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

2. Every planar graph can be drawn in the plane with straight edges Fáry-Wagner planar graph

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

4. Rubber bands and planarity Every 3-connected planar graph can be drawn with straight edges and convex faces. Tutte (1963)

5. Rubber bands and planarity outer face fixed to convex polygon edges replaced by rubber bands Energy: Equilibrium: Demo

6. G 3-connected planar rubber band embedding is planar Tutte (Easily) polynomial time computable Maxwell-Cremona Lifts to Steinitz representation if outer face is a triangle

7. If the outer face is a triangle, then the Tutte representation is a projection of a Steinitz representation. If the outer face is a not a triangle, then go to the polar. F F’ p’ p

10. G: arbitrary graph A,B  V, | A |=| B |= 3 A: triangle edges: rubber bands Energy: Equilibrium: Rubber bands and connectivity

11. ()() For almost all choices of edge strengths: B noncollinear   3 disjoint ( A,B )-paths Linial-L- Wigderson cutset

12. For almost all choices of edge strengths: B affine indep   k disjoint ( A,B )-paths Linial-L- Wigderson ()() strengthen

13.  edges strength s.t. B is independent for a.a. edge strength, B is independent no algebraic relation between edge strength

14. The maximum cut problem maximize Applications: optimization, statistical mechanics…

15. Easy with 50% error Erdős Bad news: Max Cut is NP-hard Approximations? C

16. C Easy with 50% error Erdős Bad news: Max Cut is NP-hard Approximations?

17. Easy with 50% error Erdős Bad news: Max Cut is NP-hard Approximations?

18. Easy with 50% error Erdős Bad news: Max Cut is NP-hard Approximations?

19. Easy with 50% error Erdős Bad news: Max Cut is NP-hard Approximations?

20. Easy with 50% error Erdős Bad news: Max Cut is NP-hard Approximations?

21. Easy with 50% error Erdős Bad news: Max Cut is NP-hard Approximations?

22. NP-hard with 6% error Hastad Polynomial with  12% error Goemans-Williamson Easy with 50% error Erdős Bad news: Max Cut is NP-hard Approximations?

23. How to find the minimum energy position? dim=1: Min E = 4·Max Cut Min E  4·Max Cut in any dim dim=2: probably hard  dim= n : Polynomial time solvable! semidefinite optimization spring (repulsive) Energy:

24. random hyperplane Expected number of edges cut: Probability of edge ij cut: minimum energy in n dimension

25. Stresses of tensegrity frameworks bars struts cables x y Equilibrium:

26. There is no non-zero stress on the edges of a convex polytope Cauchy Every simplicial 3-polytope is rigid.

27. If the edges of a planar graph are colored red and bule, then  (at least 2) nodes where the colors are consecutive.

28. 3 1 4 5 3 9 10 9 2 2 2 3 3

29. Every triangulation of a quadrilateral can be represented by a square tiling of a rectangle. Schramm

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

32. Representation by orthogonal circles A planar triangulation can be represented by orthogonal circles  no separating 3- or 4-cycles Andre’ev Thurston

33. Polyhedral version Andre’ev Every 3-connected planar graph is the skeleton of a convex polytope such that every edge touches the unit sphere

34. From polyhedra to circles horizon

35. From polyhedra to representation of the dual