On the existence of self-complementary and non-self-complementary strongly regular graphs with Paley parameters
type: Item
instance of: scholarly article
title: On the existence of self-complementary and non-self-complementary strongly regular graphs with Paley parameters
zbMATH Open document ID: 1360.05186
author: Nimrod Kriger
author: Mikhail H. Klin
author: Andrew J. Woldar
published in: Journal of Geometry
publication date: 2017-02-02T00:00:00Z
review text: The Paley graphs are a well-known class of strongly regular graphs, which are self-complementary. The paper at hand searches for other examples of strongly regular graphs that have the same regularity parameters as Paley. The main goal is to find examples of such graphs which are not self-complementary and which are non-Schurian. The graphs in question are in fact basis graphs of certain association schemes. An association scheme (sometimes called a homogeneous coherent configuration) consists of a set \(V\) and a set of relations \(\{R_{0}, R_{1}, \dots, R_{d}\}\) on \(V\) satisfying the properties: \(R_{0}\) is the identity relation, and the \(R_{i}\) form a partition of \(V \times V\). For any \(0 \leq i \leq d\), there is a \(j\) so that \(R_{i}^{-1} = R_{j}\). For any triple of relations \(R_{i}\), \(R_{j}\), \(R_{k}\), and for any \((x,y) \in R_{k}\), the number \[ p_{i,j}^{k} = \left|\{z \in V \mid (x,z) \in R_{i} \text{ and } (z,y) \in R_{j} \} \right| \] is independent of the choice of \((x,y)\). (These axioms can be phrased algebraically as well, in which case the numbers \(p_{i,j}^{k}\) are the structure constants of the algebra spanned by the matrices representing the \(R_{i}\).) Given an association scheme, one can potentially create other association schemes (called fusion schemes) by taking the union of several of the \(R_{i}\), creating a strictly coarser partition in the first axiom. (Of course, not all such coarser partitions will satisfy the other axioms.) The paper begins with the complete classical affine scheme \(\mathcal{A}_{p}\), built on the data of the classical affine plane of order \(p\). The underlying set \(V\) is \(\mathbb{F}_{p}^{2}\), and the relations are defined using the sets of parallel lines in the affine plane. The rest of the paper is devoted to understanding the fusion schemes of \(\mathcal{A}_{p}\). By carefully examining certain groups that act on \(\mathcal{A}_{p}\) and the resulting orbits on fusion schemes of \(\mathcal{…
Mathematics Subject Classification ID: 05E30
Mathematics Subject Classification ID: 05E18
Mathematics Subject Classification ID: 05C30
zbMATH DE Number: 6680051
zbMATH Keywords: affine plane
zbMATH Keywords: association scheme
zbMATH Keywords: classical affine scheme
zbMATH Keywords: fusion scheme
zbMATH Keywords: amorphic scheme
zbMATH Keywords: Schurian scheme
zbMATH Keywords: Paley graph
zbMATH Keywords: Peisert graph
zbMATH Keywords: strongly regular graph
zbMATH Keywords: self-complementary graph
describes a project that uses: GAP
describes a project that uses: nauty
describes a project that uses: GRAPE
MaRDI profile type: Publication
full work available at URL: s00022-015-0308-9
OpenAlex ID: W2207777136
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DOI: 10.1007/S00022-015-0308-9
description: scientific article; zbMATH DE number 6680051