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UID:223386
DTSTAMP:20250905T093107Z
DTSTART;TZID=America/New_York:20250909T160000
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URL;TYPE=URI:https://wpiedudev.wpi.edu/news/calendar/events/department-math
 ematical-sciences-discrete-math-bill-martin-wpi
SUMMARY:Department of Mathematical Sciences Discrete Math: Bill Martin, WPI
DESCRIPTION:\n\n\n      \n\n\n\n\nDepartment of Mathematical Sciences\nDisc
 rete Math Seminar\nTuesday, September 9th, 2025\nOlin Hall 126, 4:00PM- 4:
 50 PM\nSpeaker: Bill Martin, WPI\nTitle: On the nearest neighbor graph of 
 a cometric association scheme\nAbstract: In his seminal 1973 thesis, Phili
 ppe Delsarte identified two important families of association schemes dese
 rving of in-depth study: the P-polynomial association schemes and the Q-po
 lynomial association schemes. P-polynomial association schemes are essenti
 ally the same as distance-regular graphs and these have been extensively s
 tudies in the intervening years leading to a rich theory with deep results
  and a variety of applications. By contrast, Q-polynomial association sche
 mes has received very little attention in the literature with the notable 
 exception being when the scheme is also P-polynomial. Indeed, Hamming grap
 hs, Johnson graphs, and many fundamental families of distance-regular grap
 hs are both P- and Q-polynomial.\nThe class of Q-polynomial association sc
 hemes also includes the schemes determined by the shortest vectors of some
  important lattices, schemes coming from extremal error-correcting codes a
 nd 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 do
 decahdron determines a Q-polynomial association scheme with one graph corr
 esponding to each nonzero angle that occurs), our toolkit as combinatorial
 ists leads us to frame our questions in graph-theoretic terms. In this tal
 k, 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 betwe
 en the P-polynomial case and the Q-polynomial case will be highlighted.\nT
 his talk is based on joint work with Jason Williford (U Wyoming).\n
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