Γ can also be described as the graph on the 112 totally isotropic
lines of the GQ(9,3) on 280 points defined by U_{4}(3),
adjacent when they meet.

Or again: In the Steiner system S(5,8,24), take all 112 blocks starting 110... or 101...; join two blocks of the same kind when they have only two symbols in common, and join two blocks starting differently when they have four symbols in common. This yields Γ (and shows that it has a split into two Gewirtz graphs).

a) *Vertices*.
There are 112 of these, forming a single orbit.
The stabilizer of one (in the full automorphism group)
is 3^{4}:((2 × A_{6}).2^{2})
with vertex orbit sizes 1+30+81. The
second subconstituent
VO^{–}_{4}(3) (on 81 vertices) is again strongly regular,
with parameters (81,20,1,6).
It is the graph on the vectors of a 4-dimensional vector space over GF(3),
where two vectors are adjacent when the quadratic form vanishes on
their difference.

b) *Subquadrangles*.
There are 252 subquadrangles GQ(3,3) (namely, the subgraphs induced
on the perp of a nonisotropic point), forming a single orbit.
The stabilizer of one is
U_{4}(2):2×2 = PGO_{5}(3)×2.
The vertex orbit sizes are 40+72.

c) *Lines*.
There are 280 4-cliques (namely, the totally isotropic lines),
forming a single orbit. The stabilizer of one is
3_{+}^{1+4}.2_{–}^{1+4}.D_{12}.
The vertex orbit sizes are 4+108.

d) *Splits into two Gewirtz graphs*.
Γ has 324 splits into two
Gewirtz graphs, forming a
single orbit. The stabilizer of one split is
L_{3}(4).2^{2}, and the stabilizer of one Gewirtz graph
is L_{3}(4).2, of index 2 in the full automorphism group of the
Gewirtz graph. (Indeed, in the McLaughlin graph Λ we find
Γ as the set of neighbours of a vertex x, and the Gewirtz graph
as the set of common neighbours of two nonadjacent vertices x, y.
The stabilizer in McL:2 of {x,y} is the full group of the Gewirtz graph,
and the subgroup obtained by fixing both x and y has index 2 in it.)

There are 648 Gewirtz subgraphs, 324 on each point, 108 on each edge,
180 on each nonedge, 36 on each path of length 2, 12 on each K_{1,3},
4 on each quadrangle.
The intersection sizes are 0 or 56 (1x), 16 or 40 (42x), 20 or 36 (56x),
24 or 32 (105x), 28 (240x).
When two Gewirtz subgraphs C, D have nonempty intersection K,
then C and D are the only Gewirtz subgraphs on K in Γ.
If the intersection K has size u, then the graph induced on K
is regular of valency (u–16)/4.
Fixing C and varying D we find all the 16-cocliques,
half of the 10 K_{2}, all the 6 C_{4},
and half of the Coxeter graphs in C.

Equivalent to a split of the isotropic points of O^{–}_{6}(3)
into two Gewirtz graphs is a collection of 315 elliptic lines, precisely
one in each tangent plane with a single isotropic point. Thus, there are
324 such collections.

In the U_{4}(3) generalized quadrangle, this object is
equivalent to a collection of 315 K_{4,4}'s (induced on
the union of two orthogonal hyperbolic lines), precisely one
on each edge. Such objects are visible inside the Patterson graph.

e) *Regular spreads*.
A nonisotropic point in the U_{4}(3) geometry determines
a unital on its perp: 28 isotropic points, pairwise not joined
by a totally isotropic line. In the orthogonal geometry this
corresponds to a regular spread: a partition of the 112 isotropic points
into 28 totally isotropic lines, such that any two of these lines
determine a regulus: 4 lines in an O^{+}_{4}(3).
There are 540 nonisotropic points in the unitary geometry, forming
a single orbit. The stabilizer of one is (U_{3}(3) × 4):2.

There are lots of other spreads: each time we see a regulus, we can replace its four lines by the other four in their union and obtain a new spread. Or, in unitary terms, we can replace a hyperbolic line by its perp and obtain a new ovoid. However, there are also spreads without any regulus.

