New constructions of strongly regular Cayley graphs on abelian non p-groups
New constructions of strongly regular Cayley graphs on abelian non p-groups
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阿贝尔非 p 群上强正则凯莱图的新构造
DOI：
10.1016/j.jcta.2021.105514
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
Momihara Koji
中科院分区：
文献类型：
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作者：
Chen Yan-Yu;Ninomiya Hirokazu;Wu Chang-Hong;Kolar Miroslav and Yazaki Shigetoshi;Takashi Muto;Momihara Koji
Davis和Jedwab(1997)建立了一个伟大的构造理论，将许多已知的差集、相对差集和可除差集的构造统一起来。他们引入了积木的概念，这在理论中发挥了重要作用。另一方面，PolHill(2010)基于一种特殊的积木系构造了Paley类偏差集(会议图)，并证明了对于任意正整数v>；1和任意i=0，1，在9 i v 4阶交换群中都存在Paley类偏差集。他的结果涵盖了存在Paley类部分差集的所有阶交换非p-群。本文通过推广积木理论，给出了交换群上强正则Cayley图的新构造。这些结构是波尔希尔结构的一个很大的推广。特别地，我们证明了对于正整数m和初等交换群Gi，i=1，2，…，S，2 m|q i+1，在交换群G=G1×G2×⋯×G S上，存在一个具有负拉丁方型参数(u2，c(u+1)，−u+c2+3c，c2+c)的强正则Cayley图的完全图的分解，其中u=q1 2 q2 2⋯q S 2和c=(u−1)/m.这种强正则分解只有当m=2或G是p-群时才已知.此外，我们还发现了拉丁方型强正则Cayley图的完全图的另一个新的无限族分解。由此，我们得到了许多具有新参数的强正则图。
Davis and Jedwab (1997) established a great construction theory unifying many previously known constructions of difference sets, relative difference sets and divisible difference sets. They introduced the concept of building blocks, which played an important role in the theory. On the other hand, Polhill (2010) gave a construction of Paley type partial difference sets (conference graphs) based on a special system of building blocks, called a covering extended building set, and proved that there exists a Paley type partial difference set in an abelian group of order 9 i v 4 for any odd positive integer v> 1 and any i= 0, 1. His result covers all orders of abelian non p-groups in which Paley type partial difference sets exist. In this paper, we give new constructions of strongly regular Cayley graphs on abelian groups by extending the theory of building blocks. The constructions are large generalizations of Polhill's construction. In particular, we show that for a positive integer m and elementary abelian groups G i, i= 1, 2,…, s, of order q i 4 such that 2 m| q i+ 1, there exists a decomposition of the complete graph on the abelian group G= G 1× G 2×⋯× G s by strongly regular Cayley graphs with negative Latin square type parameters (u 2, c (u+ 1),− u+ c 2+ 3 c, c 2+ c), where u= q 1 2 q 2 2⋯ q s 2 and c=(u− 1)/m. Such strongly regular decompositions were previously known only when m= 2 or G is a p-group. Moreover, we find one more new infinite family of decompositions of the complete graphs by Latin square type strongly regular Cayley graphs. Thus, we obtain many strongly regular graphs with new parameters.