Citation for published version (APA):
Bussemaker, F. C., Mathon, R. A., & Seidel, J. J. (1979). Tables of two-graphs. (EUT report. WSK, Dept. of
Mathematics and Computing Science; Vol. 79-WSK-05). Technische Hogeschool Eindhoven.
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Published: 01/01/1979
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TECHNISCBE HOGESCHOOL EINDHOVEN
NEDERLAND
ONDERAFDELING DER WISKUNDE
TECHNOLOGICAL UNIVERSITY EINDHOVEN
THE NETHERLANDS
DEPARTMENT OF MATHEMATICS
TABLES OF TWO-GRAPHS
by
F.C. Bussemaker, R.A. Mathon,
J.J.
Seidel
T.R.-Report 79-WSK-OS
Ch.
i.
Introduction
Ch. 2. Two-graphs and switching classes
Ch. 3. Automorphism groups of two-graphs
Ch. 4. Regular two-graphs with
PI
+
P
2
'#
0
Ch.
5.
Conference two-graphs with integral eigenvalues
Ch. 6. Conference two-graphs with irrational eigenvalues
References
Appendix
Table 1.
Table 2.
Table 3.
Table 4.
Table 5.
Table 6.
Two-graphs of order 4, ••• ,9
Cospectral two-graphs of order 4, ••• ,9
Integral and transitive two-graphs of order 4, ••• ,9
Two-graphs with nonzero gamma of order 6,8,10
Regular two-graphs of order 16,28,36
Transitive strongly regular graphs associated with regular
two-graphs of order 16,28,36
I
3
9
14
19
21
25
27
48
49
50
62
75
Table
7.
Conference two-graphs of order 10,26,50
77
Table 8.
The 7
x7 Graeco-Latin squares, and the known S(2,4,25)
81
Table 9.
Conference two-graphs of order 6,14,18,30,38,42,46
83
Table 10. Block-valency matrices for conference two-graphs of order
94
Table 11
30,38,42
Examples of block-cireulant matrices for conference
two-qraphs of order 30,38,42
CHAPTER 1, INTRODUCTION
The first objective of the present report is to obtain table 1 of
all nonisomorphic two-graphs on n
~9 vertices, and table 4 of all
non-isomorphic two-graphs on n
~10 vertices whose full automorphism group
does not fix any graph in its switching class
(y#
0). The numbers of
these two-graphs are as follows:
n
4
5
6
7
8
9
10
y
=
e
==
0
3
7
14
54
224
2038
32728
y
#
0,
e
==
0
0
0
2
0
17
0
392
y
#
0,
S
=F
0
0
0
0
0
2
0
0
total
3
7
16
54
243
2038
33120
Here
y
and 8 denote the first and second cohomology invariant in the sense
of Cameron [6J.
The second objective is to obtain the tables 5,7,9 providing
infor-mation about the known regular two-graphs on n S 50 vertices. Their numbers
N(n) are as follows:
n
6
10
14
16
18 26
28
30
36
38
42
46
50
IS
3m 3m5
3
129
5
IS
3m
s m 5
9
129
7
1
1
1
'1
4
1
6 91
11
18
80
18
Here
P1
and
P
2
denote the eigenvalues of the regular two-graphs. A bar
denotes that N(n) is complete.
Certain strongly regular graphs are associated with a regular
two-graph. As an illustration of cospectrality, and of triviality of
auto-morphism groups, we quote from table 5 the numbers of nonisomorphic
strongly regular graphs associated with the 91 regular two-graphs of
order 36:
(36,5,5,-7)
(36,-7,5,-7)
(35,-2,5,-7)
from 5(2,3,15)
16111
57
1817
337
48
38
total
16448
105
1853
trivial Aut
15417
28
1576
Two-graphs are introduced in chapter 2. Following Cameron [6J, the
relations between two-graphs, switching classes and Euler graphs are
explained in terms of vector spaces over
F
2
, which leads to the
enumeration formulae 2.3 and 2.4, and to the Mallows-Sloane equicardinality
theorem 2.2. The construction of the tables 1,2,3 is briefly discussed,
and some comments are given. In chapter 3 the automorphism group of a
two-graph is compared and confronted with the automorphism groups of the
graphs in the corresponding switching class. Thus the significance
of the nonvanishing of the first invariant
y
is explained. The generation
of the two-graphs with y
#
0 is indicated, and some comments.on the
results in table 4 are given.
The regular two-graphs on n s 50 vertices are distinguished into
the regular two-graphs with P
1
#
-P2'
the conference two-graphs with
integral, and those with irrational eigenvalues, which are discussed
in chapters 4,5,6, and tabulated in tables 5,7,9, respectively. The
emphasis in chapter 4 is on the regular two-graphs with n
=
36; the results
of [4J are extended in the tables 5 and 6. The data of table 8 are used
for the construction of the conference two-graphs with n
=
50, the main
subject in chapter 5 and table 7. The known results [15J about the case
n
=26 are also included. The methods of construction for the conference
matrices with n
~30,38,42,46 are described in chapter 6. They are based
on partitioning of the adjacency matrix in various ways, and on filling
the blocks by circulants or multicirculants. The actual results are
obtained by use of the tables 10 and 11, and are collected in table 9.
The theory of two-graphs is surveyed in [17J and [19J. Earlier tables
are contained in [11J, [4J,
[15J, [lJ, [22J,
[7], of which the present
tables constitute a self-contained extension.
-
3
-CHAPTER 2. TWO-GRAPHS AND SWITCHING CLASSES
A two-graph (Q,8) is a pair consisting of a finite set Q and a set 8
of 3-subsets of Q, such that each 4-subset of Q contains an even number
of 3-subsets from 8. The elements of Q are the vertices, those of 8 are
the triples of the two-graph (Q,8). For any
W EQ the set 8 is determined
by its triples containing w. Indeed,
{W
1
'W
2
'W
3
}
E8 whenever an odd
number of the remaining 3-subsets from
{w,w
1
'w
2
,w
3
}
belongs to 8.
