Topic
Paley graph
About: Paley graph is a research topic. Over the lifetime, 179 publications have been published within this topic receiving 1706 citations.
Papers published on a yearly basis
Papers
TL;DR: In this article, the negacyclic C-matrices of order v = 0 (mod 4) were shown to have no multipliers, unless v = 1 + pk, p prime.
Abstract: 1. Introduction. C-matrices appear in the literature at various places; for a survey, see [11]. Important for the construction of Hadamard matrices are the symmetric C-matrices, of order v = 2 (mod 4), and the skew C-matrices, of order v = 0 (mod 4). In § 2 of the present paper it is shown that there are essentially no other C-matrices. A more general class of matrices with zero diagonal is investigated, which contains the C-matrices and the matrices of (v, k, X)-systems on k and k + 1 in the sense of Bridges and Ryser [6]. Skew C-matrices are interpreted in § 3 as the adjacency matrices of a special class of tournaments, which we call strong tournaments. They generalize the tourna ments introduced by Szekeres [24] and by Reid and Brown [21]. In § 4 we introduce the notion of negacyclic C-matrices, analogous to the similar notion introduced by Berlekamp in the setting of coding theory (cf. [4, p. 211]). Eigenvalues of negacyclic matrices are characterized and standard forms are obtained. Negacyclic C-matrices are interpreted in § 5 as the matrices of a special class of the relative difference sets introduced by Butson [7]. Exploiting some results of Elliott and Butson [10], we obtain a "multiplier theorem" for negacyclic C-matrices, and adapting a result of [2], we show that any negacyclic C-matrix has a nontrivial multiplier. Necessary conditions for the existence of a negacyclic C-matrix of order v are obtained in § 6. The nonexistence of nega cyclic C-matrices of all orders v ^ 226, v ^ 1 + Ph, with p prime, has been verified. This leads to the conjecture that they do not exist, unless v = 1 + PkPaley [19] constructed C-matrices of all orders v = 1 + pk, p prime. In § 7 it is shown that every Paley matrix is equivalent to a negacyclic C-matrix, a fact
220 citations
TL;DR: Probabilistic intuition is given for a new bound on the clique number of Paley graphs of prime order, and it is conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.
Abstract: The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.
154 citations
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TL;DR: The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing as mentioned in this paper, but deterministic constructions have found less success.
Abstract: The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.
121 citations
TL;DR: In this paper, it was shown that the automorphism group of a self-complementary symmetric graph is permutation isomorphic to a subgroup of AΓL1(pr) with three exceptions.
99 citations
TL;DR: In this article, it was shown that generalised Paley graphs are sometimes isomorphic to Hamming graphs and consequently have large automorphism groups, and precisely the parameters for which the graphs are connected, or equivalently, the schemes are primitive.
Abstract: The generalised Paley graphs are, as their name suggests, a generalisation of the Paley graphs, first defined by Paley in 1933 (see \cite{Paley}). They arise as the relation graphs of symmetric cyclotomic association schemes. However, their automorphism groups may be much larger than the groups of the corresponding schemes. We determine the parameters for which the graphs are connected, or equivalently, the schemes are primitive. Also we prove that generalised Paley graphs are sometimes isomorphic to Hamming graphs and consequently have large automorphism groups, and we determine precisely the parameters for this to occur. We prove that in the connected, non-Hamming case, the automorphism group of a generalised Paley graph is a primitive group of affine type, and we find sufficient conditions under which the group is equal to the one-dimensional affine group of the associated cyclotomic association scheme. The results have been applied in \cite{LLP} to distinguish between cyclotomic schemes and similar twisted versions of these schemes, in the context of homogeneous factorisations of complete graphs.
64 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2022 | 1 |
| 2021 | 27 |
| 2020 | 17 |
| 2019 | 15 |
| 2018 | 9 |
| 2017 | 5 |