#reguli #spreads comment 63 540 regular, 1 orbit 31 34020 obtained from regular by a single switch, 1 orbit 15 680400 510300 obtained by double switch 9 30240 7 952560 6 1088640 4 1088640 3 2721600 2 3265920 1 1905120 0 5961600 total 17729280

f) *Bases*.
If we choose the quadratic form Σ X_{i}^{2},
then we see a subgroup 2^{5}:S_{6}.
There are 567+567 = 1134 bases, of mutually orthogonal nonisotropic
points all of the same kind, forming a single orbit under
PGO*^{–}_{6}(3); for PGO^{–}_{6}(3)
they split into two orbits (all points with Q(x)=1 and all points
with Q(x)=2). The point stabilizer is 2^{5}:S_{6}.
In terms of the isotropic points these bases determine subgraphs
(of vectors of "weight" 6) of size 32. (They are the distance-3
graphs of folded 6-cubes,
distance-regular with intersection array {10,9,4;1,6,10}.)
All subgraphs with array {10,9,4;1,6,10} arise in this way.
These subgraphs are bipartite, and their halves form the 2268
maximal 16-cocliques.

g) *Elliptic lines*.
The orthogonal geometry has 2835 elliptic lines, forming a single orbit.
The stabilizer of one is 4(S_{4}×S_{4}).2^{2}.
In terms of the isotropic points the elliptic lines are the
O^{+}_{4}(3) subgeometries.

h) *Hyperbolic lines or nonedges*.
The orthogonal geometry has 4536 hyperbolic lines, forming a single orbit.
In terms of the graph on the isotropic points the hyperbolic lines are
the nonedges (and give a vertex partition 2+10+40+60).
A hyperbolic line has two isotropic points, one point with Q(x)=1
and one point with Q(x)=2. Therefore, fixing a hyperbolic line in
PGO^{–}_{6}(3) implies fixing a nonisotropic point,
and the group is not maximal. But in PGO*^{–}_{6}(3)
the stabilizer (A_{6}.2^{2} × 2).2
is maximal.

i) *Odd graphs or maximal 7-cocliques*.
Our graph contains an orbit of 5184
Odd graphs O_{4}.
This is most easily seen by viewing the orthogonal geometry
as elliptic hyperplane in the O_{7}(3) geometry.
That latter geometry can be described using the form
Σ X_{i}^{2}, and the point (1,1,1,1,1,1,1)
is elliptic. In its perp we see 112 = 7 + 35 + 70 points
(7: 1111110; 35: 1110000; 70: 1112220), where the 7-set is a maximal
coclique and the 35-set induces the Odd graph O_{4},
the unique distance-regular graph with intersection array {4,3,3;1,1,2}.
The stabilizer of a 7-coclique (or of an O_{4}) is
S_{7}. There are no other maximal 7-cocliques.
*** Are there any other Odd subgraphs? ***

j) *Triples of mutually orthogonal elliptic lines*.
There are 8505 triples of mutually orthogonal elliptic lines,
forming a single orbit. An elliptic line has two points with
Q(x)=1 and two with Q(x)=2, and hence a triple of mutually
orthogonal elliptic lines determines a unique basis consisting
of points with Q(x)=1, and a unique basis consisting of points
with Q(x)=2. It follows that the stabilizer of a triple of
mutually orthogonal elliptic lines is nonmaximal in
PGO^{–}_{6}(3). However, it is maximal in
PGO*^{–}_{6}(3), with stabilizer
4^{3}(2×S_{4}) =
2^{5}.2^{3}.S_{3}.2.

k) *Unitary elliptic quadrics and even cocliques*
The graph on the 280 totally isotropic lines, adjacent when they meet,
has 9072 10-cocliques with the property that each point outside
is adjacent to 0 or 2 points inside. These form a single orbit
(but split into two orbits of size 4536 with stabilizer M_{10}
for U_{4}(3)). These 10-cocliques can be seen as elliptic quadrics:

The U_{4}(3) geometry is defined by a nondegenerate
Hermitean form over GF(9).
When restricted to vectors with coordinates in GF(3), the
form becomes a quadratic form. This means that this geometry
contains O^{+}_{4}(3) and O^{–}_{4}(3)
substructures. The former becomes a pair of orthogonal nondegenerate
planes, but each such plane has three elliptic points, and a
pair of orthogonal nondegenerate planes is equivalent to a split
of a basis into 3+3. Thus, the corresponding subgroup is not maximal.
The latter becomes a set of ten pairwise disjoint lines such that
each t.i. line hits 0 or 2 of them.

P. J. Cameron, J.-M. Goethals & J. J. Seidel,
*Strongly regular graphs having strongly regular subconstituents*,
J. Algebra **55** (1978) 257-280.

S. Dixmier & F. Zara,
*Etude d'un quadrangle généralisé autour de deux de ses points non liés*,
preprint, 1976.

S. Dixmier & F. Zara,
*Essai d'une méthode d'étude de certains graphes liés aux
groupes classiques*,
C. R. Acad. Sci. Paris (A) **282** (1976) 259-262.

L. H. Soicher, pers. comm., Feb 2019.