We introduce certain paradigms for procuring computer-free explanations from data acquired via computer algebra experimentation. Our established context is the field of algebraic combinatorics, with special focus on coherent configurations and association schemes. All results presented here were obtained by the authors with the aid of computer algebra systems, especially COCO and GAP. A number of examples are explored, in particular of objects on 28, 50, 63, and 210 points. In a few cases, initial experimental data pointed to appropriate theoretical generalizations that yielded an infinite class of related combinatorial structures. Special attention is paid to algebraic automorphisms (of a coherent algebra), a fairly new concept that has already proved to have far-reaching consequences. Finally, we focus on the Doyle-Holt graph on 27 vertices, and some of its related structures.

Combining two concepts of regularity of graphs, namely k-isoregularity and the t-vertex condition, a generalization of a classical result by Hestenes and Higman is presented. As an application it is shown that two infinite series of graphs constructed by Brouwer, Ivanov, and Klin which are not rank 3 satisfy the 5-vertex condition. Academic Press

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The t-vertex condition, for an integer t ≥ 2, was introduced by Hestenes and Higman (SIAM Am Math Soc Proc 4:41-160, 1971) providing a combinatorial invariant defined on edges and non-edges of a graph. Finite rank 3 graphs satisfy the condition for all values of t. Moreover, a long-standing conjecture of Klin asserts the existence of an integer t 0 such that a graph satisfies the t 0 -vertex condition if and only if it is a rank 3 graph. We present the first infinite family of non-rank 3 strongly regular graphs satisfying the 7-vertex condition. This implies that the Klin parameter t 0 is at least 8. The examples are the point graphs of a certain family of generalized quadrangles.

We construct four flag-transitive symmetric designs having 96 points, blocks of size 20, and 4 blocks on each point-pair. Moreover we prove that these are the only such designs. Our classification completes the classification of flag-transitive, point-imprimitive, symmetric designs with the (constant) number of blocks on a point-pair at most four.