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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 101 536 390 (−2)160 (−5)28.

## Group

The full group of automorphisms of Γ is HJ.2 acting distance-transitively with point stabilizer 21+4: S5. 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−t2),...,s(t−td−1); 1,t2+1,...,td−1+1,t+1} (here d=4, s=2, t=4, t2=0, t3=3): if d(x,y)=i, then the ai = (s−1)(ti+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 G2(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 801 2028 1036 (−4)250. It is the second subconstituent of the strongly regular G2(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.