The complement of the two-graph (Q,8) is the two-graph (Q,6), where 8
Q
is the component of 8 in (3)'
Any graph (Q,E) gives rise to a two-graph as follows. Let 8 be
the set of the 3-subsets of Q which carry an odd number of edges from
the edge set E. Then (Q,8) is a two-graph. Indeed, for any graph on 4
vertices there is an even number of subgraphs on 3 vertices having an
odd number of edges. Conversely, with any given two-graph (Q,8) there
is associated a class of graphs on Q, each of which gives rise to (Q,8)
in the way described above. Indeed, for any disjoint partition
such a graph is defined by the edges
The class of all such graphs is called the switching class associated
with the two-graph (Q,8). The graph in the switching class defined by
Q
O
=
{w},
Q
1
=
~,Q
2
=
Q\{w}
is called the descendant with respect to
Wi
this graph has w as an isolated vertex. Associated with the void
two-graph
(Q,~)is the switching class consisting of all bipartite
graphs on Q. Clearly, the
s~perposltionmod 2 of any graphs (Q,E) and
(Q,F) in the switching class of any two-graph (Q,8) is a complete
bipartite graph on Q.
Essentially, in the above the set
n
is labeled. The labeled
graphs on
n
are represented by the elements of the vector space V
over F2 with standard basis
(~).
The complete bipartite graphs on
n
constitute a subspace B of dimension
Inl -
1. The switching classes
on
n
are the elements of the quotient space
v/B.
The Euler graphs
on
n
(the graphs having even valency at each vertex) constitute
another subspace Z, whose intersection with B has dimension
a
for odd
Inl,
and dimension
Inl -
2 for even
Inl.
The standard inner product
of any pair of graphs equals the number of their common edges mod
2.
Under this inner product the subspaces
a
and Z are maximally orthogonal.
Hence, as a consequence of a theorem from linear algebra (cf. [10],
*
p. 91), the dual vector space Z is isomorphic to
via.
The labeled two-graphs are represented as elements of the vector
n
space over F2 with standard basis
(3)'
The defining property expresses
that these elements have vanishing coboundary, cf. [21J, [6], [19].
Hence the two-graphs constitute the subspace T of the cocycles. The
correspondence between switching classes and two-graphs, mentioned earlier,
is now expressed by the isomorphism of
Via
and T, a consequence of the
triviality of the cohomology. Summarizing we have the following theorem.
*
Theorem 2.1. Z
~via
~T ,
where V,B,Z,T are the F
2
-vector spaces of the labeled
graphs, complete bipartite graphs, Euler graphs, and
two-graphs on
Q,
respectively.
By letting the symmetric group of
n
act on these vector spaces, we
obtain the theorem of Mallows and Sloane [12J, in the proof setting of
Cameron [6].
'rheorem 2.2. The isomorphism classes of the Euler graphs, the two-graphs,
and the switching classes on n vertices are equal in number.
5
-The following theorems yield explicit formulae, which are due to Robinson
[16J and to Goethals [8J, cf. Cameron [6J. Any permutation
a
of 0 extends
to a linear transformation
a
of V. The elements of V which are fixed under
a
constitute the kernel of the linear transformation
a
+
1.
Since
a
+
1
leaves B invariant, it acts on v/B. Let
2
v (a),
2
b
(a),
2
t (a), and
2
c
(a)
denote the number of fixed elements under
a
in its action on
V,
B, v/B,
and any coset C of B, respectively. Each coset of B fixed by a contains a
b(a)
.
fixed vector [12J apd hence exactly 2
f~xedvectors. It follows that
b(a) ::::
c(a) ,t(o)
=
v(a) - b(a) .
Applying Burnside's lemma we obtain:
Theorem 2.3. The number of nonisomorphic two-graphs on n vertices equals
1
n!
ad
n
2v (a)-b(a)
Theorem 2.4. The number of nonisomorphic graphs in the switching class
of a two-graph with automorphism group G equals
The interpretation of veal and b(a) in terms of the action of the
permutation
a
on the complete graph On 0 is the following, cf. [6J. v(a)
is the number of edge-cycles, and bea) is the number of vertex-cycles if
every cycle has even length and one less otherwise.
As a consequence of the isomorphism of v/B and T we can provide any
two graph with a spectrum of eigenvalues and multiplicities. For any
labeled graph (O,E), let A denote its (-1,1) adjacency matrix, defined
by the entries a
ii
:::: 0;
a
ij
:::: -1 for {i,j}
€Ei
a
ij
=
1 for {i,j}
t
E.
The switching class of (O,E) is represented by the adjacency matrices
DAD, where D runs through the diagonal matrices of order n with diagonal
entries
±
1. Clearly, A and DAD have the same spectrum. The spectrum
of a two-graph is defined to be the spectrum of the (-1,1) adjacency
matrix of any graph in its switching class.
This leads to the following geometric interpretation of two-graphs
and their switching classes of graphs. Let the (-l,l)-adjacency matrix
A have smallest eigenvalue
-P,
with multiplicity n - d. Write cos
~=
IIp,
o
~ ~ ~ ~/2,and
pI + A
=
HHt
where H is an n
xd matrix of rank d.
The rows of H denote n vectors in
d ~
m
of equal length
pand at angles
~and
~-
~,according as the
corresponding vertices of the graph are nonadjacent and adjacent,
respectively. The n lines spanned by these vectors are equiangular, that
is, the angle between each pair of lines equals
~.Thus, a two-graph
is represented by a set of equiangular lines in Euclidean space, and any
graph in the corresponding switching class is represented by a set of
unit vectors, one along each line.
