Krein Parameters of Fiber-Commutative Coherent Configurations

For fiber-commutative coherent configurations, we show that Krein parameters can be defined essentially uniquely. As a consequence, the general Krein condition reduces to positive semidefiniteness of finitely many matrices determined by the parameters of a coherent configuration. We mention its implications in the coherent configuration defined by a generalized quadrangle. We also simplify the absolute bound using the matrices of Krein parameters.

Computer algebra investigation of known primitive triangle-free strongly regular graphs

With the aid of computer algebra systems COCO and GAP withits packages we are investigating all seven known primitivetriangle-free strongly regular graphs on 5, 10, 16, 50, 56,77 and 100 vertices. These graphs are rank 3 graphs, havinga rich automorphism group. The embeddings of each graphfrom this family to other ones are described, theautomorphic equitable partitions are classified, allequitable partitions in the graphs on up to 50 vertices areenumerated. Basing on the reported computer aided resultsand using techniques of coherent configurations, a few newmodels of these graphs are suggested, which are relying onknowledge of just a small part of symmetries of a graph inconsideration.