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# The Cohen-Tits near octagon on 315 points

There is a unique distance-regular graph Γ with intersection
array {10,8,8,2; 1,1,4,5}. It was constructed in Cohen (1981),
and uniqueness (given the intersection array) was proved in
Cohen & Tits (1985). See also BCN §13.6.
It has 315 vertices and spectrum 10^{1} 5^{36}
3^{90} (−2)^{160} (−5)^{28}.

## Group

The full group of automorphisms of Γ is HJ.2
acting distance-transitively with point stabilizer
2_{−}^{1+4}: S_{5}.
This group is transitive on the 315 . 10 . 8 . 8 . 2 geodetics of length 4,
and the stabilizer of a length 4 geodetic has size 3.

## Near polygon

This graph is a near polygon: it has lines (of size 3: all triangles)
and the property that each point has minimal distance to a unique point
of each line.
(This already follows from the intersection array
of the form {s(t+1),st,s(t−t_{2}),...,s(t−t_{d−1});
1,t_{2}+1,...,t_{d−1}+1,t+1}
(here d=4, s=2, t=4, t_{2}=0, t_{3}=3):
if d(x,y)=i, then the a_{i} = (s−1)(t_{i}+1)
neighbors of y at distance i from x are already seen on lines
containing a point at distance i−1 from x, so all other lines
on y have y as their unique point closest to x.)

A near polygon of diameter 4 is called a near octagon.

## Sub generalized hexagons

The near octagon contains 100 sub generalized hexagons GH(2,2)
on 63 vertices, that arise as intersections of Γ with
vertex neighbourhoods in the G_{2}(4) graph.
If d(x,y) = 3, then y is in 4 lines that contain a point at distance 2
from x. We can pick x and any three of these four lines, and find a
unique GH(2,2) containing them.
(In particular, these sub generalized hexagons are not geodetically closed.)
## Sub generalized octagons

The near octagon contains 280 sub generalized octagons GO(2,1)
on 45 vertices, the sets of fixed points of the 3A elements of HJ.
Any two of these have 5, 15, or 45 vertices in common,
where intersections of size 5 induce two intersecting lines,
and intersections of size 15 induce five pairwise disjoint lines.
The rank 4 graph on these 280 45-sets, adjacent when they meet in 15 points,
is strongly regular with parameters (280,36,8,4).
(Anurag Bishnoi, email 2014-09-11.)
## Distance-2 graph

The distance-2 graph Δ of Γ (same vertices, adjacent
when the distance in Γ is 2) has spectrum
80^{1} 20^{28} 10^{36} (−4)^{250}.
It is the second subconstituent of the strongly regular
G_{2}(4) graph on 416 vertices.
## Cliques and clique cover of Δ

All maximal cliques of Δ have size 5. There are three orbits under G,
of respective sizes 10080, 50400, 201600.
There is a clique cover of size 63, so that the complementary graph
has chromatic number 63.
## Cocliques

Leonard Soicher (email 2020-10-30) found that the largest cocliques have
size 90, and that there is a unique orbit of these 90-cocliques.
## Spreads and resolution

Leonard Soicher (email 2020-04-24) shows that the set of 525 lines of the near polygon
can be partitioned into five spreads (of 105 lines each).
## References

BCN, pp. 408-410.
A. M. Cohen,
*Geometries originating from certain distance-regular graphs*,
pp. 81-87 in: Proc. Finite Geometries and Designs, LMS Lecture Notes,
Cambridge 1981.

A. M. Cohen & J. Tits,
*On generalized hexagons and a near octagon
whose lines have three points*,
Europ. J. Combinatorics **6** (1985) 13-27.