Strongly Regular Graphs with Strongly Regular Decomposition
W. H. Haemers
Tilburg University
Tilburg, The h&he&n&
and
D. G. Higman
University of Michigan
Ann Arbor, Michigan 48109
Dedicated to Alan J. Hoffman on the occasion of his 65th birthday.
Submitted by Alexander Schrijver
1.
INTRODUCTION
AND PRELIMINARY
RESULTS
The title refers to strongly regular graphs r, which admit a partition
{ X,, X,} of the vertex set such that each of the induced subgraphs r1 and l?,
on X, and X, respectively is strongly regular, a clique, or a coclique. A
central role is played by the design D having point set X,, block set X,, and
incidence given by adjacency in r,. If rl is a clique or a coclique and r,, is
primitive, D must be a quasisymmetric design. If rl and r, are both
strongly regular, D is a strongly regular design in the sense of D. G. Higman
[14], except possibly when r, is the graph of a regular conference matrix.
Conversely, a quasisymmetric or strongly regular design with suitable parameters gives rise to a strongly regular graph with strongly regular decomposition. Moreover, if r, and rl are strongly regular with suitable parameters,
then r, must be strongly regular too. We give several examples and some
nonexistence results. We include a table of all feasible parameter sets up to
300 vertices. For most of the cases in the table existence or nonexistence is
settled. Some of the results in this paper are old, due to M. S. Shrikhande
[17], W. G. Bridges and M. S. Shrikhande [3], and W. H. Haemers [13].
We mainly use eigenvalue techniques. We need results on interlacing
eigenvalues (see [13]). Two sequences p1 > . . . 2 p,, and u1 > . . . 2 a,
LINEAR ALGEBRA AND ITS APPLZCATlONS 114,'115:379-398
(1989;
379
0 Elsevier Science Publishing Co., Inc., 1989
655 Avenue of the Americas, New York, NY 10010
0024-3795/89/$3.50
W. H. HAEMERS AND D. G. HIGMAN
380
( n > m ) are said to interlace whenever
for
pi 2 ui 2 Pn- n, + I
Interlacing
is tight if there exists an integer
P,- n,
RESULT 1.1.
i=l
,...,m.
k such that
pi = ui
for
i=l,...,k,
ui
for
i=k+l,...,m.
+
I
=
Let A, be a symmetric
matrix partitioned
as follows:
4, =
Let B be the 2 X 2 matrix whose entries are the average rozu sums of the
blocks of A,.
The eigenvalues of A, interlace the eigenvalues
(i) Cauchy interlacing.
of A,,. If the interlacing is tight, then C = 0.
(ii) The eigenvalues of B interlace the eigenvalues of A,. Zf the interlacing is tight, then A,, A,, and C have constant row and column sums.
Conversely, if A,, A,, and C have constant row and column sums, both
eigenvalues of B are also eigenvalues of A,.
Our main tool is the following lemma. It is a kind of mixture of Theorem
5.1 in [3] and Theorem 1.3.3 in [13] (J denotes the all-one matrix).
LEMMA1.2.
A,=
For i = 0,1,2
and
let Ai be a symmetric
A,C+CA,=K+/?J
vi X vi matrix
forsome
such that
a,pER
Let A,, A,, C, and C’ have constant row sums k,, k,, r, and k respectively.
STRONGLY REGULAR GRAPHS
For i = 0,1,2
denote
381
the eigenvalues
singular values of C by fi,...,K,
the pii’s and yj’s so that:
of Ai by pi, 1,. . . , pi, “,. Denote the
where m = rank C. Then we can order
(i) pl,l = k,, pz,l = k, with all-one eigenvector, y1 = rk, k, + k, = a +
/3v,/k,
and pe,r, p0,z are the roots of (x - k,)(x - k,) = rk.
(ii) pl, j + pz, j = a with eigenvectors in the range of C and CT, respectively, and P0.2,jT
PO,Zj-1 are the roots of (x - pl, j)( x - p2, j) = yj for j =
2 ,...,m.
(iii)
pl,i
has an eigenvector
in the kernel of C’,
pl, j = P”,“,+~, for
in the kernel of C, p2, j = pO,“, + j,
j=m+l,...,
vr; p2, j has an eigenvector
fm j=m+l,...,v2.
Proof.
