph | crossing number | parameter range |
 |  |  |
 |  |  |
 |  |  |
 |  |  |
 |  |  |
 |  |  |
 |  |  |
 |  |  , 2, 3, 4;  ,  |
The excluded case 
is planar for every positive
integer 
(Gauci and Zerafa Xuereb 2024).
Ho (2012) proved that the projective plane crossing number of 
is 1 for 
and 
for 
. Zerafa Xuereb (2024) proved that the projective
plane crossing number of 
is 1 for 
and 
for 
, 
, and 
.
."
Discuss. Math. Graph Th. 32, 91-108, 2012. 
and Degree 4 or 5" [Chinese]. Dianzi Keji Daxue Xuebao 25,
272-276, 1996.