Conve~sely,any dependent set of
equi-angular lines defines a unique two-graph, cf. [17J, theorem 5.4. These
notions are to be distinguished from the set of all unit vectors along
equiangular lines, which corresponds to a double covering of the complete
graph K • In a double covering of K each vertex is replaced by a pair of
n
-
n
vertices, and two pairs are joined either direct or skew, cf. [6J, [19J.
The correspondence is established by letting vectors at acute (oblique)
angles correspond to vertices which are joined direct (skew).
In table 1 the isomorphism classes of the two-graphs on n
~9 vertices
are listed (those with n
~7 already occur in [11J). The letter s in the
last column indicates that the corresponding two-graph is self-complementary,
that is, isomorphic to its complement. If not, then only one of
(~,~)and
(n,Z)
is listed. Each two-graph in the table is given by the
~(n- 1) (n - 2)
relevant entries of the adjacency matrix of a graph in its switching class
which contains an isolated vertex. These entries are listed as the coordinates
of a vector which is the concatenation of the rows of the right upper part
of the adjacency matrix. For each two-graph, their eigenvalues are given,
7
-to 3 decimal positions. For each n, the set of the two-graphs on n vertices
is ordered lexicographically according to their eigenvalues, ending with
the complete two-graph on n vertices. Furthermore, for each two-graph
the order of its automorphism group, and the number of the nonisomorphic
graphs in its switching class is given.
Table
2
contains the cospectral two-graphs, that is, sets of
non-isomorphic two-graphs which have the same spectrum. The two-graphs having
an integral spectrum and those having a transitive automorphism group are
collected in table
3.
The two-graphs of table
1
have been generated by an exhaustive
back-tracking search. Two different strategies were adopted, depending on whether
the number n of vertices is odd or even.
For n
=
2k
+
1
the switching class of any two-graph contains a unique
Euler graph
([18]
theorem
3.5,
cf. also chapter
3).
The search proceeds in
two stages. In the first stage all feasible vertex-degree sequences are
generated (for n
=
9
there are
107
such sequences). In the second stage
we generate in lexicographical order all graphs associated with each
vertex-degree sequence. Each new graph is put into canonical form, in order to
carry out a final isomorph rejection. The canonical form of a graph on n
vertices with (l,O)-adjancency matrix A is the vector
t
max a(P
AP) IPeP
where a(A) is the vector obtained by concatenating the rows of the right
upper part of A,
P
is the set of all permutation matrices of order
n ,
and
the order is the lexicographic order for vectors.
For n
=
2k we start from the two-graphs of order 2k
+
1, and generate
all nonisomorphic descendants (deleting a vertex isolated by switching).
These are then put into canonical form as two-graphs, that is, they are
written as vectors
min
a(Q~(J-I-2A)Q)
Q
EQ
where
Q.
is the set of all permutation matrices of order n, having nonzero
entries
±
1.
These vectors are added to the list in case they have not
been generated before.
From the tables we read the following numbers of nonisomorphic
two-graphs of order n, selfcomplementary two-two-graphs, two-two-graphs with integral
spectrum, two-graphs having a transitive, and a trivial automorphism
group, the numbers of cospectral pairs, triples, and quadruples of
two-graphs (the transitive two-graphs have been determined by a special
program) •
n
4
5
6
78
9
two-graphs
3
716
54
243
2038
self-compl.
1
1
4
0
19
10
spec
€ Zl2
2
4
8
9
16
tra Aut
2
3
5
4
6
9
tri Aut
0
0
0
0
8
264
cosp. pairs
0
0
0
0
6
'160
cosp. triples
0
0
0
0
1
22
cosp. quadr.
0
0
0
0
0
2
~.
After having finished this Report, the authors became aware of
T. Sozanski, Enumeration of weak isomorphism classes of signed graphs,
J.
Graph Theory, to be published.
This paper contains a Burnside-type formula for the number of
self-complementary two-graphs.
9
-CHAPTER 3. AUTOMORPHISM GROUPS OF TWO-GRAPHS
An automorphism of the two-graph
(Q,~)is a permutation of
Q
which
preserves 8, that is, which preserves the corresponding switching class
C of graphs. Clearly, the full automorphism group Aut(c) of any graph
c in the switching class C is a subgroup of the full automorphism group
Aut (C) of the switching class C, so
Aut (c)
<
Aut (C)
Ifor all c
EC •
In terms of the F2 vector space V of all graphs this reads as follows.
Any permutation a of
Q
induces a linear transformation a of V which
leaves invariant the space
B
of all complete bipartite graphs. An
automorphism
a
of the switching class C is characterized by the property
(a
+
l)c
EB ,
for all c
EC ,
and an automorphism a of the graph c by
(a
+
l)c
=
0 •
We apply Lagrangets theorem to Aut(c)
<
Aut(C) in two different situations.
First, for any w
€Q,
let c
denote the descendant with respect to
w
W,
that is, the graph in the switching class C which has w as an isolated
vertex. The orbit of c
under Aut(C) is determined by the orbit of wunder
w
Aut(C).
Theorem 3.1. IAut(C)
I
=
IAut(c )
I .
lorbit of
W EQ
under Aut (C)
I .
WSecond, for any graph c in the switching class C
=
c + B, and any
a
EAut(C), we have
a(c)
=
c
+
b
for some b
EB. The orbit of c under
cr
a
Aut (C) is determined by the orbit of
a
EB under Aut (C) , that is, by the
orbit on B under Aut(C) as an affine transformation of B.
Theorem 3.2. IAut(C)
I
=
IAut{C)
I •
lorbit of cr on B under Aut(C) I •
As a consequence we have, cf. [17J theorem 4.7:
Theorem 3.3. Let c
1
, .•• ,c
s
denote the nonisomorphic graphs in the
switching class
C.
