Cyclic arcs of Singer type and strongly regular Cayley graphs over finite fields
Cyclic arcs of Singer type and strongly regular Cayley graphs over finite fields
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DOI：
10.1016/j.ffa.2021.101953
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发表时间：
2021-10
期刊：
影响因子：
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通讯作者：
K. Momihara;Qing Xiang
中科院分区：
文献类型：
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作者：
K. Momihara;Qing Xiang
在[32]中，第一作者利用三值Gauss周期构造了有限域的可加群上的强正则Cayley图。特别地，结合文献[4]的结果，证明了存在具有负拉丁方型参数的强正则Cayley图（q 6，r（q 3+ 1），− q 3+ r 2+ 3 r，r 2+ r），其中r= M（q 2− 1）/2，在下列情况下：（i）M= 1和q <$3（mod 4）;（ii）M= 3和q <$7（mod 24）;（iii）M= 7，q = 11，51（mod 56）。对于奇数M> 7，具有上述参数的强正则凯莱图的存在性尚不清楚。在本文中，我们证明了，如果有一个h，1 <$h <$M− 1，使得M|（h2 + h+ 1）且（Z/MZ）×中2的阶为奇数，则存在无穷多个素数q使得存在具有上述参数的强正则Cayley图.
Abstract In [32], the first author gave a construction of strongly regular Cayley graphs on the additive group of finite fields by using three-valued Gauss periods. In particular, together with the result in [4], it was shown that there exists a strongly regular Cayley graph with negative Latin square type parameters (q 6, r (q 3+ 1),− q 3+ r 2+ 3 r, r 2+ r), where r= M (q 2− 1)/2, in the following cases:(i) M= 1 and q≡ 3 (mod 4);(ii) M= 3 and q≡ 7 (mod 24); and (iii) M= 7 and q≡ 11, 51 (mod 56). The existence of strongly regular Cayley graphs with the above parameters for odd M> 7 was left open. In this paper, we prove that if there is an h, 1⩽ h⩽ M− 1, such that M|(h 2+ h+ 1) and the order of 2 in (Z/M Z)× is odd, then there exist infinitely many primes q such that strongly regular Cayley graphs with the aforementioned parameters exist.