i didnt Idi't f felsite atiids Seidel regular tetrahedron Eixample - PDF document

Spherical two-distance sets and spectral theory of signed graphs arxiv 2006.06633 Joint work with Tidor, Y ao, Zhang and Zhao (MIT team) VN E Rd 1v Spherical 2 dist set vi takes only 2 values vi Vj j 3 fit x set in RE Question Find maximum size of spherical 2 dis i didnt Idi't f felsite atiids Seidel regular tetrahedron Eixample 1251 6 111 whenever d 7 and d I did its Glazyrin Yu Nld Question What if the inner products are fixed Wfff Na p Ld I E p s L s l Given Ncd Max size of oset of unit vectors Vi VN Na p d or p in Rd s t L Vi Vj 7 ti t j L Review E ed ti Nap d C 2 unless Neuman Larman Rogers Seidel

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G kN Naaid d L Example ed 13 2 I 321 M 3 Ita FL 3 Ll 3dg 15 2 Connection between Na aid and ka Equiangular lines in Rd N vertex Graph G V E Rd VN v VN vi vz Vj Iff Lui Vj L V vi Vj EL Ivi 1 1 and At EJ Io vi Vjs ij IO AI Gram matrix PSD f f t t E d D matric rank Gram mat all ones adjmat CRANK rank XI Aat 4J ed Given d find largest NS.t I N vertex graphics with PSD Goal i RANK Given D find K vertex graph to with dpfGo Example disjoint l copies of fo l to be determined Take A X eigenvalues of G a x Check PSD l Aa c EJ Zo Take b XI rank 1 Fo Fo d I E d rank XL Aa IJ E elk 1 RANK Want Esmat ke d lik da k Ifl Better take k KIM

kind Cod Na a Cd t 0111 kata IT 44719 7 The reversed inequalityholds HARD TH m Question Is there an analog of key for Na pedi Ei of AGE adoring of fit is a coloring of the vertices using A valid t endpoints of every t edge are colored the colors sit t Cured differently are same and a sit 7 valid t color smallest t Whit Chromatic number of Gte p Eff 11 set d IITs c del Fia tepco Example th ftp with HIGH e p and ACEH d Take G i j Vj For Vi E Vi Vj C V G non edge in edge in Gt Vi Vj vs tdm v T GE G g Can be checked PSD I

Aat MJ L AGH ifAa rank 1 2 Aa MJ rank RANK EIG4 multld.AE tpwEan d Spherical 2 dist set N vertexgraph G E Rd vi Vj ift Vi n v p VN u Lor P vi Vj Vil L Agt fhJ Z O XI psp T T to Gram matrix I x p x p ed rank CGram Aat piled rank N RANK Goat Girard find largest N six 7 N rtx graph G with PSD RANK muttlx.AE tpEd IG 4 Want d told 1GHz multld.AE lG 4 I mwltcx.la Want max 16 4 Act Sp 1 i G satisfies inf Def Kpk matted Matt XAEST set D ftp.p LPIJ I a pao Edel i Cod Fi kp Then Nap d I d t old Kpk The reversed inequality holds UNI

Napld kph X d P P Exampled kin hka id 2 d d Equiangular line ish tf d km dipso p 33 pfd 1 p dich't a 2d 33 2 XE algebraic yrs Id 3 3 2d 53 2 34 method Equiangular line f ish LeZpco sit V connected G H j and O 7 C CLO g maintool mult Xj G AG E cn6g6gn with more deg Ed with White 2 This tool generalizes to G as soon as HGH This tool fails for G1 3 Counterexample multlxicail.AE n o i EI h bn tail Ti i E Gn