Van Lint–MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields

Shamil, Asgarli,; Yip, Chi Hoi

doi:10.1016/j.jcta.2022.105667

Cited by 10 publications

(22 citation statements)

References 21 publications

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“…We expected that directly extending Blokhuis' and Sziklai's proofs to a general Peiserttype graph is difficult [4,Remark 2.16]. In fact, we speculated that there might be an infinite family of Peisert-type graphs which fail to have the strict-EKR property and gave a few counterexamples of small size in [4,Example 2.18].…”

Section: Peisert-type Graphsmentioning

confidence: 99%

“…This follows from combining Theorem 4 and Theorem 17 on the strict-EKR property of the block graph of the corresponding orthogonal array. 4 The EKR-module property of Peisert-type graphs…”

Section: Peisert-type Graphs As Block Graphs Of Orthogonal Arraysmentioning

confidence: 99%

“…Theorem 1 was first proved by Blokhuis [7]. Extensions and generalizations of Theorem 1 can be found in [13,30,25,3,4]. A Fourier analytic approach was recently proposed in [34,Section 4.4].…”

Section: Introductionmentioning

confidence: 99%

“…We remark that the idea of viewing certain Cayley graphs geometrically has appeared in the past; see for example [25,Construction 5.2.1] and [4,Section 4.2] for related discussion. However, Paley graphs and block graphs of orthogonal arrays are often treated independently; see for example [17,Chapter 10], [16,Chapter 5], and [1, Section 5].…”

Section: Introductionmentioning

confidence: 99%

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The EKR-Module Property of Pseudo-Paley Graphs of Square Order

Shamil

Goryaĭnov

Lin

et al. 2022

Electron. J. Combin.

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We prove that a family of pseudo-Paley graphs of square order obtained from unions of cyclotomic classes satisfies the Erdős-Ko-Rado (EKR) module property, in a sense that the characteristic vector of each maximum clique is a linear combination of characteristic vectors of canonical cliques. This extends the EKR-module property of Paley graphs of square order and solves a problem proposed by Godsil and Meagher. Different from previous works, which heavily rely on tools from number theory, our approach is purely combinatorial in nature. The main strategy is to view these graphs as block graphs of orthogonal arrays, which is of independent interest.

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Section: Peisert-type Graphsmentioning

confidence: 99%

“…This follows from combining Theorem 4 and Theorem 17 on the strict-EKR property of the block graph of the corresponding orthogonal array. 4 The EKR-module property of Peisert-type graphs…”

Section: Peisert-type Graphs As Block Graphs Of Orthogonal Arraysmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The EKR-Module Property of Pseudo-Paley Graphs of Square Order

Shamil

Goryaĭnov

Lin

et al. 2022

Electron. J. Combin.

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“…Besides Paley graphs being generalized (see for example [4,12]), Peisert graphs have also been generalized into graphs called generalized Peisert or Peisert type graphs and their cliques have been studied in [2,3,14]. In this article, we introduce a Peisert-like graph on the commutative ring Z n , for suitable n. Computing the number of cliques in Paley, Peisert and Paley-type graphs has been of interest, for instance see [1,4,5,7].…”

Section: Introductionmentioning

confidence: 99%

Hypergeometric functions for Dirichlet characters and Peisert-like graphs

Bhowmik¹,

Barman²

2022

Preprint

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For a prime p ≡ 3 (mod 4) and a positive integer t, let q = p 2t . The Peisert graph of order q is the graph with vertex set Fq such that ab is an edge if a − b ∈ g 4 ∪ g g 4 , where g is a primitive element of Fq. In this paper, we construct a similar graph with vertex set as the commutative ring Zn for suitable n, which we call Peisert-like graph and denote by G(n). Owing to the need for cyclicity of the group of units of Zn, we consider n = p α or 2p α , where p ≡ 1 (mod 4) is a prime and α is a positive integer. For primes p ≡ 1 (mod 8), we compute the number of triangles in the graph G(p α ) by evaluating certain character sums. Next, we study cliques of order 4 in G(p α ). To find the number of 4-order cliques in G(p α ), we first introduce hypergeometric functions containing Dirichlet characters as arguments, and then express the number of 4-order cliques in G(p α ) in terms of these hypergeometric functions.

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Hypergeometric Functions for Dirichlet Characters and Peisert-Like Graphs on $$\mathbb {Z}_n$$

Bhowmik,

Barman

2023

La Matematica

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Van Lint–MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields

Cited by 10 publications

References 21 publications

The EKR-Module Property of Pseudo-Paley Graphs of Square Order

The EKR-Module Property of Pseudo-Paley Graphs of Square Order

Hypergeometric functions for Dirichlet characters and Peisert-like graphs

Hypergeometric Functions for Dirichlet Characters and Peisert-Like Graphs on $$\mathbb {Z}_n$$

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