eigenvalues and geometric representations of graphs l á szl ó lov á sz microsoft research
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Eigenvalues and geometric representations of graphs L á szl ó Lov á sz Microsoft Research One Microsoft Way, Redmond, WA 98052 [email protected]. Every 3-connected planar graph is the skeleton of a convex 3-polytope. Steinitz 1922. 3-connected planar graph. - PowerPoint PPT PresentationTRANSCRIPT
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Eigenvalues and
geometric representations
of graphs
László Lovász
Microsoft Research
One Microsoft Way, Redmond, WA 98052
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Steinitz 1922
Every 3-connected planar graphis the skeleton of a convex 3-polytope.
3-connected planar graph
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Representation by special polyhedra
Every 3-connected planar graph
is the skeleton of a convex polytope
such that every edge
touches the unit sphere
Koebe-Andreev-Thurston
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From polyhedra to circles
horizon
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From polyhedra to the polar
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Coin representation
Every planar graph can be represented by touching circles
Koebe (1936)
Discrete Riemann Mapping Theorem
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Representation by orthogonal circles:
A planar triangulation can be represented byorthogonal circles
no separating 3- or 4-cycles Andreev
Thurston
/ 2ija
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The Colin de Verdière number
G: connected graph
Roughly: multiplicity of second largest eigenvalue
of adjacency matrix
But: 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
( ) max corank ( )G M
normalization
M=(Mij): symmetric VxV matrix•
Mij
<0, if ijE
0, if ,ij E i j •
M has =1 negative eigenvalue•
( )ijX X symmetric, 0 for andijX ij E i j
00MX X •
The Colin de Verdière number of a graph
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Basic Properties
μ(G) is minor monotone
deleting and contracting edges
μk 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
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μ(G)1 G is a path
Special values
μ(G)3 G is a planar
Colin de Verdière, using pde’sVan der Holst, elementary proof
μ(G)2 G is outerplanar
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0x 0x 0x
supp ( ), supp ( )xx are connected.
Van der Holst’s lemma
Courant’s Nodal Theorem
0Mx
supp( ) minimalx
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0
0 0
0
0
0
_
_
+
+
_0
+
_
G planar corank of M is at most 3.
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representation of G in μ
Nullspace representation:
0ij jj
M u
1
2
n
u
u
u
11 12 1
21 22 2
1 2
...
...
...n n n
x x x
x x x
x x x
basis of nullspace of M1 2 .. :.x x x
corank of M is at most 3 G planar .
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Van der Holst’s Lemma, geometric form
like convex polytopes?
or…
connected
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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 L
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nullspace representationplanar embedding
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P P*
Colin de Verdière matrix M
Steinitz representationP
( )uvMp q u v
u
v
q
p
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μ(G)1 G is a path
Special values
μ(G)3 G is a planar
Colin de Verdière, using pde’sVan der Holst, elementary proof
μ(G)2 G is outerplanar
μ(G)4 G is linklessly embeddable in 3-space
L - Schrijver
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Linklessly embedable graphs
homological, homotopical,…equivalent
embedable in 3 without linked cycles
Apex graph
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G linklessly embedable
G has no minor in the “Petersen family”
Robertson – Seymour - Thomas
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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
outerplanar nullspace representation gives
outerplanar embedding in 2
G 3-connected
planar nullspace representation gives
planar embedding in 2, and also
Steinitz representationL-Schrijver; L
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μ(G)1 G is a path
…
μ(G)n-4 complement G is planar_
~
Kotlov-L-Vempala
Special values
μ(G)3 G is a planar
Colin de Verdière, using pde’sVan der Holst, elementary proof
μ(G)2 G is outerplanar
μ(G)4 G is linklessly embeddable in 3-space
L - SchrijverKoebe-Andreevrepresentation
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The Gram representation
1 1 1 1( )TA Q M Q Q MQ J pos semidefinite
, : diag( )M Q
1, ( );
1, ( ).Ti j
ij E Gu u
ij E G
if
if
Kotlov – L - Vempala
1( )T nij i j iA u u u
Gram representation
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Properties of the Gram representation
ui is a vertex of P
1: conv( ,..., )nP u u | | 1iu exceptional
Assume: G has no twin nodes, and | | 1iu
( ) i juij E G u is an edge of P
0 int P
If G has no twin nodes, and μ(G)n-4, then
is planar.G
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