Cometric Association Schemes

Some Terminology

The table describes each example in terms of these bits of information:

• the first parameter listed is the number of classes, d. The Bose-Mesner algebra of such a scheme has dimension d+1
• the number of vertices, |X|, is listed next
• the dimension, m₁, of the first eigenspace in the/a Q-polynomial ordering
• the Krein array {b₀^*, b₁^*, . . . , b_d-1^*; c₁^*, c₂^*, . . . , c_d^* } where the tridiagonal matrix L₁^* with (k,j)-entry equal to the Krein parameter q_1,j^k has diagonal entries
  a_j^*= m₁ - b_j^*- c_j^*
  and b^* above the diagonal and c^* below the diagonal
• the sequence of cosines in the first eigenspace (Q_i,1/ m₁ : i=0,1, . . . , d)
• a designation as to whether the scheme is primitive, Q-bipartite, Q-antipodal, or both
• a description of the scheme, comments or references

Parameter Files

• MAPLE input formats for the number of classes d, the number of vertices verts and the valencies v[i] (0 ≤ i ≤ d);
• MAPLE input for the second eigenmatrix Q
• MAPLE input for the first matrix of Krein parameters L₁^*. This matrix named L for simplicity and in row k, column j its entry is the Krein parameter q_1,j^k
• next come, in some sloppy format, the intersection matrices L_1,. . .,L_d. The matrix L_i, which appears as L_i in the file, has entry p_i,j^k in row k, column j
• the next data in the file are the eigenmatrices P and Q. The i^th column of P lists the eigenvalues of L_i, which are also the eigenvalues of the adjacency matrix A_i. The matrix Q is the inverse of P, scaled by the number of vertices
• finally, we give all Krein parameters in the matrices L_j^* (0 < j < d). The matrix L_j^* is denoted in the file as Lsj and has, in row k, column h, the Krein parameter q_j,h^k

Files of Vectors