Janoš Vidali: Triple intersection numbers of Q-polynomial association schemes

Povzetek. An association scheme is called P-polynomial if its intersection numbers satisfy the triangle inequality, i.e., $p^h_{ij}$ \ne 0$ implies $|i-j| \le h \le i+j$ for some ordering of its relations. Dually, an association scheme is called Q-polynomial if its Krein parameters $q^h_{ij}$ satisfy the triangle inequality for some ordering of its eigenspaces. P-polynomial association schemes correspond precisely to distance-regular graphs, and their parameters can be derived from a subset of the intersection numbers which are usually written as the intersection array. Similarly, the parameters of a Q-polynomial association scheme can be computed from the Krein array. A distance-regular graph is bipartite iff $p^h_{ij} = 0$ whenever h+i+j is odd. Dually, a Q-polynomial association scheme is said to be Q-bipartite if $q^h_{ij} = 0$ whenever h+i+j is odd.

Recently, Williford has published lists of feasible Krein arrays for primitive 3-class Q-polynomial association schemes on up to 2800 vertices, and for Q-bipartite 4- and 5-class Q-bipartite association schemes on up to 10000 and 50000 vertices, respectively. Gavrilyuk has suggested that the sage-drg package be used to compute triple intersection numbers for the open cases in the lists. We have been able to use these computations to rule out many open cases.

This is joint work with Alexander Gavrilyuk.