Examples of computer experimentation in algebraic combinatorics

Abstract

Summary: 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.