Designs with a Polarity
A polarity of a symmetric design is a pair of permutations, one that permutes
the rows and the other the columns of the incidence matrix in such a way that
the resulting matrix is symmetric. It is easy to see that if a design has a
polarity then it has a polarity in which only the points (blocks) are
permuted, the blocks (points) remaining fixed. The number of absolute points of
a polarity is the trace of the symmetric incidence matrix. If A is the
incidence matrix of a symmetric (v, k, λ) design and
k - λ is not a square, the only possible number of fixed points
is k. On the other hand, if k - λ = n2, say,
A has eigenvalues k, +n or -n and it follows that
the number of fixed points is of the form k± 2mn, where
m = 0, 1, 2, ... when v is odd, and k±
(2m+1)n, where m = 0, 1, 2, ... when v is
even.
A symmetric design with a null polarity, i.e., a polarity with no absolute
points, is a strongly regular
graph. Some of these on few vertices (i.e. at most 64) are listed
in
here.
The "small" designs for which k - λ is a square have
parameters (15, 7, 3), (16,6,2), (21,5,1), (35,17,8), (36,15,6), (40,13,4) and
(45,12,3). Designs with the first three of these paramter sets are completely
determined whereas the numbers in the other sets are far from being
classified. However, in each of these four cases I have been able to find
polarities having all possible numbers of fixed points. These are listed
below.
A 45-12-3 design with a null polarity is a strongly regular graph, of which
there are 78. However, some of these graphs, considered as designs, are in
fact isomorphic. It turns out that the number of 45-12-3 designs with a null
polarity is 75. It is known that
a 45-12-3 design cannot have a polarity with 42 absolute points, but all
other candidates are possible, namely 6, 12, 18, 24, 30 and 36. Designs
with these respective numbers of absolute points are listed below. The orders
of the automorphism groups of these designs are, 18, 18, 2, 3, 3, 2.
| 6 abs pts | 12 abs pts | 18 abs pts | 24 abs pts | 30 abs pts | 36 abs pts |
As far as (40-13-3) designs are concerned, the possible numbers of absolute points for a polarity are 4, 10, 16, 22, 28, 34 and 40. In fact all these numbers are achievable which perhaps should not come as a surprise, since there are literally millions of these designs. Those with a polarity having 40 absolute points are equivalent to SRG's (40-12-2-4) of which there are 28. Some of these 28 designs have polarities achieving 6 out of the 7 possible numbers of absolute points. In the table below we give examples of designs, none of which comes from a strongly regular graph, having a polarity with the appropriate number of absolute points. The orders of the automorphism groups of these designs are 8, 1, 1, 3, 1, 1 respectively.
| 4 abs pts | 10 abs pts | 16 abs pts | 22 abs pts | 28 abs pts | 34 abs pts |
There are 3854 SRGs with parameters (35, 16, 6, 8) and by replacing the zeros on the diagonal of the adjacency matrices by ones, these give rise to symmetric (35, 17, 8) designs with a polarity having 35 absolute points. The other possible numbers of absolute points are 5, 11, 17, 23 and 29, and all of these are achievable.
| 5 abs pts | 11 abs pts | 17 abs pts | 23 abs pts | 29 abs pts |
There are 32,548 SRGs with parameters (36, 15, 6, 6) and considered as designs they all have a null polarity. Also the 180 (36, 14, 4, 6) SRGs give rise to (36, 15, 6) designs by replacing the zero diagonal of the adjacency matrix with all ones, and all these designs have a polarity with 36 absolute points. Since all these designs are pairwise non-isomorphic considered as designs it is clear that they constitute the complete set of (36, 15, 6) designs having a polarity with 36 absolute points. The other possible numbers for absolute points are 6, 12, 18, 24 and 30. All of these are achieved and some corresponding designs are given below.
| 6 abs pts | 12 abs pts | 18 abs pts | 24 abs pts | 30 abs pts |
In the table below we have listed some designs for which k - λ is not a square and which have a polarity.
| 19-9-4 | 23-11-5 | 25-9-3 | 27-13-6 | 31-15-7 | 37-9-2 |