Eigenvalues and geometric representations of graphs L á szl ó Lov á sz Microsoft Research

1. Eigenvalues and geometric representations of graphs László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052 lovasz@microsoft.com

2. Every 3-connected planar graph is the skeleton of a convex 3-polytope. Steinitz 1922 3-connected planar graph

3. 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

4. From polyhedra to circles horizon

5. From polyhedra to the polar

6. Coin representation Koebe (1936) Every planar graph can be represented by touching circles Discrete Riemann Mapping Theorem

7. A planar triangulation can be represented by orthogonal circles  no separating 3- or 4-cycles Andreev Thurston Representation by orthogonal circles:

8. 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

9. M=(Mij): symmetric VxV matrix • <0, if ijE Mij 0, if • M has =1 negative eigenvalue • symmetric, • The Colin de Verdière number of a graph Miiarbitrary normalization Strong Arnold Property

10. μk is polynomial time decidable for fixed k deleting and contracting edges for μ>2, μ(G) is invariant under subdivision for μ>3, μ(G) is invariant under Δ-Y transformation Basic Properties μ(G) isminor monotone

11. μ(G)3 G is a planar Colin de Verdière, using pde’s Van der Holst, elementary proof Special values μ(G)1 G is a path μ(G)2 G is outerplanar

12. are connected. Van der Holst’s lemma Courant’s Nodal Theorem

13. 0 _ + 0 0 _ + 0 + 0 _ 0 0 _ G planar  corank of M is at most 3.

14. basis of nullspace of M representation of G in Rμ corank of M is at most 3 G planar . Nullspace representation:

15. connected Van der Holst’s Lemma, geometric form or… like convex polytopes?

16. G 3-connected planar  nullspace representation can be scaled to convex polytope L G 3-connected planar  nullspace representation, scaled to unit vectors, gives embedding in S2 L-Schrijver

17. nullspace representation planar embedding

18. u q p v P P* Colin de Verdière matrix M Steinitz representation P

19. μ(G)3 G is a planar Colin de Verdière, using pde’s Van der Holst, elementary proof μ(G)4 G is linklessly embeddable in 3-space L - Schrijver Special values μ(G)1 G is a path μ(G)2 G is outerplanar

20. homological, homotopical,… equivalent Apex graph Linklessly embedable graphs embedable in R3 without linked cycles

21. G linklessly embedable  G has no minor in the “Petersen family” Robertson – Seymour - Thomas

22. properly normalized nullspace representation gives outerplanar embedding in R2 G 2-connected outerplanar  G 3-connected planar nullspace representation gives planar embedding in S2, and also Steinitz representation  L-Schrijver; L ? G 4-connected linkless embed. nullspace representation gives linkless embedding in R3   G path nullspace representation gives embedding in R1

23. μ(G)3 G is a planar Colin de Verdière, using pde’s Van der Holst, elementary proof μ(G)4 G is linklessly embeddable in 3-space L - Schrijver … _ ~ μ(G)n-4  complement G is planar Kotlov-L-Vempala Special values μ(G)1 G is a path μ(G)2 G is outerplanar Koebe-Andreev representation

24. Gram representation The Gram representation Kotlov – L - Vempala pos semidefinite

25. Assume:G has no twin nodes, and exceptional is an edge of P  If G has no twin nodes, and μ(G)n-4, then is planar. Properties of the Gram representation  ui is a vertex of P  0 intP