V 3 lines maximum size of equi lines in IR Question 23 41 42 6 7 - PDF document

Equiangular lines with a fixed angle Joint with Alexandr Polyanskii, Jonathan Tidor, Yuan Y ao, Shengtong Zhang and Yufei Zhao arxiv arxiv 1708.02317 1907.12466 through 0 pairwise separated by same angle Lines in IR rt IRB blines V 3 lines maximum size of equi lines in IR Question 23 41 42 6 7 14 3 4 5 2 h 6 10 16 28 276 276 288 3 Max E NY Max E Cn de Caen Gerzon 1973 2000 Question What if the angle is fixed Max size of equiangular lines Each in IR with anglearccos 2 for n Lemmens Seidel 21h 15 Eyz n l 1973 Zz n for not il 1989 Neumaier Eys n unless ya is odd Exch E Zn Neumann 1973 depends on d Ex E Ca Bakh n 2016 Balla Droixler Sudator i'f n 2018 Edu E 1.93 n hold Keevash 13 and at

Ey Cult In EI Conjecture 1 Butch a n n Conjecture 2 J Polyanskii where K Kal EL Cnl 2 X n Spectral radius order D smallest k sit I k vertex graph Gat X G Kali a 31kcal X CA dock 3 the eigenvalues of adjacency matrix h d f k Eun 1 3 I Too 2 2h In 2 do 3 1 5 9 n 3 DX 4 1 7 ii a F IR J polyanskii THM Spectral radii of graphs conj 2 holds for all X E J is dense in a as 4 Barrier JTYZZI TMM L il when Kal Eun CA n uz noch when KCN D Es n t o n n When 2 217 k Kat can show Remark L hence Es h n hold n n

nin vertex graph G Equiangular lines in IR V V Set of vertex set unit vectors each vector represents a line A EJ Z 0 pgp XI E E L Y J Vi Vz all ones matrix adj mat 2 G ram matrix vi Vj 7 i j to rank R2 A EJ CRANK n rank Gram mat En Think as if rank 1dL A En Given n find largest m Goal m v.to graph G with PSD t CRAE NK Sx an Alternative goal Given on find smallest ns.t m Vfx graph G RANK Et PSD t an minimize rank 42 In other words given me A mult d G maximize We need to deal with 2 cases 7 PSD o G c where G U G completely reducible each connected component G i satisfies X Xilai malt Cd G i Matt X G I mkay C it is optional to take KCA lait In this case

G is connected where Klas X Irreducible mult Ka G 0 m n vertex connected graph G with JTYZH Given Tum an If a cat then max deg of a E O X is h Watt X G f Gg n o login I l I Ll 4 SRG id I it need connectedness dual D max deg so E malt Cd G I e Is it n Question that true