It can also be described as the graph on the 77–21 = 56 blocks
of the (unique) Steiner system S(3,6,22)
not containing a fixed symbol, adjacent when they are disjoint.
Or, equivalently, as the graph on the 56 hyperovals from one
L_{3}(4) orbit in PG(2,4), adjacent when they are disjoint.
Or again, equivalently, as the graph on the 56 octads
that contain two given symbols a,b but not a third symbol c, where
two octads are adjacent when they meet in just {a,b}.

b) *An edge*.
There are 280 of these, forming a single orbit.
The stabilizer of one is 3^{2}:Q_{8}.2×2
with vertex orbit sizes 2+18+36.
In PG(2,4) these can be identified with the unitals (AG(2,3) subplanes):
the complement of the union of two disjoint hyperovals is a unital.
The graph on the 280 edges, where two edges are adjacent when they
are opposite edges of a quadrangle, is distance-transitive with
intersection array {9,8,6,3;1,1,3,8} and spectrum
9^{1} 4^{64} 1^{105} (–3)^{90}
(–5)^{20}. Its automorphism group Aut (L_{3}(4))
is three times larger than that of the Gewirtz graph.
The uniqueness of this graph (given its parameters) was shown by Lambeck.

c) *A 16-coclique*.
There are 42 of these, forming a single orbit.
The stabilizer of one is 2^{4}.S_{5}
with vertex orbit sizes 16+40.
In the Higman-Sims graph these are visible as point neighbourhoods.
Also in PG(2,4) these 16-cocliques are visible: each point is
in 16 of our hyperovals, and these hyperovals are mutually nonadjacent.
Moreover, each hyperoval has six exterior lines forming a dual hyperoval,
and disjoint hyperovals have disjoint dual hyperovals; it follows that
the 21 lines of PG(2,4) also give 16-cocliques.
The graph on the 16-cocliques, adjacent when disjoint, is the point-line
incidence graph of PG(2,4). Distance 0,1,2,3 in this graph corresponds
to an intersection of size 16,0,4,6, respectively.

d) *6 C _{4}*.
There are 105 of these, forming a single orbit.
The stabilizer of one is 2

e) *10 K _{2} and Sylvester*.
There are 112 of these, forming a single orbit.
The stabilizer of one is M

A hyperoval of a non-chosen orbit meets our hyperovals in 1 or 3 points;
20 in 3 (each triple occurs once) and 36 in 1.
This induces a partition of the graph into two subgraphs, one isomorphic
to 10 K_{2}, the other to the
Sylvester graph (distance-regular
with intersection array {5,4,2;1,1,4} on 36 vertices).

A different way to see this is to look at the Higman-Sims graph.
It has splits into two
Hoffman-Singleton graphs.
Choosing an edge in one of the Hoffman-Singleton graphs we find that
the subgraph of the Higman-Sims graph far away from that edge is split
into a Sylvester graph and a 10 K_{2}.

A different way to see the presence of subgraphs 10 K_{2}
is by looking at the generalized quadrangle GQ(3,9) on 112
vertices. It has splits into two Gewirtz graphs, and the ten lines
on a point on one side meet the other side in 10 K_{2}.

f) *co-Heawood graph*.
(The co-Heawood graph is the point-line
nonincidence graph of the Fano plane.)
There are 120 of these, forming a single orbit.
The stabilizer of one is L_{2}(7):2×2
with vertex orbit sizes 7+7+42.
Their presence can be seen from the M_{24} construction:
if our Gewirtz graph consists of the octads starting 110, then
fix an octad B starting 001. There are 7, 42, 7 octads in our graph
that meet B in 4, 2, 0 vertices, respectively, and the 7+7 induces
a co-Heawood graph. The involution in M_{24} that fixes B
pointwise, and fixes the pair contained in all Gewirtz octads
fixes the co-Heawood graph pointwise. There is no automorphism
here interchanging the two 7-cocliques in the co-Heawood graph.

g) *Splits into two Coxeter graphs*.
The Gewirtz graph has 240 splits into two Coxeter graphs, forming
a single orbit.
The stabilizer of one split is L_{2}(7):2 with vertex orbit
sizes 28+28.

These splits can be seen inside the Higman-Sims graph. It has splits into two Hoffman-Singleton graphs. Choosing an edge that meets both sides we find that the subgraph of the Higman-Sims graph far away from that edge is split into two Coxeter graphs.

A different way to see these is to look at the M_{24} construction:
if our Gewirtz graph consists of the octads starting 110, then fix an
octad B starting 100 or 010. There are 28, 28 octads in our graph that
meet B in 3, 1 vertices, respectively, and thus each B gives a split.

h) *Extended bipartite double of the Petersen graph*.
There are 336 of these, forming a single orbit.
The stabilizer of one is S_{5}×2
with vertex orbit sizes 6+20+30.
The extended bipartite double of the Petersen graph has 20 vertices
and spectrum 4^{1} 2^{5} 1^{4}
(–1)^{4} (–2)^{5} (–4)^{1}.
Inside the Higman-Sims graph it can be seen by taking a path
a ~ b ~ c ~ d of length 3, and taking the 20 vertices adjacent to
one of a, d and none of b, c.

size # 7 240 9 2520 10 43960 11 20160 12 5460 13 1680 16 42A colouring with 4 colours must use two disjoint 16-cocliques. The complement of their union is the union of six quadrangles, in particular bipartite. Any 2-colouring of this 6 C

A. Gewirtz,
*Graphs with maximal even girth*,
Canad. J. Math. **21** (1969) 915-934.

A. Gewirtz,
*The uniqueness of g(2,2,10,56)*,
Trans. New York Acad. Sci. **31** (1969) 656-675.

M. Hall, jr., R. Lane & D. Wales,
*Designs derived from permutation groups*,
J. Combin. Th. **8** (1970) 12-22.

A.E. Brouwer & W.H. Haemers,
*The Gewirtz graph - an exercise in the theory of graph spectra*,
Europ. J. Comb. **14** (1993) 397-407.

E.W. Lambeck,
*The uniqueness of a distance-regular graph on 280 vertices*,
in:
*Contributions to the theory of distance regular graphs*,
Ph.D. Thesis, Techn. Univ. Eindhoven, Netherlands, 1990.