Department of Mathematical Sciences
Discrete Math Seminar
Tuesday, September 9th, 2025
Olin Hall 126, 4:00PM- 4:50 PM
Speaker: Bill Martin, WPI
Title: On the nearest neighbor graph of a cometric association scheme
Abstract: In his seminal 1973 thesis, Philippe Delsarte identified two important families of association schemes deserving of in-depth study: the P-polynomial association schemes and the Q-polynomial association schemes. P-polynomial association schemes are essentially the same as distance-regular graphs and these have been extensively studies in the intervening years leading to a rich theory with deep results and a variety of applications. By contrast, Q-polynomial association schemes has received very little attention in the literature with the notable exception being when the scheme is also P-polynomial. Indeed, Hamming graphs, Johnson graphs, and many fundamental families of distance-regular graphs are both P- and Q-polynomial.
The class of Q-polynomial association schemes also includes the schemes determined by the shortest vectors of some important lattices, schemes coming from extremal error-correcting codes and combinatorial designs, and real mutually unbiased bases, whose study is motivated by questions about measurements in quantum information theory. While it may be more natural to view a Q-polynomial association scheme as a certain type of spherical code (e.g., every platonic solid except the dodecahdron determines a Q-polynomial association scheme with one graph corresponding to each nonzero angle that occurs), our toolkit as combinatorialists leads us to frame our questions in graph-theoretic terms. In this talk, we study the graph determined by the smallest non-zero angle appearing among pairs of unit vectors in this spherical code. As we build the basic theory of this nearest neighbour graph, similarities and differences between the P-polynomial case and the Q-polynomial case will be highlighted.
This talk is based on joint work with Jason Williford (U Wyoming).