Cyclotomy, Gauss Sums, Difference Sets and Strongly Regular Cayley Graphs

Xiang, Qing

doi:10.1007/978-3-642-30615-0_23

Cited by 6 publications

(6 citation statements)

References 33 publications

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“…Recently, Feng and Xiang [FX12] gave constructions of strongly regular Cayley graphs by using union of cyclotomic classes and index 2 Gauss sums. For other constructions we refer to the survey [Xia12].…”

Section: 3mentioning

confidence: 99%

The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes

Shamil¹,

Yip²

2021

Preprint

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Blokhuis showed that Paley graphs with square order have the Erdős-Ko-Rado (EKR) property in the sense that all maximum cliques are canonical. In our previous work, we extended the EKR property of Paley graphs to certain Peisert graphs and generalized Peisert graphs. In this paper, we propose a conjecture which generalizes the EKR property of Paley graphs, and can be viewed as an analogue of Chvátal's Conjecture for families of set systems. As a partial progress, we prove that maximum cliques in pseudo-Paley graphs have a rigid structure.

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Section: 3mentioning

confidence: 99%

The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes

Shamil¹,

Yip²

2021

Preprint

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“…This family of SRGs has been studied extensively by many authors; see [24,5,19,12]. We refer the reader to section 4 of [26] for a survey on these graphs. If H is the multiplicative group of a non-trivial subfield of K, then Cay(K, H) is a cyclotomic SRG.…”

Section: Introductionmentioning

confidence: 99%

“…Further assume that there exists an integer s such that p s ≡ −1 (mod N). These arithmetic restrictions on H ensure that the adjacency matrix of the regular graph Cay(K, H) has exactly 3 eigenvalues and thus is a cyclotomic SRG (see for example, Section 4 of [26]). A graph of this form is called a semi-primitive cyclotomic SRG.…”

Section: Introductionmentioning

confidence: 99%

“…A graph of this form is called a semi-primitive cyclotomic SRG. According to a conjecture by Schmidt and White (Conjecture 4.4 of [19]/ Conjecture 4.1 of [26]), other than the above mentioned classes, there are only 11 sporadic examples of cyclotomic SRGs. In this paper we consider a class of semi-primitve cyclotomic SRGs discovered in [24].…”

Section: Introductionmentioning

confidence: 99%

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Critical groups of van Lint-Schrijver Cyclotomic Strongly Regular Graphs

Pantangi

2018

Preprint

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The critical group of a finite connected graph is an abelian group defined by the Smith normal form of its Laplacian. Let q be a power of a prime and H be a multiplicative subgroup of K = F q . By Cay(K, H) we denote the Cayley graph on the additive group of K with "connection" set H. A strongly regular graph of the form Cay(K, H) is called a cyclotomic strongly regular graph. Let ℓ > 2 and p be primes such that p is primitive (mod ℓ). We compute the critical groups of a family of cyclotomic strongly regular graphs for which q = p (ℓ−1)t (with t ∈ N) and H is the unique multiplicative subgroup of order k = q−1 ℓ . These graphs were first discovered by van Lint and Schrijver in [24].

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“…One sees their use in coding theory [37], [116]; point counting on elliptic curves [97], [117]; determining the number of solutions to polynomial equations [7], [95]; determining the value of cyclotomic numbers [114]; constructing difference sets [39]; and the evaluation of various related functions [85]. In particular, the finite field Gauss sums are widely used in the construction of Cayley graphs [38], [40], [43], [87], [88], [113], [115].…”

Section: Current Applicationsmentioning

confidence: 99%

Quadratic Form Gauss Sums

Doyle¹

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Cyclotomy, Gauss Sums, Difference Sets and Strongly Regular Cayley Graphs

Cited by 6 publications

References 33 publications

The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes

The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes

Critical groups of van Lint-Schrijver Cyclotomic Strongly Regular Graphs

Quadratic Form Gauss Sums

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