ISRAEL JOURNAL OF MATHEMATICS, Vol. 41, No. 4, 1982
STRONGLY REGULAR GRAPHS
DEFINED BY SPREADS
BY
WILLIAM M. KANTOR'
ABSTRACT
Spreads of finite symplectic, orthogonal and unitary vector spaces are used to
construct new strongly regular graphs having the same parameters as the
perpendicularity graphs of the underlying vector spaces. Some of the graphs are.
related to partial geometries, while others produce interesting symmetric
designs.
1. Introduction
Let V be a vector space over GF(q), equipped with a symplectic, orthogonal
or unitary geometry. A
spread
of V is a family E of maximal totally isotropic or
singular subspaces which partitions the set P of totally isotropic or singular
points. Let m = dim M for M E E, and assume that m -> 3. Using E, we will
construct a strongly regular graph G(E). The parameters of G(E) are the same as
those of the classical strongly regular graph G(V) = (P, • where • denotes the
relation "distinct but perpendicular"; however, the strongly regular graphs G(X)
and G(V) need not be isomorphic.
When V has type
f~+(2m,
2) or f~+(2m, 3), G(E) is the line-graph of the partial
geometry found by DeClerck, Dye and Thas [2, 14]. When V has type
Sp(2m, q), G(E) determines a symmetric design having the same parameters as
PG(2m - 1, q). Our goal is not just to define these graphs, partial geometries and
designs: we will also indicate some of their properties.
I am grateful to A. M. Cohen and H. Wilbrink for their numerous helpful
comments and corrections.
Permanent address: Department of Mathematics, University of Oregon, Eugene, OR 97403
USA.
Received June 6, 1981 and in revised form November 26, 1981
298
Vol. 41, 1982 GRAPHS DEFINED BY SPREADS 299
2. Spreads
We begin by summarizing some of the properties of V which will be needed
later. Proofs can be found in Dieudonn6 [4].
If V is as in w then it is equipped with a form (bilinear, quadratic or
hermitian) and a notion of perpendicularity. A subspace on which the form
vanishes is called
totally isotropic
(for symplectic or unitary V) or
totally singular
(for orthogonal V); such a subspace is perpendicular to itself, but the converse is
false for orthogonal geometries in characteristic 2. If X is any subspace then
dim X • -- dim V - dim X; in particular, if X < X • then dim X =< 89 V. All
maximal totally isotropic or singular subspaces have the same dimension
m _-< 89 V.
Throughout this paper, V, E, P, G(V) and m will have the same meaning as in
w tPI = (q"§ + 1)(q" - 1)/(q - 1), where m and e are related to V as in the
following table:
TypeofV Sp(2m, q) fl(2m+l,q)
f~+(2m, q) fl-(2m+2, q) U(2m, q ''2) U(2m+l,q m)
0 0 -1 1 -89 89
In [4] the orthogonal geometries of type fl+(2m, q) and fF(2m + 2, q) are those
having maximal and non-maximal index, respectively.
By definition, IPI = IEl(q" - 1)/(q - 1). Consequently, IEI = q"+" + 1.
We are only interested in the case m _-__ 3. The only spaces in which spreads are
known
to exist are then as follows (Dillon [5]; Dye [6]; Thus [13]; Kantor [9,
101):
Sp(2m, q). All
m,q.
l~(2m + 1, q). All m if q is even; m =3, q =0 or 2 (rood 3).
l~+(2m, q). All even m if q is even; m =4, q =0 or 2 (rood 3).
I~-(2m +2, q). All m if q is even.
Examples of spreads in some of these cases will be given later. No examples
exist in fY(2m, q) spaces if m is odd. Nothing is known about existence or
nonexistence for unitary spaces.
3. The graphs G(E)
Let V, P and E be as before.
Let l~ be the set of all hyperplanes of members of E. If X E f~, let E(X) denote
that member of E containing X.
300 w.M. KANTOR Isr. J. Math.
Write X~YC:~xnY~0, where X, YE~andX~Y.
Set G(E) = (f~, ~ ).
THEOREM 3.1. G(V) and G(E) are strongly regular graphs having the same
parameters.