We have
A,CCT=&CT+PrJ--CAaC’.
The right-hand side is a symmetric matrix; hence A ,CC T = CC ‘;1 r. So A 1
and CC T commute, and therefore they have a common orthonormal bases of
eigenvectors ur, . . . , uol (say), ordered so that A,uj = pl, juj for j = 1,. . . , vr,
CCTuj = yjuj for j = l,..., m,CCTuj=Oforj=m+l,...,v,,andu,isthe
a&one vector. Now the first two equations of (i) are obvious. Furthermore
for
A,CT~j=~CTuj+~Juj-CCA1uj=(~-P1,j)CTuj
j=2,...,m,
proving the first equation of (ii). Define
1
Yj"j
wj=i (~-p,,~)ck~
for
j = l,...,m.
Then it is easily verified that A,wj = lcwj whenever (r - pr, j)(x - pa,j) = y,.
Thus (i) and (ii) are proved. Next define
wj =
ui
i 0
1
for
j=m+l,...,v,.
Then A,wj = pr, jwj, proving the first part of (iii). The second part of (iii)
follows by interchanging A, and A,.
n
W. H. HAEMERS AND D. G. HIGMAN
382
We assume the reader to be familiar with the theory of designs and
strongly regular graphs. Some references are Beth, Jungnickel, and Lenz [l],
Cameron and Van Lint [9], and Seidel [16]. We recall some result about
strongly regular designs (see Higman [14]).
DEFINITION 1.3.
A design D with ui points and us blocks and incidence
matrix C is strongly regular whenever there exist graphs Ii and I, (not
complete or void) with adjacency matrices A, and A, respectively, such that
the following hold:
(i) CC’= w,Z + yiJ+ .ziA, for integers wi, yi, and zi (z, z 0),
(ii) C TC = w,Z + y,J + “aA, for integers wg, y2, and zs (zs # 0),
(iii) CC’C = yC + SJ for integers y and 6.
It is easily seen that C has constant row sum r = wi + yr and column
sum k = wg + y,, and that 6 = k(kr - y)/ul.
The graph l?i is the point
graph of D, and I’, is the block graph of D. It is straightforward that Ii
(i = 1,2) is strongly regular with eigenvalues
ki =
kr - yiui - wi
Y - wi
Pi =
‘i
zi
- w,
0., = 2,
of multiplicity 1, m - 1, and ui - m, respectively, where m = rank C. The
eigenspaces of the eigenvalues ui and ua are the kernels of C and C ‘,
respectively. (The point and block graph are determined up to taking
complements. To avoid this ambiguity one often requires that zi > 0. However, for our purposes it is not convenient to do so.) Bose, Bridges, and
Shrikhande [2] proved that (iii) may be replaced by:
(iii’) The singular values 6,.
yI=rk,
. . , qz
Yz=
of C satisfy
. . . =y,,,=y.
In case zr = 0, D is a quasisymmetric block design. A strongly regular
design is the same as a quasisymmetric special partially balanced incomplete
block design (see Shrikhande [Ml).
We finish this section with some notation. For a graph r,, ui denotes the
number of vertices, and the adjacency matrix is denoted by A,. If Ai has
eigenvalues p 1, . . . , pn with respective multiplicities ‘pi,. . . , (p,, we write
383
STRONGLY REGULAR GRAPHS
If Ii is regular,
write
the degree is denoted
spec
r, = { ki , rif’, sf’
by k,, and if I, is strongly regular, we
}
with
ri>O>si.
Throughout the paper r, denotes a graph decomposed into subgraphs Ii and
r,, that is, the respective adjacency matrices A,, A,, and A, satisfy
A,=
where C is the incidence matrix of some structure D (say). For regular I, the
decomposition
is called regular if also Ii and I, are regular. For strongly
regular I,, the decomposition
is strongly regular if Ii and I, are strongly
regular, a clique, or a coclique.
2.
THEORY
If I’, or the complement is the disjoint union of two or more cliques of
equal size, then I, is a so-called imprimitive strongly regular graph. In this
case the strongly regular decompositions
are obvious. Therefore we restrict
ourselves to a primitive
r,.
LEMMA 2.1.
