Small strongly regular graphs and corresponding partial difference sets - a survey
by
Aiso Heinze
Carl von Ossietzky-University Oldenburg, Germany

Let (H, ·) be a group of order n and D be a subset of H with k elements. The set D is called a (n, k, l, m)-partial difference set, if g^-1h, for distinct elements g, h of D, represents each nonidentity element not contained in D exactly l times and each nonidentity element contained in D exactly m times. A partial difference set D is called regular, if D^-1=D and D does not contain the identity element.

An undirected graph G with n vertices is called a strongly regular graph with parameters (n, k, l, m), if it is regular of degree k, any pair of adjacent vertices have exactly l common neighbours and any pair of nonadjacent vertices have exactly m common neighbours.

It is known that a regular partial difference set D in a finite group H represents a strongly regular graph which is generated by H and D as a Cayley graph (e.g. [Ma94]).

In this talk a method for the creation of partial difference sets by strongly regular graphs will be presented. We used lists of strongly regular graphs up to 41 vertices (e.g. [Spe00]) to determine all regular partial difference sets in groups up to order 41. In some cases the determination was done by the computer package GAP [GAP99], in other cases it was done on a pure theoretical level.

References:

[Ma94] Ma, S.L., A Survey of Partial Difference Sets. Designs, Codes and Cryptography 4 (1994), pp. 221 - 261.

[GAP99] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.2; Aachen, St Andrews, 1999. (http://www-gap.dcs.st-and.ac.uk/ gap)

[Spe00] Spence, E. Strongly Regular Graphs on at most 64 vertices. http://www.maths.gla.ac.uk/ es/srgraphs.html.

Date received: January 2, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafo-44.