PROOF. Clearly, IPI =
I~l(q m - 1)/(q - 1) = If~l. It is
well-known that G(V)
is strongly regular; its parameters are
[PI, k = q(q,.-~- 1)(q .... l+ 1)/(q - 1),
h = q - 1 + q~(q,.-:- 1)(q .... 2+ 1)/(q - 1), /z = (q,.-' - 1)(q,.+'-~ + 1)/(q - 1).
We will check that G(~) is a strongly regular graph with these parameters,
proceeding in several steps. The letters X, Y and Z will always denote members
of 1-/, while x and y will belong to P.
(1) If E(X)~MEE, then X • For, dimX l=dimV-dimX=
dim V - (m - 1), so that dim(X l n M) = (dim V - (m - 1)) + m
- dim(X l, M) _-> 1. The maximality of m now shows that dim(X 1 O M) = 1.
(2) If X- Y then Y-X. For, this is clear if E(X)= E(Y), so assume that
E(X)~E(Y). Set x=XAY• and y=X • Then yEX ~<x~, so
y ~x~n(x~nE(Y))=x~n Y.
(3) Let X E ~. Clearly, X - Y whenever X~ Y < E(X). Let
M E E - {E(X)}, and set y = X l O M. Then X - Y whenever y E Y < M. Thus,
G(E) has valence (q" - q)/(q - 1) + q"+" 9 (qm-~ _ 1)/(q - 1) = k.
(4) Let X - V with E(X) = E(Y). If Z~ X, V and Z < E(X) then Z - X, Y.
This accounts for (q"-1)/(q- 1)-2 members of l~. We now search for all
Z ~ X, V with E(Z) ~ E(X).
Let MEE-{E(X)}. Form X lnM=x and Y~AM=y. If Z<M, then
Z ~X, Y precisely when x,y E Z. This accounts for qm+E (q ,,-2_ 1)/(q- 1)
members of fl.
Thus, there are q"*E (q,.-2 _ 1)/(q - 1) + (q '~ - 1)/(q - 1)- 2 = h subspaces
Z E f~ such that Z - X, Y.
(5) Let X - Y with E(X) ~ E(Y). If Z~ X and V~ n E(X) E Z < E(X), then
Z ~ X, Y. Reversing the roles of X and Y, we obtain 2(q m-~- q)/(q -1) such
subspaces Z ~ X, Y.
Next, let M EE-{E(X),E(Y)}, and set x = X • O M and y = Y~ n M. If
Z < M, then Z ~ X, Y precisely when x, y ~ Z.
If x = y then x E (X, Y)~. Then (x, X, X • n Y) is .contained in P and has
dimension _-> m, so that x E (X, Y)I O (X, X • O Y) = (X • n Y, x n Y~). Con-
VOI. 41, 1982 GRAPHS DEFINED BY SPREADS
301
versely, each of the q -1 members of E-{E(X),~s meeting the latter line
produces an instance of x = y. Consequently, exactly (q - 1). (q'-~- 1)/(q - i)
subspaces Z-X, Y arise in this manner. Similarly, if x~ y we obtain
(q .... q). (q"-2-1)/(q- 1) subspaces Z. Thus, the number of Z- X, Y is
2(q m-' -
q)/(q -
1) + (q - 1)(q m ' - 1)/(q - 1) + (qm+,
_ q)(qm-2 _
1)/(q - 1) = h.
(6) Let X;zy. If Y~AE(X)<Z<E(x) then Z-X,Y. This produces
2(q'-'-1)/(q-
1) subspaces Z ~ X, Y lying in Z(X) or E(Y).
If MEE-{E(X),E(Y)},set x=X lnM and y=YInM. If Z<M, then
Z ~ X, Y precisely when x, y E Z.
Suppose that x = y. Then x EX• Y~=(X, Y)~. This is a nonsingular
subspace of dimension dim V-2(m- 1). Checking all cases, we find that it
contains exactly q"*~+ 1 members of P. This produces
subspaces Z.
