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# Cyclotomic Graph

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.

 graph 4 2-ladder rung graph 7 cycle graph 16 Clebsch graph 25 rook graph

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

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## References

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.

Cyclotomic Graph

## Cite this as:

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