Zf I, is strongly regular with a regular decomposition, then
Cl= (k, - k,)J,
CTl=(ko-k&L
A”1+ CC T = (r, + so)A, - ros,Z + (k, + q,sO)J,
A2, + C ‘C = (r, + so)A,
- rOsOZ-+ (k, + rOsO)J,
A,C + CA, = (r,, + sa)C + (k, + q,q,)J.
The first line reflects the fact that the decomposition is regular. If
Proof.
Ia is strongly regular, then A$ - (ra + sa)A, + r,s,Z = (k, + rOsO).Z.Thus the
m
block structure of A, gives the remaining formulas.
384
W. H. HAEMERS AND D. 6. HIGMAN
THEOREM
2.2.
Suppose I’, is strongly regular and r, is regular.
sg <
k,% - k,v,
vo - v1
Then
6 r,.
The decomposition
is regular if and only if equality holds on the left- or
right-hand side. Ifthe left-hand [right-hand ] inequality is met, then
k, = k, - k, + s0
Proof.
We apply Result l.l(ii).
[k,=k,-k,+r,]
The matrix of the average row sums,
k1
h-k,
(kc,-k,)u,/‘v,
kc,-(kc,-k,)v,/u,
has eigenvalues k, (row sum) and p (say). From k, + p = trace B it follows
that p = (k,v, - k,v,)/(
v0 - ul), which gives the desired inequalities. Equality on either side means that the interlacing is tight, and hence the decomposition must be regular. If the decomposition is regular, the eigenvalues of R
are k, and p = k, + k, - k,. These are also eigenvalues of A,; hence p = sg
n
or p = rO.
It is easily verified that if equality holds on one side, then the corresponding decomposition
of the complement of r, satisfies equality on the other
side. If rl is a coclique (i.e. k, = 0) the above result gives
This is Hoffman’s
THEOREM
2.3.
coclique
bound. Another bound is the following one.
If rl is a coclique
and r, is primitively
strongly regular,
then
vl i min{ f,, g,}.
Proof.
Define A = A, - vi- ‘(k, - sO)J - s,Z. Then rank A = fO. Since
A, = 0, A has a submatrix - v;‘(k,
- s,)J - s,Z of size zjl X vl, which is
nonsingular ( s0 # 0, since r, is primitive). Hence vuI< f,.Similarly we get
n
271 ~&I.
Theorems 2.2 and 2.3 are special cases of theorems
Cvetcovib [lo], respectively.
of Haemers
[13] and
STRONGLY REGULAR GRAPHS
38.5
Suppose F, and IF1 are strongly regular, let I?, be
THEOREM 2.4.
primitive, and suppose the decomposition is regular. Put E equal to 0 or 1,
according to whether the lej& or the right-hand side is tight in Theorem 2.2
(e.g. k, = k, - k, + ET,,+(l - E)s~).
Then one of the following holds:
(i) sr > sa, f1 < ra, vr < min{ f0 + 1 - s, g, + .s},
(iii) sl>so,
r,=ro,
speck,=
vlGfO+l-E,
{ k,,(r,
+ so - s~)~‘, rofo-Ol+l-E, ~$(-gl-l+~}.
Proof.
By Lemmas 1.2 and 2.1 it follows that k,, r, + so - rl, r. + so - sI,
and so are the only possible eigenvahres of T,, and that r, + so - rr
[r, + so - sr] has multiplicity f, [g,] whenever r,f r, [s,f so]. From
trace A, = 0 one finds that the multiplicity of so [r,] equals g, - v1 + E
[ f, - v1+ 1 - E],which must be a nonnegative number. The inequalities
sr >, so and rl < r. follow from Cauchy interlacing [Result 1.1(i)]. What
remains to be proved is that sr = so and rr = r. do not both occur. Suppose
they do. Define a = (k, - ETA- (1- E)s~)/v~; then the matrix A, - cr./,
which has eigenvahres r. and so only, has principal submatrix A, - aJ,
having only eigenvahres r, and so too. So, by Result 1.1(i), C - aJ = 0 and
hence F. is imprimitive: a contradiction.
=
r,,
The regular graph I?, is strongly regular, a clique, or a coclique whenever
it has at most two distinct eigenvalues, except for the degree k,. This leads to
the following result.
