Abstract
We describe a new class of maximal cliques, with a vector space structure, of Cayley graphs defined on the additive group of a field. In particular, we show that in the cubic Paley graph with order \(q^3\), the subfield with q elements forms a maximal clique. Similar statements also hold for quadruple Paley graphs and Peisert graphs with quartic order.
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Acknowledgements
The author would like to thank Joshua Zahl for his valuable suggestions, and Greg Martin, József Solymosi, and Ethan White for helpful discussions. The author also would like to thank the anonymous referees for a careful reading of the draft.
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Appendix: Sage code
Appendix: Sage code
Sage code for finding the clique number of a Peisert graph (for example, of order 81):
Sage code for checking whether \(\mathbb {F}_q\) is a maximal clique in the Peisert graph with order \(q^4\), where q is a power of a prime \(p \equiv 3 \pmod 4\) (for example, \(q=23\)):
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Yip, C.H. On maximal cliques of Cayley graphs over fields. J Algebr Comb 56, 323–333 (2022). https://doi.org/10.1007/s10801-021-01113-y
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DOI: https://doi.org/10.1007/s10801-021-01113-y