January 2016 Spectra of graphs and geometric representations Lszl Lovsz Hungarian Academy of...
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Transcript of January 2016 Spectra of graphs and geometric representations Lszl Lovsz Hungarian Academy of...
January 2016
Spectra of graphs and geometric representations
László Lovász Hungarian Academy of Sciences
Eötvös Loránd [email protected]
January 2016
Happy Birthday, Noga!
January 2016
Extreme graphs?Shannon capacity?Strong regularity lemma?Property testing?Combinatorial Nullstellensatz?Anti-Hadamard matrices?Optimization?Eigenvalues?Eigenvalues!
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
January 2016
Graphs and the eigenvalue gap
Gap between 1 and 2 expander graph
Alon - MilmanAlon
1 < 2 graph is connected
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
January 2016
UM = 0
nullspace representation
M U: nullspace representation
unique up to linear transformation
cycle fixed toconvex polygon
edges replaced byrubber bands
MU: rubber bands
G is 3-connected planar, fixed cycle a face
planar embedding
Tutte
MU: rubber bands
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
January 2016
MU: rubber bands
Mij: stress
define stress Mij so that
equilibrium condition
holds at all nodes
January 2016
UM: bar-and-joint structures
--+
+ + +
0
0
ij jj
ijj
M u
M
M has corank 3 and is positive semidefinite.
Connelly
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
January 2016
Braced stresses
UM = 0
nullspace representation
M’
MU 0
U’U’M’=0
January 2016
Braced stresses
P P*
( )uvMp q u v
u
v
q
p
January 2016
UM: canonical stress on 3-polytopes
Canonical braced stress
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)
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
Mii arbitrary
Strong Arnold Property( ) maxcorank( )G M
normalization
M=(Mij): well-signed G-matrix•
M has =1 negative eigenvalue•
January 2016
[(G)-connected]
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
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
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
like convex polytopes?
or…
connected
January 2016
Van der Holst’s lemma for nullspace representation
S+
S-
Corank bound
January 2016
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
January 2016
The eigenvalue gap
Gap between 2 < 3 G[supp+(v2)], G[supp-(v2)]
are expanders
expander expander
?
Use (v2)i2 as weights!
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
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
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
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
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
January 2016
M well-signed h is a counterclockwise circulation
M has one negative eigenvalue ?
UM: circulations
January 2016
Shifting the origin
ui: nullspace representation, |ui|=1
M: really good G-matrix with corank 2
January 2016
Many more nice theorems,Noga!