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

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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


slide-1
SLIDE 1

Spherical two-distance sets and spectral theory of signed graphs

Joint work with Tidor, Y ao, Zhang and Zhao (MIT team)

arxiv 2006.06633

Spherical 2 dist set

vi

VN ERd 1v

vi Vj

takes only 2 values fit

j 3

Question Find maximum sizeof spherical 2 dis

x set in RE

i didnt Idi't f

felsite atiidsSeidel

Eixample

regular tetrahedron

6

1251

Glazyrin Yu

Nld

I did

111 whenever d 7 and d

its

Question What if the inner products are fixed

Given

I E p s L s l

Ncd

Wfff Na pLd

Nap d

Max size of oset of unit vectors Vi

VN

in Rd s t

L Vi Vj 7 L

  • r p

ti

tj

Review

Neuman

Nap d

E ed ti

unless

C 2

Larman Rogers Seidel

slide-2
SLIDE 2

Napull

Cd4 if

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p

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Na pLdI E

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if

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i

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Eia

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4 Determine NapLd for larged

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d

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vertex graph G with dich

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largesteigenvalueof

set KIN

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adj mat of G

THI lTTY

YZ ft

Fix

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a Cd

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slide-3
SLIDE 3

Example

L

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G

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15

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vi

vz

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V

v

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Grammatrix

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i

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find K vertex graphto with dpfGo

Take A

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l to bedetermined Check PSD

eigenvalues of G a x

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XI

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d

RANK

rank XL Aa IJ

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slide-4
SLIDE 4

Cod

Na aCd

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kata

t 0111

THm IT44719 7 The reversed inequalityholds

HARD

Question Isthere an analogofkey for Napedi

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a coloringofthe vertices using

t

colors sit endpointsof every t edge are coloredthe

same and

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smallest t

sit 7 valid t color

Example

Fia tepco

c del

set d IITs

p Eff 11

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with HIGH e p and ACEH d th ftp

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For Vi EVi VjC

Vj

i j

ViVj edgeinGt

non edgein

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v

vs

tdm

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GE

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I

slide-5
SLIDE 5

RANK

rank1 2 Aa MJ

rank

L AGHifAa

AatMJ

EIG4 multld.AE tpwEan d

Spherical 2 distset

N vertexgraph G

u

VN

ERd

Vi n v

ift

vi Vj

p

Vil L

vi Vj

Lor P

psp

XI

Agt fhJ Z O

T

T

Gram matrix

to

I

x p

rankCGram

ed

x p

RANK

rank N Aatpiled Goat Girard find largestN six 7 N rtx graphG with PSD

RANK

Want

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muttlx.AE tpEd 1GHz

d

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mwltcx.la

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i

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kp

Kpk

I d t old

UNI

The reversed inequality holds

slide-6
SLIDE 6

Exampled

d

P

X

P

kph

Napld

Equiangular

d

d 2

kin

hkaid

line ish

dipso p

km

tf

d

a

dich't 1 33

p

pfd

XE

33

2

2d

algebraic yrs

3

3

Id

method

53

34 2

2d

Equiangular line f ish

LeZpco

maintool

H j and O 7 C

CLO g

sit V connected G

with more deg Ed

mult Xj G AG E cn6g6gn

This tool generalizes to G

with White 2

This tool fails for G1

as soon as HGH

3

Counterexample

multlxicail.AE

n

  • h

i EI

tail

bn

Ti

i

E

Gn