COROLLARY2.5.
tion is strmgly
(i)
(ii)
(iii)
(iv)
With the hypotheses of Theorem 2.4, the decomposiregular if and only if one of the following holds:
Or= f,+l-E=go+E,
so = s1 and fo = fi + E,
so =sr andv,=g,+s,
r,=r, andg,=g,+l-E,
(v) ro=r,
a&v,=
fo+l-e.
W. H. HAEMERS AND D. G. HIGMAN
386
A strongly regular decomposition is called improper if lr or I’, is a clique
or a coclique. Without loss of generality we may assume then that r, is a
coclique. If l?, is strongly regular and IL is a coclique, then also Theorem
2.4(i) holds with rl = 0 and g, = 0. Thus we find the following result of
Haemers [ 131:
THEOREM 2.6. Let r, be primitively strongly regular, and let rl be a
coclique. Then v1 = g, = ,- voso/(k,
- sO) (i.e., both HoffiTlan’s bound and
Cvetcovid’s bound are tight) if and only if I?, is strongly regular.
Proof.
Hoffman’s bound is tight if and only if the decomposition is
regular. Theorem 2.4(i) gives
spec
r, = { kz,(r,,
+ s~)“~-‘,
rOf~O-“lll, sgg”-‘l),
since s=Oif rl is a coclique. By Theorem 2.3 we have fo - vr f 1> 0; hence
n
r, is strongly regular if and only if g, = vi.
We call a proper strongly regular decomposition exceptional if si # s,,
and rl # rO, which is by Theorem 2.4(i) equivalent to ss # sa and rz z ra.
THEOREM 2.7. If r, is primitively
strongly regular and admits an
exceptional strongly regular decomposition, then I’, is the graph of a regular
symmetric conference matrix, that is, I’,, or its complement satisfies
v,=4r~+4rO+2,
Moreover,
k,=2rt+r0,
one of the following
for integer rO.
sO= -r,-1
holds:
(i) Ii and r, are so-called conference graphs, that is,
VI =
v2
=
rl = r2=
and D is a symmetric
2rOz+ 2r, + 1,
-1+J;T
2
’
k, = k, = r,” + rO,
s1=s2=
-1-G
2
’
2-(v,, TO”,
rO(rO- 1)/2) design, or the complement.
387
STRONGLY REGULAR GRAPHS
(ii) We have
k, = k, = ~0”+ r,,
v,=v,=2r(+2ro+1,
k,-r,
rl f
r1 <
r2,
and rl, r,, and (2kH + k,)/(k,
sz=
-rz-1,
sl=
5=2r,+l’
ro,
-r,-1,
rz < rot
+2rT +2r, + 1) are integers.
Take without loss of generality E = 0. Then Corollary 2.5(i) gives
and the remaining parameters of I’, follow straightforwardly (see
&+I=&
[9]). Also by 2.5(i) we have 2v, = v,, so vi = v2 and k, = k,. By Theorem 2.2
it follows that k, = k, = (k, + so)/2 = ro2+ ro. If Pr is a conference graph,
then so is P2 (by Theorem 2.4) and by Lemma 2.1 we find
Proof.
CC'=
ro(ro+ 1)z+ r,(ro- 1)
2
2
I,
proving (i). If Pi and I, are not conference graphs, then rl z - 1 - s1 and
the eigenvalues are integers. The remaining formulas of (ii) follow easily from
n
Theorem 2.4 and the well-known identities for strongly regular graphs.
The Petersen graph partitioned into two pentagons is an example for (i).
We give another example in the next section. For case (ii) it seems hopeless
to find an example: The smallest feasible solution has rl = 554, r, = 731,
v. = 2,140,370.
The next theorem relates strongly regular decompositions to strongly
regular designs. The result is due to W. G. Bridges and M. S. Shrikhande [3].
isprimitively strongly regular with a strongly regular
THEOREM 2.8. I',,
decomposition which is proper and not exceptional, if and only if D is a
strongly regular design with point graph rl and block graph r,, whose
parameters satisfy
(i) k,+r=k,+k,
(ii) k, - k E { u19p1 + ~2 - al},
(iii) p2 = ui + z,, pl = a2 + z2.
W. H. HAEMERS AND D. G. HIGMAN
388
Proof.
