The cyclotomic graph of order with a prime power is a graph on
nodes with two nodes adjacent if their
difference is a cube in the finite field GF(). This graph is undirected when . Simple cyclotomic graphs
therefore exist for orders 4, 7, 13, 16, 19, 25, 31, 37, 43, 49, 61, 64, 67, 73,
79, 97, ... (OEIS A137827).

The cyclotomic graphs are cubic analogs of the Paley graphs.

For
a prime, cyclotomic graphs are also circulant graphs with parameters given by the cubes (mod ).

Special case of cyclotomic graphs are summarized in the table below.

Sloane, N. J. A. Sequence A137827 in "The On-Line Encyclopedia of Integer Sequences."van Dam,
E. R. "Graphs with Few Eigenvalues: An Interplay Between Combinatorics
and Algebra." Ph.D. dissertation. Tilburg, Netherlands: Tilburg University,
pp. 51-52, October 4, 1996.