Folded 6-cube and graphs with the same parameters

There are precisely three distance-regular graphs with intersection array {6,5,4;1,2,6}. They have 32 vertices and spectrum 6¹ 2¹⁵ (−2)¹⁵ (−6)¹.

Such a graph is the point-block incidence graph of a square 2-(16,6,2) design. But Hussain found that there are precisely three such designs, all of them self-dual.

Since blocks meet in two points, if we fix a block B then for each point x outside, the blocks on x determine edges on B, so that x determines a graph on B, regular of degree 2, that is, a union of polygons. These graphs are called Hussain chains. On 6 points the only possibilities for Hussain chains are 3+3 (two triangles), or 6 (a hexagon). Each path of length two on B is in a unique Hussain chain, so in order to describe the design it suffices to give the hexagons - then the paths of length two not covered yet are in triangles. The three designs all have a transitive group (so that the choice of B does not matter), and are characterized by having 0, 4 or 6 hexagons on B. Here 0 occurs for the folded 6-cube.

Group

The full group of automorphisms of the folded 6-cube is 2⁵.S₆ of order 23040, acting distance-transitively with point stabilizer S₆. The other two designs have transitive groups of order 1536 and 768, respectively.

Supergraphs

The folded 6-cube is found as subgraph of the M₂₂ graph.

p-rank

The 2-ranks of the point-block incidence matrices are 6, 7, 8, respectively.

Construction

The folded 6-cube is the result of identifying the antipodes in the 6-cube. It is the coset graph of {000000,111111}. It is the bipartite double of the 4x4 grid.

The second design can be described in terms of the generalized quadrangle GQ(2,2), with as points the pairs from a 6-set, and as lines the partitions of the 6-set into three pairs. Fix a line L. It is on four dual hyperbolic lines (triples of disjoint lines in a 3x3 grid) {L,M,N}. Each of the four pairs {M,N} determines six points of the GQ(2,2), that is, six edges (forming a hexagon) of the 6-set. Take these four hexagons to construct the second design. This construction shows that the stabilizer of B has order 6!/15 = 48.

The third design, described in a similar way: Fix a point P and a cyclic order (L,M,N) on the three lines on P. We find six hexagons by taking, for each ordered pair (L,M), (M,N), (N,L), the two dual hyperlines on the first element that have the second element as transversal. This construction shows that the stabilizer of B has order 6!/(15.2) = 24.

Remarks

The complementary designs are square 2-(16,10,6) designs. Their incidence graphs, the distance-3 graphs of the graphs studied above, are the three distance-regular graphs with intersection array {10,9,4;1,6,10}. The distance-3 graph of the folded 6-cube is found as subgraph of the O⁻₆(3) graph on 112 vertices.

Biplanes

A square 2-(v,k,2) design is called a biplane, and k−2 is called the order of the biplane. Only finitely many biplanes are known. There are unique biplanes of orders 1, 2, 3, three biplanes of order 4, three biplanes of order 7, and five biplanes of order 9. A single biplane of order 11 is known.

References

Q. M. Hussain, On the totality of solutions for the symmetrical incomplete block designs: λ=2, k=5 or 6, Sankhya 7 (1945) 204-208.

[BCN], Theorem 9.2.7.