Let D be a strongly regular design. From Definition 1.3 it follows
that A,C + CA, E (C, 1); hence Lemma 1.2 applies. Clearly r, is regular (of
degree k, = k 1+ r ) whenever k, + r = k, + k. For the remaining eigenvalues
of r, we get k, + k, - k, = k, - r [by Lemma
l.Z(iii)], and the roots of
1.2(i)],
ur and u2 [by Lemma
(x - Pl)(X - P2)= Y
(*I
(by 1.2(ii)). These five eigenvalues take only two values if and only if ur, ua,
and k 2 - r are roots of ( * ). By use of pi - ui = y, /z i we find that u, is a root
of ( *) for i = 1,2 if and only if (iii) holds. Suppose ur is a root of ( *); then
pr + pz - u1 is the other root; hence k, - r is a root of ( *) if and only if (ii)
holds. The decomposition is clearly proper, and it is not exceptional, since ur
and u2 are eigenvalues of I’,.
Next assume r, has the required properties. Then r, = r, or sg = sr, since
the decomposition
is not exceptional. Take without loss of generality r, = rr.
Then Lemma 1.2(ii) gives
for
Yj=(r~-sO)(sl-s~)
So D satisfies (iii’) of Definition
of r, and r, we have
j=2,...,m.
1.3. By Lemma 2.1 and the strong regularity
CC’ E (A,, 1, J),
CTC E (A,, 1, J).
Moreover, the coefficient of Ai equals r, + s0 - r, - si # 0 for i = 1,2. Hence
also (i) and (ii) of Definition 1.3 are satisfied, so D is a strongly regular
W
design.
From the above proof we have that a strongly regular I?, has eigenvalues
o1 and p1 + ,02 - el; one of the two must be equal to ua. The following result
can be regarded as a special case of the above theorem (therefore a proof is
superfluous).
THEOREM 2.9. I’, is primitively
strongly regular with an improper
strongly regular decomposition (where rl is a coclique) if and only if D is a
quasisymmetric
2design, having r, as block graph, whose parameters
satisfy
r=k+k,,
a,+-
b2 --0,
k = - y/z,.
389
STRONGLY REGULAR GRAPHS
From k = - y/z, it follows that za < 0. This means that if (as usual)
adjacency in the block graph corresponds to the larger intersection number,
then I, is the complement of the block graph of D. M. S. Shrikhande [17]
(see also 131) proved that the conditions for D in Theorem 2.9 are equivalent
to the following: D is a quasisymmetric 2-(1-t z&k - l)/(k - zz), k, k( k z,“)/( z2 + 1)) design with intersection numbers k - ~2” and k - z,” - z2.
3.
CONSTRUCTIONS
In this section we give constructions and some nonexistence results for
strongly regular graphs with strongly regular decompositions. With the help
of the results of the previous section we have made a table of feasible
parameters up to 300 vertices (Table 1). For all cases in the table we indicate
existence or nonexistence if known (to us).
EZXAMPLE
3.1. The vertices of the triangular graph T(m) are all pairs of
a given set M of cardinality m (m > 3); two vertices are adjacent whenever
the pairs are not disjoint. T(m) is strongly regular with
specT(m)
= {2(m
- 2), (m - 4)“‘-r,
( - 2)‘ric”1+3)‘2}.
For a fixed x E M, partition the vertices into the pairs containing x and pairs
not containing x. It is easily seen that this gives an improper strongly regular
decomposition of T(m) into a clique of size m - 1 and T(m - 1).
The next result has often been observed before.
THEOREM3.2.
The block graph of a quasisymmetric 3design E admits a
strongly regular decomposition. The decomposition is improper if and only if
E is the extension of a symmetric
2design.
Proof.
Fix a point x of E. Partition the blocks of E to the blocks
containing x and the blocks not containing x. This gives a partition of the
block graph of E into the block graphs of the derived and the residual design
of E (with respect to x) respectively. The derived or residual design of E is
symmetric whenever E is the extension of a symmetric design; otherwise
both designs are quasisymmetric. This proves the result.