Then
s
I
2
n-l
•
i::::1
We now approach the key problem of the present chapter. Mallows and Sloane
([12J, cf [6] theorem 3.4) proved that any automorphism of a switching
class is an automorphism of some graph in the switching class, that is,
However,
~~efollowing property
(*)
is not generally
~for any switching
class
C
and any
G
<Aut(C):
3
V
«a
+
1)
c
=
0) •
eEC
aEG
If this property holds for G
=Aut(C) , then Aut(C)
=Aut(c) for some c
EC.
However, there exist switching classes C whose Aut(C) is larger than
Aut(c) for each c
EC. In fact, this makes switching classes and two-graphs
interesting, and this is why they have been invented. They provide
combinatorial structures whose Aut
ma~'go beyond the Aut of their graphs
ithey even may have 2-transitive automorphism groups. It is the aim of the
present chapter to determine all two-graphs on n
~10 vertices whose full
automorphism group does not satisfy the property
(*)mentioned above.
Cameron [6J phrased the validity of property
(*)
for a switching class
1
C and G
<Aut{C) in terms of the one-dimensional cohomology group H (G,B).
Indeed, for any c
+
b
EC the derivation d : G
+B, defined by
d (a) = (a
+
1) (c+
b) ,a
EG ,
reduces to an inner derivation
deal
=
(a
+
l)b ,
a
EG ,
whenever c satisfies
(a +
1)c
=
0 for all
a
EG. This implies that the
switching class C defines an element
y
EH
I
(G , C), called the first invariant
of C and G, and
y
=
0 if and only if property
(*)
holds.
11
-Cameron [6J also introduces the second invariant
a
of C and G, an element
2
of the two-dimensional cohomology group H
(G'~2)'He shows that
a
=
a
if
and only if G can be realized as a group of automorphisms of the double
covering associated with C. We shall not enter further details, since
for n
~10 there exists only one pair of complementarv two-araohs with
B
~O. The switching class of one of these contains the octagon graph. For
details we refer to [19J •
In our search for all two-graphs on n
~10 vertices with
y
~0, and
those with
S
~0, the following necessary conditions, taken from [6J, will
prove useful.
Theorem 3.4. Let C be a switching class of graphs on n vertices, and let
G
<
Aut(C). Suppose
y
~0, that is, suppose there is no graph
c
€C with G
<Aut(c). Then
Ca) n
°
(mod 2) ,
(b)
IGI -
a
(mod 4),
(c) all orbits of G on
n
have even size.
Theorem 3.5. Let C be a switching class of graphs on n vertices, let
G
<Aut(C), and let
a
~O. Then
(a) y
~0 ,
(b) n - 0 (mod 8) ,
(c) the largest and the smallest eigenvalue of C have even
multiplicity.
In table 4 the two-graphs with
y
~0, n
~10 are displayed. As before,
the self-complementary two-graphs are indicated by the letter s, and from
the remaining two-graphs only one of
(n,~)
and (n,X) is listed. Those with
n
=4,6,8 were selected from table 1 on the basis of !Aut(c)
I
<
IAut(C)
I
for
y
~0, and theorem 3.5(c) for
B
~0.
The two-graphs with
y
F
0, n
=
10 were generated by a backtracking
search. The underlying idea for an efficient search is the experimental
fact that the switching classes contain only few graphs having a minimum
number of edges, and that these graphs have relatively large automorphism
groups. In fact, most often such a graph c has !Aut(C)!
=
IAut(C)! and
hence is rejected. Together with the criteria of theorem 3.4, this reduces
the number of candidates for the switching classes with
y
F
0 by a factor
of about 50. There are 471 valency sequences (all vertex-degrees
~4),
yielding 24423 graphs. From these, only 271 satisfy the necessary
conditions. The latter are put into canonical form and are tested for
isomorphism. This leads to 206 non-isomorphic two-graphs with
y
F
0, up
to taking complements. Since 20 of them are selfcomplementary, we finally
arrive at the total of 392 two-graphs with y
F
0, n
=10.
Summarizing we find the following numbers of nonisomorphic two-graphs
on n
~10 vertices having the indicated properties for the invariants
y
and
S.
n
4
5
6
7
8
9
10
Y
B
=
0
3
7
14
54
224
2038
32728
Y
F
0,
B
0
0
0
2
0
17
0
392
Y
F
0,
B
F
0
0
0
0
0
2
0
0
total
3
7
16
54
243
2038
33120
For n
=
6 there are 2 two-graphs with y
F
0, both selfcomplementary.
One is represented by the switching class of the pentagon graph with an
additional isolated vertex and has the alternating group on 5 symbols
as its full automorphism group. The other is represented by the path of
length 4 with an additional isolated vertex and has the Klein group as
its full automorphism group.
For n
=
8 there are 2 complementary two-graphs with
B
F
0, listed
under identification number
~6.
One is represented by the switching
class of the octagon graph and has the holomorph of
Za
as
its full
automorphism group, cf. [19J. The 17 two-graphs with
y
F
0,
B
=
0 fall
into 8 complementary pairs and one selfcomplementary two-graph. The
13
-two-graphs
:#
1, # 2,
:#
7, # 10 have the same I Aut
I,
number of graphs,
and numbers [x,yJ
=(x graphs with IAutl
=
y),
and so do the two-graphs
#3, #8, #9.
For n
=10, Y
~0, the 206 two-graphs listed in table 4 are ordered
following their eigenvalues; this shows that there are many cospectral
pairs, and even 2 cospectral triples, namely # 27, # 28, # 29 and # 151,
# 152, # 153. Those with integral spectrum are ## 10, 11, 191, 193.
The two-graph #191, represented by the Petersen graph, has the largest
automorphism group, namely Sp(4,2)
~L6 of order 720. As above for n
=8
the 206 two-graphs of table 4 may be grouped into sets of two-graphs
having the same
i,I
Aut
I,
number g of graphs, and numbers [x, y J
==
(x graphs with IAutl
=
y).
