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Cyclotomic Graph

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CyclotomicGraphs

The cyclotomic graph of order q, for a prime power q satisfying q=1 (mod 3), is the simple graph on the elements of the finite field GF(q) in which two distinct vertices are adjacent iff their difference is a nonzero cube in the finite field. These undirected graphs have vertex degree (q-1)/3. Their orders begin 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 number, cyclotomic graphs are also circulant graphs Ci_q(l_1,...) with parameters l_i given by the nonzero cubes modulo q, using one representative from each pair {c,-c}.

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

qgraph
42-ladder rung graph
7cycle graph C_7
16Clebsch graph
25rook graph L_(5,5)

See also

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

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References

House of Graphs. Cyclotomic Graphs. Clebsch Graph, C_7, Circulant C_13 (1,5), Circulant C_19 (1,7,8), Circulant C_31 (1,2,4,8,15), and others.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." PhD dissertation. Tilburg, Netherlands: Tilburg University, pp. 51-52, 1996. https://research.tilburguniversity.edu/en/publications/graphs-with-few-eigenvalues-an-interplay-between-combinatorics-an/.

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Cyclotomic Graph

Cite this as:

Weisstein, Eric W. "Cyclotomic Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CyclotomicGraph.html

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