Strongly Regular Graphs on at most 64 vertices

  • Over the years I have been attempting to classify all strongly regular graphs with "few" vertices and have achieved some success in the area of complete classification in two cases that were previously unknown. These are (a) (29,14,6,7) and (b) (40,12,2,4). In the first of these, a non-exhaustive computer search by several authors unearthed 41 graphs, and this I managed to confirm as being the complete number (as did Frans Bussemaker in an independent search). It is perhaps interesting to point out that the complete search on a Pentium Pro 200 MHz now takes less that 12 hours! In case (b) an incomplete search had discovered 27 such graphs, but on returning to the problem I managed to complete the search, in the course of which one further graph was found. This is described in the paper
  • The Strongly Regular (40,12,2,4) Graphs Elec. Journ. Combin. Vol 7(1) 2000.

    As time permits I shall make these graphs available, along with others that I have on file. In the meantime, the two families referred to above can be obtained below as zero-one matrices.

    There are four sets of parameters for strongly regular graphs on 36 vertices, two of which give rise to unique graphs. In the case of the other two, the exact numbers were not known at the time I wrote the paper

  • Regular Two-Graphs on 36 Vertices, Lin.Alg. and its App., 226-228 (1995) 459-497

    There it was found that there are at least 32,548 srg's with parameters (36,15,6,6) (too many to include even here) and at least 180 with parameters (36,21,12,12) (and consequently the same number of complementary graphs (36,14,4,6) which are the ones listed here). The graphs with parameters (36,15,6,6) all arise in the switching classes of the 227 regular two-graphs that I found in the paper cies above. Since each switching class contains at least one such graph, I have listed one from each, namely the one that is lexicographically the greatest in its switching class. Associated with the (36,15,6,6) graphs are those on 35 vertices obtained by isolating a vertex by switching and deleting it. The resulting graphs are srg's (35,18,9,9) of which there are at least 3,854, again too many to list in full here. What I have done instead is to tabulate the lexicographically greatest descendant of each of the 227 known regular two-graphs on 36 vertices. Recently Brendan McKay and I have completed the search for all these strongly regular graphs and have found that the lower bounds mentioned here are in fact also upper bounds. This follows from the fact that we have found there are precisely 227 regular two-graphs on 36 vertices. This result has been written up and now appears in the Australasian Journal of Combinatorics. For those interested I have included here a list of the 32,548 graphs with parameters (36-15-6-6) (4.5Mb) compressed using bzip2. Also, there are 3,854 descendants of the 227 regular two-graphs on 36 vertices. These are obtained by isolating a vertex by switching and then deleting it to get a strongly regular graph with parameters (35-18-9-9). These can be obtained either as a texfile of (0,1) incidence matrices, below (4.64Mb), or its compressed form (36-18-9-9.bz2) (286Kb)

    In a further classification, Willem Haemers and I have determined all SRG's(64,18,2,6). The paper is entitled The Pseudo-Geometric Graphs for Generalised Quadrangles of Order (3,t), European J. Combin. 22 (2001), no. 6, 839-845, while the graphs found are all listed here.

    Finally, I have succeeded in completing the search for the (45,12,3,3) strongly regular graphs. There are precisely 78 of these and I have listed them here. The final results were written up with K. Coolsaet and J. Degraer

  • Strongly regular graphs on 5 vertices
  • Strongly regular graphs on 9 vertices
  • Strongly regular graphs on 10 vertices
  • Strongly regular graphs on 13 vertices
  • Strongly regular graphs on 15 vertices
  • Strongly regular graphs on 16 vertices
  • Strongly regular graphs on 17 vertices
  • Strongly regular graphs on 21 vertices
  • Strongly regular graphs on 25 vertices
  • Strongly regular graphs on 26 vertices
  • Strongly regular graphs on 27 vertices
  • Strongly regular graphs on 28 vertices
  • Strongly regular graphs on 29 vertices
  • Strongly regular graphs on 35 vertices
  • Strongly regular graphs on 36 vertices
  • Strongly regular graphs on 40 vertices
  • Strongly regular graphs on 45 vertices
  • Strongly regular graphs on 64 vertices