These sets Si' i
=
1, ••• ,18, ordered according
to their IAutl, are the following.
1
Isil
IAutl
g
[x,yJ
1
1
720
10
[2,120J, [2,72J, [2,12J, [4,8J
2
4
144
48
[12,72J, [2,36J, [8,24J, [12,12J, [12,8J, [2,4J
3
10
64
60
[12,32J, [14,16J, [18,8J, [14,4], [2,2J
4
1
60
20
[2,10J, [4,6], [2,3J, [10,2J, [2,lJ
5
3
48
34
[1,24J, [4,12J, [7,8J, [1,6J, [7,4J, [13,2J, [l,lJ
6
8
32
70
[16,16J, [18,8J, [23,4J, [12,2J, [l,lJ
7
4
24
62
[12,12J, [2,6J, [20,4J, [26,2J, [2,lJ
8
28
16
102
[32,8J, [32,4J, [36,2J, [2,lJ
9
16
16
100
[32,8J, [28,4J, [38,2], [2,lJ
10
3
16
82
[20,8], [30,4J, [20,2], [12,lJ
11
6
16
68
[8,8J, [20,4J, [28,2J, [12,1J
12
1
12
56
[8,3], [16,2J, [32,1J
13
12
8
116
[32,4J, [56,2J, [28,1]
14
8
8
116
[40,4J, [44,2J, [32,1]
15
4
8
108
[28,4J, [46,2J, [34,1J
16
56
4
176
[96,2], [80,1]
17
31
4
168
[80,2], [88,1]
18
10
4
152
[48,2], [104,1]
CHAPTER 4. REGULAR TWO-GRAPHS WITH P 1
+
p2
'#
0
We give two equivalent definitions for regular two-graphs.
A two-graph
(Q,~)is regular whenever each pair of elements of
Q
is contained in the same number of triples of
~.A two-graph is regular whenever it has only two eigenvalues, that
is, whenever its switching class consists of graphs whose (-1,1) adjacency
matrix A satisfies
The equivalence of these definitions is seen as follows, cf. [17J. For
any adjacent vertices x and y of a graph, let p(x,y) denote the number
of vertices which are adjacent to x and nonadjacent to y. For any
non-adjacent vertices u and v, let q(u,v) denote the number of vertices
which are adjacent to u and nonadjacent to v. For any graph in the switching
class of a regular two-graph, either definition amounts to the independence
of p(x,y)
+
p(y,x) and q(u,v)
+
q(v,u) of the choice of {x,y} and {u,v},
respectively, and
n
1 - PI P2 '
p(x,y)
+
p(y,x)
-1:1
(p1 -
1)
(p2 -
1)
q(u,v)
+
q(v,u)
=
-1:I(Pl
+
1) (P2
+
1)
Also a
~tEon2lyre~~r ~ra2hadmits two equivalent definitions, cf. [20J.
It is a regular graph for which p(x,y)
=
p(y,x) and q(u,v)
=
q(v,u) are
independent of the choice of {x,y} and {u,v}. Equivalently, its (-1,1)
adjacency matrix C of size n satisfies
with 01
>
02' say. The numbers (n,00,01,02) are the parameters of the
strongly regular graph. They satisfy
-
15
-The switching class of a regular two-graph may contain strongly regular
graphs. Their (-1,1) adjacency matrix A should satisfy
The descendent of any vertex of a regular two-graph on n vertices yields
a strongly regular graph on n - 1 vertices, after removal of the isolated
vert:ex. Its (-1,
1)adjacency matrix B satisfies
The eigenvalues of regular two-graphs are subject to restrictions.
From trA
=0 and trA
2
~
n - 1 it follows that PI and P
2
are odd integers
if Pl
+
P
2
F
0, cf. [17J. In the present chapter we investigate nontrivial
regular two-graphs with n
~50 and eigenvalues satisfying P1
+
P2
F
O.
up to taking complements there are 3 possibilities, namely
n
=
16, PI
=3, P2
=-5; n
=
28, P
l
=
3, P
2
=-9; n
=36, PI
=
5, P2
=-7.
For n
16 there is only one regular two-graph, namely the complement
of the symplectic two-graph E(4,2), cf. r17]. Its switching class contains
strongly regular graphs with valencies 6 and 10. Valency 6 is realized by
the lattice graph L
2
(4) and by the exceptional Shrikhande graph. valency
10 is uniquely realized by the Clebsch graph. All descendants yield the
triangular graph T(6), after removal of the isolated vertex.
For n
=28 there is only one regular two-graph, namely the complement
of the orthogonal two-graph Q-(6,2), whose automorphism group is isomorphic
to Sp(6,2). Its switching class may contain strongly regular graphs with
valencies 12 and 18. Valency 12 is realized by the triangular graph T(a)
and by the 3 exceptional Chang graphs. Valency 18 is impossible, cf. [20J.
All descendants yield the Schlafli graph on 27 vertices, after removal of
the isolated vertex.
For n
=
36 a total of 91 nonisomorphic regular two-graphs is known,
cf. [4J. We conjecture that no further such two-graphs exist. The possible
parameter sets for strongly regular graphs in the switching class are
(36,5,5,-7) and (36,-7,5,-7), with valencies 15 and 21, respectively. From
table 5
i t
follows that the first set is realized by 16448 graphs, and
that the second set is realized by 105 graphs. By deleting an isolated
vertex we obtain 1853 strongly regular graphs with parameters (35,-2,5,-7).
We first discuss the generation and the analysis of the known regular
two-graphs on 36 vertices in more detail.
A Steiner system S(2,k,v) 1s a collection of k-subsets (called blocks)
of a v-set V such that every 2-subset is contained in exactly one k-subset.
