Paley type partial difference sets in abelian groups

Wang, Zeying

doi:10.1002/jcd.21691

Cited by 7 publications

(4 citation statements)

References 10 publications

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“…Moreover, Polhill in [8] proved that there exist Paley type partial difference sets in the group Z 2 3 ×Z 4t p , where t is a natural number and p is an odd prime number that answered the first part of this question. Finally, the order of abelian groups which admit Paley type partial difference sets has been discovered in [10]. It is proven that if an abelian group admits a Paley type partial difference set and its order is not a prime power, then its order is n 4 or 9n 4 , where n > 1 is an odd integer, but, however, Paley type partial difference sets which come from self-complementary strongly regular Cayley graphs over abelian groups could be restricted than this.…”

Section: Introductionmentioning

confidence: 99%

Self-complementary distance-regular Cayley graphs over abelian groups

Jazaeri¹

2023

Preprint

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In this paper, we study self-complementary distanceregular Cayley graphs over abelian groups. We prove that if a regular graph is self-complementary distance-regular, then it is selfcomplementary strongly regular. We also deal with self-complementary strongly regular Cayley graphs over abelian groups and give an example of a self-complementary strongly regular Cayley graph over a non-elementary abelian group. 2020 Mathematics Subject Classification. 05E16 and 05E30.

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Section: Introductionmentioning

confidence: 99%

Self-complementary distance-regular Cayley graphs over abelian groups

Jazaeri¹

2023

Preprint

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“…For further applications of partial difference sets to coding theory and finite geometry, see the survey of Ma [17]. The case when G is abelian has been thoroughly studied; see [17] for a survey of older results and [5,6,7,8,11,18,19,20,23,24] for a number of very recent results. On the other hand, comparatively little is known in the case when G is nonabelian.…”

Section: Introductionmentioning

confidence: 99%

Restrictions on parameters of partial difference sets in nonabelian groups

Swartz

Tauscheck

2020

J of Combinatorial Designs

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A partial difference set S in a finite group G satisfying ∉ S 1 and S S = −1 corresponds to an undirected strongly regular Cayley graph G S Cay(,). While the case when G is abelian has been thoroughly studied, there are comparatively few results when G is nonabelian. In this paper, we provide restrictions on the parameters of a partial difference set that apply to both abelian and nonabelian groups and are especially effective in groups with a nontrivial center. In particular, these results apply to p-groups, and we are able to rule out the existence of partial difference sets in many instances. K E Y W O R D S partial difference set, strongly regular Cayley graph −1. If the set S is a v k λ μ (, , ,)-PDS, then the Cayley graph G S Cay(,) is a v k λ μ (, , ,)-strongly regular graph (SRG) [17, Proposition 1.1], which means that G S Cay(,) has v vertices, G S Cay(,) is regular of degree k, any two adjacent vertices in G S Cay(,) have exactly λ common neighbors, and any two nonadjacent vertices in G S Cay(,) have exactly How to cite this article: Swartz E, Tauscheck G. Restrictions on parameters of partial difference sets in nonabelian groups.

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“…On the other hand, Wang [18] proved that a Paley type partial difference set in a nonelementary abelian group G exists only when |G| = v 4 or 9v 4 for an odd integer v > 1. Hence, Polhill's result covers all orders of nonelementary abelain groups in which Paley type partial difference sets exist.…”

Section: Introductionmentioning

confidence: 99%

New constructions of strongly regular Cayley graphs on abelian groups

Momihara¹

2020

Preprint

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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 nonelementary abelian 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 4 i such that 2m | 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 + 3c, c 2 + c), where u = q 2 1 q 2 2 • • • q 2 s 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.

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Paley type partial difference sets in abelian groups

Cited by 7 publications

References 10 publications

Self-complementary distance-regular Cayley graphs over abelian groups

Self-complementary distance-regular Cayley graphs over abelian groups

Restrictions on parameters of partial difference sets in nonabelian groups

New constructions of strongly regular Cayley graphs on abelian groups

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