Strongly regular Cayley graphs with λ − μ = −1

Arasu, K. T.; Jungnickel, Dieter; L., S.; Pott, Alexander

doi:10.1016/0097-3165(94)90007-8

Cited by 20 publications

(22 citation statements)

References 8 publications

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“…Below we cite three results on abelian regular PDSs. The first of these was proved by Ma [7], the second one was proved by Arasu et al [1], and the last one, the local multiplier theorem, was proved by De Winter, Kamischke, and Wang [3].…”

Section: Proof Of the Main Resultsmentioning

confidence: 93%

“…PDSs with parameters

(v, (v - 1) ∕ 2, (v - 5) ∕ 4, (v - 1) ∕ 4)

are called Paley‐type PDSs . Over the last three decades, this subject has seen active research (see, eg, , (Theorem 3.2) ). There are two key problems on Paley‐type PDSs in abelian groups: …”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…Theorem 2.2 (Arasu et al [1]). Let Γ be a strongly regular Cayley graph based on an abelian group G, with parameters v k λ , , , and μ satisfying β λ μ = − = −1.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

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New necessary conditions on Paley‐type partial difference sets in abelian groups

Wang

2019

J of Combinatorial Designs

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Let G be an abelian group of order v, where v = p 1 2 k 1 p 2 2 k 2 ⋯ p n 2 k n , n ≥ 2 , p 1 , p 2 , … , p n are distinct odd prime numbers. In this paper, we prove that if G contains a regular Paley‐type partial difference set (PDS), then for any 1 ≤ i ≤ n , p i is congruent to 3 modulo 4 whenever k i is odd. These new necessary conditions further limit the specific order of an abelian group G in which there can exist a Paley‐type PDS. Our result is similar to a result on abelian Hadamard (Menon) difference sets proved by Ray‐Chaudhuri and Xiang in 1997.

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Section: Proof Of the Main Resultsmentioning

confidence: 93%

“…PDSs with parameters

(v, (v - 1) ∕ 2, (v - 5) ∕ 4, (v - 1) ∕ 4)

Section: Introduction and The Main Resultsmentioning

confidence: 99%

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New necessary conditions on Paley‐type partial difference sets in abelian groups

Wang

2019

J of Combinatorial Designs

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“…Although many further examples of Paley type PDSs in

p

‐groups were found (eg, and ), not much was known about Paley type PDSs in abelian groups of order not a prime power. In 1994, Arasu et al asked the following question.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Theorem 2.1 (Arasu et al [1]). Let D be a nontrivial regular v k λ μ ( , , , )-PDS in an abelian group G, and assume that Δ is a square, say…”

Section: Let D Be a Regular Paley Type Pds In An Abelian Groupmentioning

confidence: 99%

Paley type partial difference sets in abelian groups

Wang

2019

J of Combinatorial Designs

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Partial difference sets with parameters (v,k,λ,μ)=(v,(v−1)/2,(v−5)/4,(v−1)/4) are called Paley type partial difference sets. In this note, we prove that if there exists a Paley type partial difference set in an abelian group of order v, where v is not a prime power, then v=n4 or 9n4, n>1 an odd integer. In 2010, Polhill constructed Paley type partial difference sets in abelian groups with those orders. Thus, combining with the constructions of Polhill and the classical Paley construction using nonzero squares of a finite field, we completely answer the following question: “For which odd positive integers v>1, can we find a Paley type partial difference set in an abelian group of order v?”

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A survey of partial difference sets

1994

Des Codes Crypt

204

254

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Abstract. Let G be a finite group of order v. A k-element subset D of G is called a (v, k, 3,,/z)-partial difference set if the expressions gh -1, for g and h in D with g ~ h, represent each nonidentity element in D exactly ), times and each nonidentity element not in D exactly # times. If e t~ D and g E D iff g-1 E D, then D is essentially the same as a strongly regular Cayley graph. In this survey, we try to list all important existence and nonexistence results concerning partial difference sets. In particular, various construction methods are studied, e.g., constructions using partial congruence partitions, quadratic forms, cyclotomic classes and finite local rings. Also, the relations with Schur tings, two-weight codes, projective sets, difference sets, divisible difference sets and partial geometries are discussed in detail. Strongly Regular Graphs and Partial Difference SetsA graph I TM with v vertices is said to be a (v, k, k, #)-strongly regular graph if (i) it is regular of valency k, i.e., each vertex is joined to exactly k other vertices; (ii) any two adjacent vertices are both joined to exactly 3, other vertices and two nonadjacent vertices are both joined to exactly/z other vertices. A Cayley graph is defined as a graph r = (v, E) which admits an automorphism group G acting regularly on the vertex set V (see Yap [7] for background on Cayley graphs). If we identify the vertices of I ~ with the elements of the regular automorphism group G, then r can be generated by a subset D of G such that two vertices g, h E G are joined nonabelian) if the group G is abelian (resp. nonabelian). For a subset S of a group G, let S it) = {g~ : g E S} for any integer t.

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Strongly regular Cayley graphs with λ − μ = −1

Cited by 20 publications

References 8 publications

New necessary conditions on Paley‐type partial difference sets in abelian groups

New necessary conditions on Paley‐type partial difference sets in abelian groups

Paley type partial difference sets in abelian groups

A survey of partial difference sets

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