Finally, if x/y
X, Y. Since
(q~*'-l)-(q ~-l- 1)/(q -1)
we obtain (q .... q,+,). (q,~-2_ 1)/(q - 1) subspaces Z
iz = 2(q"-' - 1)/(q - 1)+ (q'+' - 1)(q"-' - 1)/(q - 1)
+ (q .... q,+,)(q,,-2 _
1)/(q - 1),
this completes the proof of the theorem.
DEHNIT1Or~S 3.2. (i) If xEMEE then M*={ZEf~IZ<M} and x*=
{Z EfIIx
EZ<M}.
(ii) E* = {M* I M E
E}.
(iii) Let X-Y. If Y.(X)= ~(Y), set
XY={Z~flIXN Y<Z<E(X)};
if
~(X) ~ :~(Y), set
XY={ZEI~[ZN(XA Y• Y)/O,Z• Y~,x• v)}.
In the latter case, every member of E meeting (X n yl, X• y) nontrivially
contains a unique member of
XY
(compare step (5) of the proof of (3.1)). Thus,
I XYI
= q + 1 in any case.
(iv) If
X-Y, L(X,Y)={X,Y}U{ZEf~
IWEI~-{Z} and
W-X,Y
w~z}.
(v) Let X~ Y. If E(X)=E(Y), set
Lo(X, Y)=XY;
if Z(X)~(Y), set
Lo(X,Y)={X,Y}U{ZEf~IWEf~-{Z}, W-X,Y,
and
Wl>X~n Y
or
XN Y~- ~ W~Z}.
302 w.M. KANTOR Isr. J. Math.
Evidently, XY, L(X, Y) and Lo(X, Y) are all versions of "lines". Only
L(X, Y) depends strictly upon the graph G(E). In computational situations,
Lo(X, Y) is easier to deal with than L(X, Y).
These definitions are related, in view of the following simple lemmas.
LEMMA 3.3. (i) ~* is a partition of ~ into maximal cliques.
(ii) If ~,(X) ~ M E ~ then {Z E M* IX ~ Z} = x* for a unique point x of M.
Thus, the subsets x* of M* can be recovered from G(E), so that each clique M*
inherits from G(E) a natural structure as a projective space PG(m - 1, q).
(iii) If X ~ g and ~(X) = E(Y), then L (X, Y) = Lo(X, Y) = XY.
(iv) If X ~ Y then L (X, Y) C_ Lo(X, Y) C_ XY.
LEMMA 3.4. Assume that ~* contains every clique C of size (q m _ 1)/(q - 1)
such that X, Y E C and X~ Y imply that L(X, Y)C C and [L(X, Y)I = q + 1.
Then Aut G(E) is induced by the group of automorphisms of G(V) which send
to itself.
PROOF. By (3.3i) and hypothesis, G(E) uniquely determines E*. Then (3.3ii)
can be used to recover G(V) from G(E): simply interchange the roles of P, E and
t, E* in the construction described at the beginning of this section.
REMARKS. Aut G(V) is well-known (Dieudonn6 [4, ch. II, w it consists of
all invertible semilinear transformations of V which preserve the underlying
form projectively.
It seems likely that the hypothesis of (3.4) holds whenever V does not have
type fl+(8,q) (compare (4.1), (4.2)). In those cases we have computed,
[ L (X, Y) [ = 2 whenever X - Y and ~(X) # E(Y); in fact, except in the situation
of the next lemma it appears that [Lo(X, Y)[ = 2 when X(X) ~ E(Y).
LEMMA 3.5. Let V have type ~+(2m, q).
(i) If x E M E ~, then x* lies in exactly two maximal cliques: M* and
C(x*)= x* U{x • n N[N EE-{M}}.
(ii) If X ~ Y then [Lo(X, Y)[ = q + 1.
PROOF. (i) Let S be a clique containing x*, and let Y ~ S - M*. If X ~ x*
then Y~ n M ~ X, and hence Y• n M = x. Thus, S C_ C(x*).
Let Y, Z E C(x*)- x*, Y~ Z. Then V = (~(Y), E(X)). Let x E (y, z) with
y ~ E(Y) and z E ~(Z). Then (y, z) is totally singular since it has at least three
singular points. Then y E x ~ N ~(Y) = Y, while x, z E Z ~, so that y ~ Y N Z ~.
Thus, C(x*) is a clique.