From S(2,k,v) the Steiner graph
S~(v)is obtained as follows. The verti.ces
are the b
=
v(v - l)/k(k - 1) blocks, and two vertices are adjacent whenever
the corresponding blocks have one element in common. In particular,
St
k
(2k
2
-
k) extended by an isolated vertex belongs to the switching class
of a regular two-graph with parameters
2
n
=
4k
,
P
1
=
2k -
1 ,
P
2
=
-2k - 1 •
A transversal design TD(k,v) is a collection of k-vectors with coordinates
in a v-set V such that for each pair of coordinates every pair of elements
occurs exactly once. A TD(k,v) is equivalent to a set of k - 2 mutually
orthogonal Latin squares of size v. From TD(k,v) the Latin square graph
2
~(v)
is obtained as follows. The vertices are the v
vectors, and two
vertices are adjacent whenever the corresponding vectors agree in one
coordinate. In particular,
~(2k)belongs to the switching class of a
regular two-graph with parameters
2
n
=
4k
IPi
=
2k - 1
tP2
=
-2k -
1 •
The known regular two-graphs on 36 vertices are constructed either from
Steiner triple systems 8(2,3,15) or from Latin squares TD(3,6) of order 6.
There are 80 nonisomorphic S(2,3,15) and 12 nonisomorphic TD(3,6). These
yield 80
+
11
91 nonisomorphic regular t.wo-graphs with n
=
36,
Pi
=
5,
P2 "" -7, of Steiner
~and Latin squa:t'e
~Irespectively. Indeed, in
[4J it was shown that only two of the Latin square graphs are switching
equivalent.
I
- 17
-In order to count the strongly regular graphs with parameters
(35,-2,5,-7) we use the following consequence of theorem 3.1.
Theorem 4.1. Let the action of
Aut(S2,~)on Q have the orbits nl, ••• ,Qs'
The descendants c
and c • are isomorphic iff the corresponding
w
w
vertices wand w' are in the same orbit, and
In order to count the st:rongly regular graphs with parameters (36,5,5,-7)
and (36,-7,5,-7) we use the following consequence of theorem 3.2. Let
p E
{P
1
,P
2
}
be one of the eigenvalues of the regular two-graph
(Q,~).Let
c, with adjacency matrix A, be any graph in
the
switching class of
(Q/A).We wish to switch
cinto a regular graph with
Po ::
p.
This amounts to a
search for eigenvectors d, with entries
±
1, to the eigenvalue p of A.
Indeed,
(A - pI)d= 0
iff
DADj :: c j , D
~diag(d) ,
and DAD is the adjacency matrix of a regular graph, to be denoted by Cd'
Let
P
denote the set of all such eigenvectors d.
p
Theorem 4.2. Let the action of Auten/A) on
Vp
have the orbits
Vl/ ••• ,V
t
•
The strongly regular graphs Cd and Cd' are isomorphic iff
the corresponding eigenvectors d and d' are in the same orbit,
and
IAut(Q,fI)I
if d
EVi .
For a given matrix A and eigenvector P, we construct the set
V
by using
p
the backtracking procedure Eigenvector of Paulus [15].
Table
contai.ns the information about the regular two-graphs with
n
=
16,28,36, and their strongly regular graphs. Each two-graph is
re-presented in canonical form by use of the upper triangle of the (-1,1)
adjacency matrix A of a descendant. We use octal representation, that
iS
Iwe
con-vert binary vectors into octal numbers (for instance, the vector
-++j+_+I+ __
into the octal number 423). For each two-graph (n,A), the table gives
the order and the generators of
Aut(n/~),and the automorphism partition.
Furthermore, the total number of strongly regular graphs with a given
parameter set , and their distribution according to the order of their
automorphism group is given.
In table 5, the numbering of the two-graphs with n
=
36 is the
same as in [4],
#- 1 through
7ft
80 being of Steiner type, and
#
81 through
#92 being of Latin square type (#87 does not occur since the two-graphs
# 87 and # 99 are isomorphic; we choose #- 99 since the corresponding Latin
square graph has an automorphism group which is 3 times larger than that
of
7ft
97). We mention .in passing that the Steiner graphs and the Steiner
triple systems agree in the order of their automorphism groups, except
for # 1 where a factor 2 occurs.
We make some observations concerning table 5. The automorphism groups
of the two-graphs ## 1,89,90,92 act transitively on the vertex set
Q.
The groups of the two-graphs ##8 through 90 and 91 through 92 are the
same as the groups of the corresponding Steiner and Latin square graphs,
hence these two-graphs all have
y
=
O. For # # 1 through 7 these groups
differ by a factor 36,8,2,2,2,2,2, respectively. This makes it conceivable
that the two-graphs # # 1 through 7 have
y ,.O.
All Steiner graphs can be switched regular with valency 15, but only
23 of them can be switched regular with valency 21 (namely # # 1 through
22 and #61). All Latin square graphs can be switched both IS-regular and
21-regular. A total of 36 (out of 91) two-graphs has trivial automorphism
group. For the strongly regular graphs (36,5,5,-7), (36,-7,5,-7), (35,-2,5,-7)
the totals having trivial automorphism group are 15417 (out of 16448),
28 (out of
lOS), 1576 (out of 1853), respectively.
In table 6 all transitive strongly regular graphs are listed which
are associated with the regular two-graphs under consideration. The
2-+
transitive regular two-graph n (6,2), #- 1 in the tables 5 and 6, contains
three rank 3 graphs in its switching class, cf. [4J, namely (36,5,5,-7)
with group 0-(6,2), (36,-7,5,-7) with group G(2,2), and (35,-2,5,-7) with
group L(4,2). The two-graphs##-89,90,92 of Latin square type yield
transitive strongly regular graphs with parameters (36,5,5,-7) and
(36,-7,5,-7); these are not rank 3 graphs.
