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( de Gruyter 2002 New prolific constructions of strongly regular graphs
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@MISC{Fon-der-flaass_(de,
author = {Dmitry G. Fon-der-flaass and Communicated W. Kantor},
title = {( de Gruyter 2002 New prolific constructions of strongly regular graphs},
year = {}
}
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Abstract
Abstract. We present a purely combinatorial construction of strongly regular graphs with geometric parameters. The construction produces hyperexponentially many graphs, and uses neither linear algebra nor groups. Among the graphs constructed, there are graphs with pa-rameters of a generalized quadrangle GQðn 1; nþ 1Þ which further produce distance regular covers of complete graphs. In this paper we present an exceptionally prolific construction of strongly regular graphs. We use the word ‘‘prolific’ ’ informally, but in a rather strict sense. Thus, there are prolific constructions of strongly regular graphs from Latin squares, from Steiner triple systems, from collections of mutually orthogonal Latin squares (of prime power order), etc. What is especially interesting about our construction is that it produces graphs with geometric parameters (in the sense of [1]); in particular, with those of generalized quadrangles GQðn i; nþ iÞ for i 0;G1. Moreover, the graphs with parameters of GQðn 1; nþ 1Þ, by virtue of the construction, will have spreads of n-cliques; removing the edges of these cliques one obtains distance regular n-covers of the complete graph Kn2. In this way one can find, for instance, distance regular graphs of diameter 3 with the trivial group of automorphisms. As building blocks of the construction we shall use certain 2-designs: resolvable de-signs with constant intersection of non-parallel lines. In this paper we shall call them a‰ne designs. Definition 1. An a‰ne design is a 2-design with the following two properties: (i) every two blocks are either disjoint or intersect in a constant number r of points; (ii) each block together with all blocks disjoint from it forms a parallel class: a set of n mutually disjoint blocks partitioning all points of the design. Examples of a‰ne designs: all lines of an a‰ne plane of order n ðr 1Þ; all hy-
Keyphrases
regular graph ne design new prolific construction generalized quadrangle gq geometric parameter prolific construction constant number combinatorial construction orthogonal latin square distance regular cover constant intersection prime power order steiner triple system complete graph disjoint block building block trivial group complete graph kn2 non-parallel line distance regular n-covers strict sense many graph latin square word prolific distance regular graph linear algebra ne plane resolvable de-signs parallel class
