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January 2016
Spectra of graphs and geometric representations
László Lovász Hungarian Academy of Sciences
Eötvös Loránd [email protected]
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January 2016
Happy Birthday, Noga!
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January 2016
Extreme graphs?Shannon capacity?Strong regularity lemma?Property testing?Combinatorial Nullstellensatz?Anti-Hadamard matrices?Optimization?Eigenvalues?Eigenvalues!
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January 2016
The eigenvalue gap
Laplacian:
1
2
0 ... 1 0 10 ... 0 1 0
1 00 1
1 0 n
dd
L
d
M M
O
adjacent positions
degrees
Eigenvalues: 1 2 ... n
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January 2016
Graphs and the eigenvalue gap
Gap between 1 and 2 expander graph
Alon - MilmanAlon
1 < 2 graph is connected
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January 2016
G-matrix: , symmetric, 0 ( , )
V V
ij
M MM ij E i j
¡
G = (V,E): simple graph, V=[n]
well-signed G-matrix: 0 ( )ijM ij E
Graphs, matrices, geometric representations
Want to understand: UM=0, M: G-matrix, Udxn
d=rank(U)=corank(M)
really good G-matrix: well-signed, one negative eigenvalue
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January 2016
UM = 0
nullspace representation
M U: nullspace representation
unique up to linear transformation
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cycle fixed toconvex polygon
edges replaced byrubber bands
MU: rubber bands
G is 3-connected planar, fixed cycle a face
planar embedding
Tutte
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MU: rubber bands
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2( )ij i jij E
M u u
EEnergy:
MU: rubber bands
0ij ij jj i
iij
uM M M
Equilibrium:( )
( ) 0ij j ij N i
M u u
(j free node)
stress matrix
stress in rubber bandorstrength of rubber band
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January 2016
MU: rubber bands
Mij: stress
define stress Mij so that
equilibrium condition
holds at all nodes
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January 2016
UM: bar-and-joint structures
--+
+ + +
0
0
ij jj
ijj
M u
M
M has corank 3 and is positive semidefinite.
Connelly
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January 2016
UM: bar-and-joint structures
--+
+ + +
2
,
2
,
( )
( ) ( ) ( )
ij i ji j
ij i i j ji j
u M u u
u x M u x u x u x
E
E E E
ui
Mij
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January 2016
Braced stresses
UM = 0
nullspace representation
M’
MU 0
U’U’M’=0
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January 2016
Braced stresses
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P P*
( )uvMp q u v
u
v
q
p
January 2016
UM: canonical stress on 3-polytopes
Canonical braced stress
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P P*
u
v
q
p
January 2016
UM: canonical stress on 3-polytopes
The canonical braced stress matrixhas 1 negative and 3 zero eigenvalues. L
(really good G-matrix)
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January 2016
MU: the Colin de Verdière number
G: connected graph
Roughly: multiplicity of second largest eigenvalue
of adjacency matrix
And: non-degeneracy condition on weightings
Largest has multiplicity 1.
But: maximize over weighting the edges and diagonal entries
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Mii arbitrary
Strong Arnold Property( ) maxcorank( )G M
normalization
M=(Mij): well-signed G-matrix•
M has =1 negative eigenvalue•
January 2016
[(G)-connected]
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μ(G) is minor monotone
deleting and contracting edges
μ k is polynomial timedecidable for fixed k
for μ>2, μ(G) is invariant under subdivision
for μ>3, μ(G) is invariant under Δ-Y transformation
January 2016
Colin de Verdière number Basic properties
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μ(G)1 G is a path
μ(G)3 G is a planar
Colin de Verdière, using pde’sVan der Holst, elementary proof
μ(G)2 G is outerplanar
January 2016
Colin de Verdière number Special values
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0x 0x 0x
supp ( ), supp ( )xx are connected.
discrete Courant Nodal TheoremJanuary 2016
M: really good G-matrix
Mx = 0
supp(x) minimal
Van der Holst’s lemma
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like convex polytopes?
or…
connected
January 2016
Van der Holst’s lemma for nullspace representation
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S+
S-
Corank bound
January 2016
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January 2016
The eigenvalue gap
Gap between 1 and 2 expander graph
Alon - MilmanAlon
1 < 2 graph is connected
2 < 3 G[supp+(v2)], G[supp-(v2)] are connected
van der Holst
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January 2016
The eigenvalue gap
Gap between 2 < 3 G[supp+(v2)], G[supp-(v2)]
are expanders
expander expander
?
Use (v2)i2 as weights!
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G 3-connected planar
nullspace representation,scaled to unit vectors,gives embedding in S2 L-Schrijver
G 3-connected planar
nullspace representationcan be scaled to convex polytope
LJanuary 2016
MU: Steinitz representations
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μ(G)1 G is a path
μ(G) 3 G is a planar
μ(G)2 G is outerplanar
μ(G)4 G is linklessly embeddable in 3-spaceL - Schrijver
January 2016
Colin de Verdière number Special values
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G 4-connected linkless embed.
nullspace representation gives
linkless embedding in 3
?
G path nullspace representation gives
embedding in 1
properly normalized
G 2-connected nullspace representation gives
outerplanar outerplanar embedding in 2
G 3-connected nullspace representation gives
planar planar embedding in 2, and also
Steinitz representation
L-Schrijver; L
January 2016
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January 2016
Computing G-matrices
Input: A 2-connected graph G=(V,E).
Output: Either an outerplanar embedding of G,
or a really good G-matrix with corank 3.
Special case: G 3-connected planar
Steinitz representation of G
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January 2016
UM: circulations
h: circulation on edges ij with ui and uj not parallel
i ui 2
, ( , ) 0ij j i ij j i ijj i j i j i
area M u u M area u u h
( , ) ( , ),area( , )
arbitrary ( , )
i ji j
i j
ij
h i j ij E u uu uM
ij E u u
P
P
ij j i ij j ij i j i
iiM u u MM u u
P Every G-matrix arises this way
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January 2016
M well-signed h is a counterclockwise circulation
M has one negative eigenvalue ?
UM: circulations
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January 2016
Shifting the origin
ui: nullspace representation, |ui|=1
M: really good G-matrix with corank 2
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January 2016
Many more nice theorems,Noga!