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].

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.

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