Cyclotomic Graph


The cyclotomic graph of order q with q a prime power is a graph on q nodes with two nodes adjacent if their difference is a cube in the finite field GF(q). This graph is undirected when q=1 (mod 3). 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 q a prime, cyclotomic graphs are also circulant graphs Ci_q(l_1,...) with parameters l_i given by the cubes (mod q).

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

See also

Brouwer-Haemers Graph, Circulant Graph, Finite Field, Paley Graph

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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.

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Cyclotomic Graph

Cite this as:

Weisstein, Eric W. "Cyclotomic Graph." From MathWorld--A Wolfram Web Resource.

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