Pseudo‐random properties of self‐complementary symmetric graphs

Kisielewicz, Andrzej; Peisert, Wojciech

doi:10.1002/jgt.20035

Cited by 14 publications

(9 citation statements)

References 14 publications

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“…First, if m | (n − 1) then, rather than quadratic residues in F n , one may determine adjacency by means of the multiplicative subgroup of m-th powers and its cosets. Having similar quasirandom properties [2,27], such graphs proved useful in Ramsey theory [15,24,42]. They appear also in the classification of graphs with strong symmetries [32,37].…”

Section: Explicit Construction Of Ample Complexesmentioning

confidence: 99%

Ample simplicial complexes

Even-Zohar

Farber

Mead

2022

European Journal of Mathematics

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Motivated by potential applications in network theory, engineering and computer science, we study r-ample simplicial complexes. These complexes can be viewed as finite approximations to the Rado complex which has a remarkable property of indestructibility, in the sense that removing any finite number of its simplexes leaves a complex isomorphic to itself. We prove that an r-ample simplicial complex is simply connected and 2-connected for r large. The number n of vertexes of an r-ample simplicial complex satisfies $$\exp \bigl (\Omega \bigl (\frac{2^r}{\sqrt{r}}\bigr )\bigr )$$ exp ( Ω ( 2 r r ) ) . We use the probabilistic method to establish the existence of r-ample simplicial complexes with n vertexes for any $$n>r 2^r 2^{2^r}$$ n > r 2 r 2 2 r . Finally, we introduce the iterated Paley simplicial complexes, which are explicitly constructed r-ample simplicial complexes with nearly optimal number of vertexes.

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Section: Explicit Construction Of Ample Complexesmentioning

confidence: 99%

Ample simplicial complexes

Even-Zohar

Farber

Mead

2022

European Journal of Mathematics

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“…Note that M q is not closed under multiplication since g • g = g 2 / ∈ M q . Kisielewicz and Peisert [KP04] showed that Paley graphs and Peisert graphs are similar in many aspects. However, we know very little about the structure of cliques of Peisert graphs other than Theorem 2.5.…”

Section: Background and Overview Of The Papermentioning

confidence: 99%

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

Asgarli,

Yip

2021

Preprint

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A well-known conjecture due to van Lint and MacWilliams states that if A is a subset of F q 2 such that 0, 1 ∈ A, |A| = q, and a − b is a square for each a, b ∈ A, then A must be the subfield F q . This conjecture is often phrased in terms of the maximum cliques in Paley graphs. It was first proved by Blokhuis and later extended by Sziklai to generalized Paley graphs. In this paper, we give a new proof of the conjecture and its variants, and show this Erdős-Ko-Rado property of Paley graphs extends to a larger family of Cayley graphs, which we call Peisert-type graphs, resolving conjectures by Mullin and Yip.

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“…We will introduce Peisert graphs formally in Section 5. Peisert graphs form a nice family of Cayley graphs defined on finite fields with square order, and they are similar to Paley graphs in many aspects (see the discussion on [32]). However, little is known about the cliques of Peisert graphs and they seem to be much harder to study compared to (generalized) Paley graphs.…”

Section: Theorem 14 ([40]mentioning

confidence: 99%

“…Theorem 5.2 (Theorem 5.1 in [32]). Let q = p 2s , where p is a prime such that p ≡ 3 (mod 4) and s is a positive integer.…”

Section: Maximum Cliques In Peisert Graphsmentioning

confidence: 99%

Gauss sums and the maximum cliques in generalized Paley graphs of square order

Yip

2021

Preprint

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Let GP (q, d) be the d-Paley graph defined on the finite field F q . It is notoriously difficult to improve the trivial upper bound √ q on the clique number of GP (q, d). In this paper, we investigate the connection between Gauss sums over a finite field and maximum cliques of their corresponding generalized Paley graphs. In particular, we show that the trivial upper bound on the clique number of GP (q, d) attains if and only d | ( √ q + 1), which strengthens the previous related results by Broere-Döman-Ridley and Schneider-Silva, as well as improves the trivial upper bound on the clique number of GP (q, d) when d ∤ ( √ q + 1).

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Pseudo‐random properties of self‐complementary symmetric graphs

Abstract: There are some results in the literature showing that Paley graphs behave in many ways like random graphs G(n; 1=2). In this paper, we extend these results to the other family of self-complementary symmetric graphs. ß

Cited by 14 publications

References 14 publications

Ample simplicial complexes

Ample simplicial complexes

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

Gauss sums and the maximum cliques in generalized Paley graphs of square order

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