Geometric Representations of Graphs and Energy Equilibrium: A Study

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

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

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

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

7. p F F’ p’ 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.

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

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 no algebraic relation between edge strength for a.a. edge strength, B is independent

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

34. From polyhedra to circles horizon

35. From polyhedra to representation of the dual