-
19
-CHAPTER 5. CONFERENCE TWO-GRAPHS WITH INTEGRAL EIGENVALUES
A conference two-graph is a regular two-graph on n vertices having
eigenvalues P1
=
In - 1, P
2
= -
In - 1. With each conference two-graph
there is associated a switching class of symmetric conference matrices.
A conference matrix A of order n is an n
x
n matrix A, having zero's on
the diagonal and entries
±
1 elsewhere, and satisfying AAt
=(n - 1)
I.Conference matrices have been introduced by Belevitch [2J in connection
with the construction of networks for conference telephony. For the
existence of a conference two-graph on n vertices it is necessary that
n :: 2 (mod
4)and that n - 1 is a sum of squares of two integers, cf.
[11].
In the present chapter we shall discuss conference two-graphs with
parameters
n
=
(2k - 1)2
+
1,
P
1
=
-P
2
=
2k - 1,
k
~2 •
Such two-graphs can be constructed from Steiner systems and from transversal
designs. It is easily checked that both St
k
(2k
2
- k
+
1) and L
k
(2k - 1)
u {w}
where
w
is an isolated vertex, belong to the switching class of a conference
2
two-graph of the order n
=
(2k - 1)
+
1, cf. [17J and chapter
4.
Further-more, the complement of any conference two-graph again is a conference
two-graph.
We recall two further constructions for conference two-graphs, the
first of which (to be called the PN construction) is due to Goethals and
Delsarte and to Turyn
(unpublished, cf. [17J). Let V
=
V(2,q) denote
the vector space of dimension 2 over GF(q), where q is an odd prime power.
Partition the one-dimensional subspaces into any two disjoint sets P and
N of equal size
~(q+
1). Then a selfcomplementary conference two-graph
(V U
{oo},~)
on q2
+
1 vertices is defined by
v
V ( ( {00,
x,
y}
E ~) : ~«x -
y>
E P)) •X,YE
Let D : V
x
V
~GF(q) be a nondegenerate bilinear form on V, and let
q - 1 (mod 4). The Paley two-graph
(n,~)with parameters
is defined as follows.
Q
is the set of the projective pOints of PG(l,q),
and 6 is the set of all triples {x,y,z} from
Q
for which D(x,y)D(y,Z)D(z,x)
is a non-square in GF(q). The Paley two-graph is selfcomplementary and
admits PSL(2,q) as a 2-transitive automorphism group.
There are 3 possibilities for conference two-graphs with integral
eigenvalues on n
~50 vertices, namely
n
=
10,
Pi
=
-P
2
3; n
=
26,
P
1
=
-P2
=
5; n
=
50,
P
1
=
-P2
=
7.
For n
10 there is only one conference two-graph, namely the
orthogonal two-graph Q+(4,2), whose automorphism group is Sp(4,2}. Its
switching class contains the Petersen graph and its complement St
2
(S)
=
T(5).
Deleting any isolated vertex we obtain L
2
(3).
For n
=
26 there are exactly 4 conference two-graphs, cf. Weisfeiler [22].
Two of these are constructed from Latin squares of order 5, and two from
Steiner triple systems of order 13. They have been investigated in [15J,
[lJ, [22J, [7]. They are displayed in table 7. All four are selfcomplementary.
They admit 15600, 72, 39, 6 automorphisms, respectively (the first case
corresponds to the Paley two-graph). The 4 switching classes contain 10
strongly regular graphs with parameters (26,-5,5,-5), one of which has a
trivial automorphism group. Deleting an isolated vertex we obtain 15
strongly regular graphs (25,0,5,-5), two of which have a trivial group.
For n
=
50 we have constructed a total of 18 conference two-graphs.
They arise from the 7 nonequivalent Graeco-Latin squares of order 7
(Norton [14J) and from the 4 known Steiner systems S(2,4,25) (Brouwer [3J).
These are displayed in table 8. Notice that one of Norton' • •
~elhas
been corrected. Following Brouwer there are no further S(2,4,25) with
IAutJ
z
5. Table 7 contains the pairwise complementary two-graphs
##3,4; 6,7; 8,9; 10,11, and the selfcomplementary #5, #1 (the Paley
two-graph), # 2 (the other PN two-graph) arising from the Graeco-Latin
squares, and the pairwise complementary ## 12,13; 15,16; 17,18, and the
selfcomplementary #14 arising from S(2,4,25). There ar.e, strongly
regular graphs (49,0,7,-7), derived from the 18 conferen . . tw8*,raphs by
deleting an isolated vertex. 36 of these have a trivial ._tomorphism
group. Due to difficulties with computer time, we have
only
partial results
concerning the strongly regular graphs (50,-7,7,-7)
1n the18.1tchlnq
classes. Each switching class contains at least one
stl'~y.,l"tlqul.rgraph
with these parameters.
-
21
-CHAPTER 6. CONFERENCE TWO-GRAPHS WITH IRRATIONAL EIGENVALUES
For conference two-graphs of order n, where n - 1 is not an integral
square, the following possibilities with n
~50 exist:
n
=
6,14,30,38,42,46 •
Notice, that the paley two-graph exists for all these orders, except
for n
=
46. However, many further conference two-graphs of these orders
have been constructed. Table 9 lists the known ones and their descendants
(n - 1, 0, In - 1,
-In -
1). For irrational eigenvalues there are no
strongly regular graphs on n vertices in the switching class.
For n
=
6 there is only one conference two-graph, namely n-(4,2)
with automorphism group Sp(4,2)
~AS" For n
14 and n
=
18 only the Paley
two-graph exists [18]. For n
=
30 there are 6 nonisomorphic conference
two-graphs known, and 41 descendants. This follows from the incomplete
backtracking search in
[1],
we have found no further two-graphs. For
n
38 we found 11 conference two-graphs and 82 descendants, and for
n
=42 we found 18 conference two-graphs and 120 descendants. For n
=
46,
there have been constructed 80 conference two-graphs, yielding 1856
descendants (1344 of which have trivial automorphism group). It is
interesting to observe that all known conference two-graphs of order 46
have small automorphism groups,
IAutl
~
10, and that none of them is
self-complementary. For n
=
38 we find nonisomorphic pairs of complemented
two-graphs, and for n
=
42 we find a non-Paley transitive two-graph.
