Extremal Peisert-type graphs without the strict-EKR property
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914840471142400 |
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| author | Goryainov, Sergey Yip, Chi Hoi |
| author_facet | Goryainov, Sergey Yip, Chi Hoi |
| contents | It is known that Paley graphs of square order have the strict-EKR property, that is, all maximum cliques are canonical cliques. Peisert-type graphs are natural generalizations of Paley graphs and some of them also have the strict-EKR property. Given a prime power $q \geq 3$, we study Peisert-type graphs of order $q^2$ without the strict-EKR property and with the minimum number of edges and we call such graphs extremal. We determine number of edges in extremal graphs for each value of $q$. If $q$ is a a square or a cube, we show the uniqueness of the extremal graph and classify all maximum cliques explicitly. Moreover, when $q$ is a square, we prove that there is no Hilton-Milner type result for the extremal graph, and show the tightness of the weight-distribution bound for both non-principal eigenvalues of this graph. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_00391 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Extremal Peisert-type graphs without the strict-EKR property Goryainov, Sergey Yip, Chi Hoi Combinatorics It is known that Paley graphs of square order have the strict-EKR property, that is, all maximum cliques are canonical cliques. Peisert-type graphs are natural generalizations of Paley graphs and some of them also have the strict-EKR property. Given a prime power $q \geq 3$, we study Peisert-type graphs of order $q^2$ without the strict-EKR property and with the minimum number of edges and we call such graphs extremal. We determine number of edges in extremal graphs for each value of $q$. If $q$ is a a square or a cube, we show the uniqueness of the extremal graph and classify all maximum cliques explicitly. Moreover, when $q$ is a square, we prove that there is no Hilton-Milner type result for the extremal graph, and show the tightness of the weight-distribution bound for both non-principal eigenvalues of this graph. |
| title | Extremal Peisert-type graphs without the strict-EKR property |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2306.00391 |