n
The design whose blocks are just all pairs of points can be seen as a
degenerate quasisymmetric 3-design. This leads to Example 3.1. M’e know of
-7
-6
-3
21
16
14
76
77
85
3450
55
56
21
19
49
-3
27
70
20
-3
20
69
45
27
35
-5
30
63
23
27
35
30
63
-5
5
10
5
16
-3
5
3
0
2
12
7
13
13
0
1
28
35
15
21
56
35
50
2
19
57
7
4
21
49
24
45
9
5
27
36
8
8
2
15
6
9
GRAPHS
3
VERTICESa
REGULAR
TABLE 1
300
STRONGLY
9
SETS FOR PRIMITIVE
6
ALL FEASIBLE
P ARAMETER
1
1
3
4
4
7
0
10
0
14
6
18
5
12
10
15
4
2
2
2
6
4
5
3
1
3
1
3
1
1
0
1
STRONGLY
15
18
WITH
20
15
-5
-3
-1
-3
-4
-5
-1
-3
6
28
35
38
2
18
3
20
20
14
-5
-3
-1
-3
3
3
4
4
28
21
20
18
18
30
20
24
6
20
7
20
4
4
1
4
DECOMPOSITION
-1
-1
-3
-2
FiEGULAFi
5
6
0
3
1
6
4
3
5
7
0
6
10
9
6
9
0
0
0
3
2
0
1
0
UP TO
2
56
162
176
217
220
15
16
17
18
19
+
+
_
-
?
208
12
60
126
14
_
84
66
75
70
14
4
10
2
2
30
112
13
+
2
22
100
12
2
40
+
2
95
40
11
95
_
-
-4
- 11
- 5
- 18
- 16
-3
-10
-8
- 10
- 10
44
154
64
154
140
21
90
77
75
75
175
62
143
21
21
104
21
22
19
19
38
15
30
18
10
33
2
0
12
12
28
22
25
34
24
24
10
6
20
20
45
175
63
154
64
144
56
120
81
81
21
105
56
56
50
50
19
76
20
75
14
66
22
48
30
55
10
42
20
20
20
52
10
10
7
7
0
30
10
32
14
11
1
4
10
7
2
2
2
2
10
2
2
2
2
2
0
2
55
98
2
42
- 11
-8
-1
-4
8
55
35
99
-4
- 12
-2
-5
60
60
20
35
35
28
28
57
18
56
-7
-7
-3
-4
-4
-3
-3
-8
-8
- 10
42
132
7
55
55
88
20
20
20
20
84
20
20
21
21
18
1
18
0
13
29
1
12
18
22
0
8
1
1
29
0
0
0
0
8
10
0
22
11
16
10
20
2
18
6
6
22
2
2
1
1
14
10
16
22
?
26
+
28
29
30
31
?
?
?
3
27
25
+
-
24
+
23
21
?
_
20
+
287
287
266
265
261
255
255
253
246
245
236
221
126
126
45
96
84
126
126
112
85
52
55
64
3
3
3
6
21
7
7
2
3
3
11
12
- 21
- 21
- 12
- 10
-3
- 9
- 9
- 26
- 17
- 13
-4
-4
245
245
209
159
29
135
135
230
204
195
59
51
41
41
56
105
231
119
119
22
41
49
176
169
45
45
0
32
39
61
61
36
20
3
18
24
63
63
9
36
21
63
63
60
34
13
11
16
41
246
42
245
56
210
105
160
29
232
119
136
120
135
77
176
41
205
49
196
60
176
52
169
0
105
21
108
0
33
32
54
28
77
54
63
63
70
16
70
0
68
0
39
11
40
12
48
3
0
3
3
2
6
19
3
7
3
7
2
2
3
3
11
8
12
9
~ 18
21
- 18
-9
~ 10
- 6
-3
- 9
-5
-9
-5
- 6
- 18
- 14
- 10
-1
-4
-1
-4
205
40
204
154
84
75
28
84
51
84
50
55
154
164
147
4
55
3
48
40
1
40
55
20
84
203
34
84
35
84
21
21
40
48
55
120
48
120
36
0
39
0
4
18
36
21
30
30
37
0
18
15
2
10
12
11
17
0
51
21
54
6
12
18
20
27
28
36
35
4
34
26
9
0
8
12
“The triangular graphs (Example 3.1) and the exceptional decompositions (Theorem 2.7) are omitted. For each case it is
indicated whether the decomposition exists (+ ), does not exist ( - ), or is not settled (?).
