Two-graphs
The files that appear below can be regarded as updated versions of
some of the tables that appeared in [1]. The additional material
concerns two-graphs on 10 vertices and regular two-graphs on 36,
46 and 50 vertices.
For full details of the references involved the reader is referred to
the author's chapter in [2].
The files two-graph.04,
two-graph.05,
two-graph.06,
two-graph.07,
two-graph.08,
two-graph.09 (132Kb) and
two-graph.10 (2.4Mb),
contain tables of two-graphs on 4, 5, 6, 7, 8, 9 and 10 vertices,
respectively. The format of the tables is the same as that referred
to above [1]. However, since these tables were generated by the author
independently
there is not an exact correspondence between the graphs in the tables.
Occasionally I have listed graphs complementary to those of [1].
Each row comprises the upper half of the (+,-) adjacency of a
descendant (with its first row deleted), the eigenvalues, the order of
the automorphism group, the number
of graphs in the switching class, and if a two-graph is
self-complementary this is denoted by the letter S.
Alternatively, you may wish to download all these graphs as a
gzipped
tar file two-graph.tar.gz, which is around
500K.
The further files
reg2graph.06,
reg2graph.10,
reg2graph.14,
reg2graph.16,
reg2graph.18,
reg2graph.26,
reg2graph.28,
reg2graph.30,
reg2graph.36,
reg2graph.38,
reg2graph.42,
reg2graph.46 and
reg2graph.50,
as their titles suggest, contain all regular two-graphs on at most
50 vertices that are known to me at this time. Again the format mimics
that of [1]. Here each two-graph is identified with the hexadecimal
form of
its greatest descendant, where the ordering involved is the natural
one obtained
by expressing as a hexadecimal number the binary integer obtained by
concatenating the rows
of the upper triangular part of the adjacency matrix. I also list
the order of its automorphism group, and the orbits under the action of
this group. In addition, I include the numbers of strongly regular
graphs that
are obtained either as descendants of the regular two-graph or, where
appropriate, in its switching class. The strongly regular graphs
are identified by their parameters which comprise the number of
vertices and the eigenvalues of their Seidel Spectrum. This
is simply the spectrum of its (-,+) adjacency matrix. The notation
[x,y] means that there are x graphs with automorphism group of order y.
Perhaps the updated files concerning regular two-graphs on 36, 46 and 50
vertices need some explanation. There are now 227 known regular two-graphs
on 36 vertices [5], Mathon [6] found 17 self-complementary strongly
regular graphs on 45 vertices that give rise to the same number of
self-complementary regular two-graphs on 46 vertices, and the
additional work [3], [4] and [7] increased the known number of
regular two-graphs on 50 vertices to 54 (6 self-complementary and
24 complementary pairs).
Many of the these files were compiled with the help of
Rudi Mathon
and Frans Bussemaker. I am grateful to
them both for their invaluable assistance.
- [1] F.C. Bussemaker, R. A. Mathon , J.J. Seidel,
"Tables of Two-graphs", Technical Report 79-WSK-05, Technical
University of Eindhoven
- [2] E.Spence, "Two-Graphs" in: CRC Handbook of Combinatorial Designs,
(C.J. Colbourne and J.H. Dinitz, eds) pp 686-695
- [3] E.S. Kramer, S.S. Magliveras and R. Mathon, The Steiner
systems S(2,4,25) with non-trivial automorphism group, Discrete
Math. 77 (1989), 137-157.
- [4] E. Spence, The complete classification of Steiner systems
S(2,4,25), Journ. Comb. Designs, 4 (1996), no. 4,
295-300.
- [5] E. Spence, Regular two-graphs on 36 vertices, Lin. Alg. Appl.
226-228 (1995), 459-497.
- [6] R, Mathon, On self-complementary strongly regular graphs,
Discrete Math. 69 (1988), 263-281.
- [7] E. Spence, Unpublished computer result, 1995