Our construction method for n
=
30,38,42 is based on the assumption
that an essential part of the adjacency matrix A of the conference
two-graph can be written as an n
1
x
n
1
block matrix [AijJ, where each block
Aij is a regular n
2
xn
2
matrix with constant row and column sums r ij •
We consider such block-regular partitions for the matrices AO, A1,A2
defined by
~
jJ
0
1
0
1
jt
jt _jt
jt
A
=
AO
= =A1
j
jA2
j
- j2
and denote them by type 0,1,2. respectively. Since A • (n - 1)1, the
block-valencies r •. satisfy certain conditions. For example, for a type
1J
a
matrix we have
4(t
ii
+
r
1i
) - (n
2
- 1)(4s
i
- n + 2) + 4s
i
, 1 s i s n 1 •
4(t
ik
+
r
ik
) ... n
2
(2s
i
+
2s
k
- n
+
2) ,
l s i
<
k
~ n 1 '
n]
s.'"
i:
r ..
t 1 •I
1JJ-Similar relations hold for matrices of type
I
and 2. The search for
conference two-graphs with n - 30,38,42 proceeeds in two stages. In
the first stage we generate block-valency matrices which in
non-trivial cases are of a form exemplified by
a
b
c
c
c
c
d
e
f
a
c
c
c
c
e
d
f
g
h
i
h
j
j
k
g
h
i
j
j
k
g
h
j
j
k
g
j
j
k
R,m
n
R.
n
0
to be denoted by (a,b) (c,c,c,c) (d,e) (f) (g,h,i,h)
(j,j)(k) (R.,m) (n)
(0) •(the
square superblocks are circulants, and the nonsquare superblocks are
con-stants), In the second stage we attempt to fill each block A .. by a
circu-~J
lant matrix of the appropriate order and valency, by use of a systematic
backtracking procedure. The various block-valency matrices which have been
used are listed in table 10, and table 11 contains some of the
block-circu-lant matrices.
23
-For n
=
46 we follow the constructions of [13J. We consider graphs
in the switching class whose A is of type 1, where AI is partitioned
into 5
x5 regular blocks of size
9.
Define
~-J
~-j
[+j
~+j
D ..
- 0 -
E
=
+ -
+
F -
+ -
+
p •
~
0
+
- - 0
+ -
+
+ + -
+
0 0
BO -
~
Fp2
FPj
B] _
~2
FP
W
J
~P
EP
EP
D
:p
2
,
F
FP
,
B2 • :p2
Ep2
Ep2
.
Fp2 FP
FP
Fp2
F
E
E
We construct a basic set of 8 conference two-graphs by adding an isolated
vertex to the graphs which are represented by the adjacency matrices
BO
B] -B2
_BT
2
BT
1
BO -B I BI
BT _BT
2
)
BT
BO
BI
-B -B
T
_BT
BO -B
B2
BT
)2
2
I
I
2
A •
I
_BT
2
BT
t
BO
BI -B2
,
AI! -
BT _BT
2
] BO -B}
B2 '
T BT
BO
B)
B2
BT _BT BO -B)
-B -B
2
2
I
2
1
B)
-B -B
T BT
BO
-B
B2
BT _BT
BO
2
2
] )2
)
BO
B) -B
2
BT _BT
2
)
BO -B 1
B2 -B2
T BT
1
BT
)BO
B} -B2
_BT
2
_BT
I
BO -B)
B2
BT
2
AlII •
_BT
2
BT
}BO -B 1 B2
,
AlV -
BT _BT
2
1 BO
B] -B2
tT
T
BO
Bl
-B
BT
BT
BO -B I
B2 -B2 -B 1
2
2
)-B -B
1
2
BT
2
BT
t
BO
B)
B2
_BT
2
_BT
)BO
and their complements. From these 8 basic conference two-graphs we
obtain many more new two-graphs by permuting the diagonal blocks
([13J, theorem 4.2). Let
Q denote the permutation matrix
t
Let b
=
(b
1
, ••• ,b
S
) denote the socalled pointer vector, with integral
components b
i
=
±
k iff
l~i~5, l~k~3.
Hence each pointer vector b represents a permutation of the diagonal
blocks of A. In [13J, by an exhaustive search the admissable permutations
are determined as follows:
(1,-1,3,-2,3)
(1,2,2,-1,3)
(2,2,2,2,2)
(3,3,3,3,3)
(1,-3,-1,-2,-2)
(2,-1,3,1,2)
(2,2,-1,3,1)
(2,-3,-1,1,-3)
(1,-3,2,-3,-1)
(3,1,-1,3,-2)
(3,1,2,2,-1)
(3,-2,3,1,-1) •
The first column of block permutations is applied to AI' -AI' All' -All'
yielding 8 two-graphs with IAutl
=
10 and 8 two-graphs with IAutl
=
2.
If all 12 block permutations are applied to AlII' -AlII' AIV ' -AIV' then
another 48 conference two-graphs are obtained, all with IAutl
=
2.
Many more two-graphs can be obtained by permuting off-diagonal blocks
of the basic set as well. For example, putting
and applying the block permutations (3,3,3,3,3) to all matrices and
(2,-3,-1,1,-3), (1,-3,2,-3,-1) to AlII' -AlII' A
IV
' -A
rv '
we obtain
16 new two-graphs with jAutl
=3. The 40 pairs of complementary two-graphs
are displayed in table 9 .
25
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