Relation Algebras andt-vertex Condition Graphs

The scheme associated with a graph is an association scheme iff the graph is strongly regular. Consider the problem of extending such an association scheme to a superscheme. The obstacles can be expressed in terms oft-vertex conditions. If a graph does not satisfy thet-vertex condition, a presuperscheme associated with it cannot be erected beyond the (t − 3)rd level. We give an example of an association scheme which is not extendible to a superscheme: it cannot be extended beyond the bottom level of a presuperscheme.