The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes
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| Format: | Preprint |
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2021
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| _version_ | 1866913468203925504 |
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| author | Asgarli, Shamil Yip, Chi Hoi |
| author_facet | Asgarli, Shamil Yip, Chi Hoi |
| contents | Blokhuis showed that all maximum cliques in Paley graphs of square order have a subfield structure. Recently, it has been shown that in Peisert-type graphs, all maximum cliques are affine subspaces, and yet some maximum cliques do not arise from a subfield. In this paper, we investigate the existence of a clique of size $\sqrt{q}$ with a subspace structure in pseudo-Paley graphs of order $q$ from unions of semi-primitive cyclotomic classes. We show that such a clique must have an equal contribution from each cyclotomic class and that most such pseudo-Paley graphs do not admit such cliques, suggesting that the Delsarte bound $\sqrt{q}$ on the clique number can be improved in general. We also prove that generalized Peisert graphs are not isomorphic to Paley graphs or Peisert graphs, confirming a conjecture of Mullin. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_07176 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes Asgarli, Shamil Yip, Chi Hoi Combinatorics Number Theory Primary 05C25, 11T22, Secondary 11T24, 11B30, 05E30, 05C60 Blokhuis showed that all maximum cliques in Paley graphs of square order have a subfield structure. Recently, it has been shown that in Peisert-type graphs, all maximum cliques are affine subspaces, and yet some maximum cliques do not arise from a subfield. In this paper, we investigate the existence of a clique of size $\sqrt{q}$ with a subspace structure in pseudo-Paley graphs of order $q$ from unions of semi-primitive cyclotomic classes. We show that such a clique must have an equal contribution from each cyclotomic class and that most such pseudo-Paley graphs do not admit such cliques, suggesting that the Delsarte bound $\sqrt{q}$ on the clique number can be improved in general. We also prove that generalized Peisert graphs are not isomorphic to Paley graphs or Peisert graphs, confirming a conjecture of Mullin. |
| title | The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes |
| topic | Combinatorics Number Theory Primary 05C25, 11T22, Secondary 11T24, 11B30, 05E30, 05C60 |
| url | https://arxiv.org/abs/2110.07176 |