Eigenvalues%20and

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Eigenvalues and geometric representations of
graphs László Lovász Microsoft Research One
Microsoft Way, Redmond, WA 98052
lovasz_at_microsoft.com
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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
Koebe (1936)
Every planar graph can be represented by touching
circles
Discrete Riemann Mapping Theorem
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Representation by orthogonal circles
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The Colin de Verdière number
G connected graph Roughly multiplicity of
second largest eigenvalue of
adjacency matrix

Largest has multiplicity 1.
But maximize over weighting the edges and
diagonal entries
But non-degeneracy condition on weightings
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The Colin de Verdière number of a graph
Mii arbitrary
normalization
Strong Arnold Property
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Basic Properties
µ(G) is minor monotone
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Special values
µ(G)?1 ? G is a path
µ(G)?2 ? G is outerplanar
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Van der Holsts lemma
Courants Nodal Theorem
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G planar ? corank of M is at most 3.
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corank of M is at most 3 ? G planar .
Nullspace representation
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Van der Holsts Lemma, geometric form
or
like convex polytopes?
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• G 3-connected planar
• ?
• nullspace representation,
• scaled to unit vectors,
• gives embedding in S2

L-Schrijver
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planar embedding
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Colin de Verdière matrix M
Steinitz representation P
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Special values
µ(G)?1 ? G is a path
µ(G)?2 ? G is outerplanar
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Linklessly embedable graphs
embedable in R3 without linked cycles
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?
G path
nullspace representation gives embedding in R1
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Special values
µ(G)?1 ? G is a path
µ(G)?2 ? G is outerplanar
Koebe-Andreev representation
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The Gram representation
Kotlov L - Vempala
pos semidefinite
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Properties of the Gram representation
? ui is a vertex of P
? 0 ? int P
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