-
TABLE 1 (Continued)
393
STRONGLY REGULAR GRAPHS
just three other quasisymmetric 3designs (up to taking complements and
except for the Hadamard 3-designs, which have imprimitive block graphs):
the famous 4-(23, 7, 1) design (see [9]), its derived design, and its residual
design. These three 3-designs are cases 8, 16, and 24 in the table. The first
one provides an improper decomposition (D is the extension of the projective
plane of order 4). In fact, this decomposition and the ones of Example 3.1 are
the only improper decompositions we know.
Let rl and D be as in Theorem 2.7(i). Suppose their
THEOREM 3.3.
matrices A, and C commute, and let I’, be the complement of lYl. Then r, is
strongly regular with an exceptional strongly regular decomposition.
Proof.
We have
At = - Ai + $,,( r, + l).J + $a( r, + l)Z
for
i=1,2,
CC’ =CTC=&(rO--l)J+&(rO+l)Z,
ArC+CA,=
Arc-CA,+CJ-C=$_Z-C.
This yields
A: = - A, + rt_Z+ rO(r, + l)Z,
which proves the result.
n
If r, = 1, then Ii is the pentagon, D is the degenerate 2-(5, 1, 0) design,
and I?, is the Petersen graph. For rO= 2, the desired graph and design are
known:
A,=cycle(l
0
1
1
0
0
0
0
1
1
0
1
O),
C=cycle(l
1
0
1
0
0
0
0
0
1
0
0
0).
Since A, and C are both cyclic, they commute. Thus by the above theorem
we find a strongly regular I,, with (u,, k,, r,, sO) = (26,10,2, - 3), decomposed into the strongly regular Ii and I, with (or, k,, ri, sr) = (os, k,, r,, sz)
= (13,6, ( - 1 + m)/2,
( - 1 - m)/2).
These are all the exceptional
strongly regular decompositions we know. More graphs and designs with
suitable parameters are known, but it is not known whether there exists a pair
with commuting matrices.
W. H. HAEMERS AND D. G. HIGMAN
394
Zf q is the order of a projective plane, and if there exist
THEOREM 3.4.
q - 1 mutually orthogonal Latin squares of order q2 + q + 1, then there exists
a strongly regular decomposition with
(vo~kO~r,~so~=((q2+q+l)(q2+2q+2),(q+1)3,q2+q,
-q-l),
(“1~kl~~,~sl)=((q2+q+l)(q+1)~q(q+l),q(q+1),
-l),
(~2,
k,>r,>
Proof.
transversal
~2)
=
((q2
+
q +
I)‘>
q(q
+
1)2,
q2,
-
q -
11.
A set of q - 1 mutually orthogonal Latin squares is the same as a
design with q + 1 groups of size q2 + q + 1. Let
B,=(NC
N,T
N;+L)l
...
be the incidence matrix of the transversal design, where the Ni’s correspond
to the groups. Let M be the incidence matrix of a projective plane of order q,
and define Z?, = I@ M (B denotes the tensor product), and B = (B, B,).
Then B is the incidence matrix of a 2-(( q2 + q + l)( q + l), q + 1,l) design
(which is obviously quasisymmetric) with block graph r,. Clearly the block
graph rr of B, is imprimitively strongly regular. Also the block graph of a
transversal design is strongly regular. So the decomposition is strongly regular
n
and the eigenvahres readily follow.
For many values of q
instance, if q and q2 + q +
also (see Brouwer [4]) if
1,2,3,4,7,8).
Cases 1, 6,
manner. We do not know
following
example,
of Theorem 3.4 are fulfilled-for
1 are both prime powers (e.g. q = 1,2,3,5,8),
but
q
and q + 1 are both prime powers (e.g. q =
and 20 in the table can be constructed in this
if the theorem provides an infinite family. The
however,
the conditions
does give infinitely
many proper strongly regu-
lar decompositions.
EXAMPLE 3.5.
For every integer
m > 1, the symplectic
( vo, k,, r,, so) = (22m - 1,22”-‘-
2,2”-l-
graph r. with
1, - 2”-’
- 1)
STRONGLY
REGULAR GRAPHS
395
admits two strongly regular decompositions:
(u2,
k2,r2,
s2)
=
(22-l
_ 2”‘-1,22”-2
=
(22nlFl
+y-1,22”‘~~2
one with
_ 1,2”‘-”
_
1,
_
y-1
_
I),
and one with
(02. k 2,r2,
s2>
-
1,277’
-I_
1,
_
2”’
2 _
1).
In both cases l1 is the orthogonal graph, defined on the points of an
orthogonal quadric in PG(2m - 1,2). The symplectic and orthogonal graphs
are described in Seidel [15]. For m = 2 the decompositions
coincide with
Theorem 3.4 (9 = 1) and Example 3.1 (m = 6) respectively. For larger m,
the decompositions
are proper and P, and l?, are both primitive. Cases 1, 3,
4, 25, and 26 in the table are of this type.
Next we shall give some sporadic examples (making use of the table).
EXAMPLE 3.6.
strongly
Case 2 in the table exists, that is, the Clebsch
regular decomposition with
A,=A,=(;
EXAMPLE 3.7.
admits a strongly
;j,
C=iJTI
graph has a
llr).
Case 12 in the table exists, that is, the Higman-Sims graph
regular decomposition into two Hoffman-Singleton
graphs
(see Sims [19]).
EXAMPLE 3.8.
A hemisystem (see Cameron, Delsarte, and Goethals [8])
is a strongly regular decomposition of the point graph of a generalized
quadrangle of order (92, 9). The only known hemisystem has 9 = 3, where
the point graph is decomposed into two Gewirtz graphs. This produces case
13 of the table.
EXAMPLE 3.9.
Goethals and Seidel [ll] give a construction
15 from which the strongly regular decomposition is obvious.
of r, of case
W. H. HAEMERS AND D. G. HIGMAN
396
Finally some nonexistence results are considered. Case 7 in the table is
impossible, since Wilbrink and Brouwer [22] showed that I, does not exist.
For cases 17 and 18, rl does not exist because of the absolute bound. By
Theorem 2.9 the existence of an improper strongly regular decomposition is
equivalent to the existence of a quasisymmetric 2design with suitable parameters. For quasisymmetric designs many nonexistence results are known.
These results lead to nonexistence of cases 11, 14, 23 (due to Calderbank
[6, 7]), and 27 (due to Haemers [12]; see also Tonchev [20]) in the table.
The remaining cases are more complicated.
THEOREM3.10.
position exist
forthe
No strongly regular graphs with strongly regular decomparameter sets numbered 5 and 9 in Table 1.
Proof. In both cases I1 is imprimitive. Therefore D is a group divisible
design. Take C in canonical form, that is,
...
c=((y
C$
where the Ci’s correspond to the groups. For case 5 we define
I
B,=cycle(J-Z
wherein the
follows that
design with
showed that
I),
Z
blocks are 6 X 6 matrices. Then by straightforward verification it
(C B,) is the incidence matrix of a quasi-symmetric 2-(24, 8, 7)
intersection numbers 4 and 2. Brouwer and Calderbank [5]
such a design does not exist. Similarly, for number 9 we define
B,=cycle(J-Z
Z
Z
0
Z
0
O),
wherein the blocks have size 5 X 5. Then (C B,) is the incidence matrix of a
quasisymmetric 2-(35, 7, 3) design with intersection numbers 3 and 1.
Calderbank [6] has proved the nonexistence of such a design.
n
THEOREM 3.11.
Stnmgly regular decomposition
10 in Table 1 does not
exist.
Proof.
We use Seidel switching (see [9] or [16]). Since k, = - 2r,s,, we
obtain a strong graph by extending Ia with an isolated vertex. Next we
STRONGLY
397
REGULAR GRAPHS
isolate, by switching,
a vertex of rl. The graph on the remaining vertices is
strongly regular with the same parameters as the original r,. However, the
remaining vertices of rl now induce a coclique of size 19. Therefore, by
Theorem 2.6 we have constructed strongly regular decomposition 11, which
is impossible.
n
The smallest unsettled case in the table is 19. Tonchev constructed a
strongly regular graph with the parameters of r,, (see [21]), but it has no
cliques of size 15, so it does not admit a strongly regular decomposition. (In
fact, Tonchev’s graph has maximal clique and coclique size equal to 10.)
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1
2
3
4
5
6
7
8
9
10
11
12
13
14
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18
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20
21
22
W. H. HAEMERS
AND D. G. HIGMAN
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Received March 1988; final munuscript accepted 15 August 1988