Eigenvalues and geometric representations of graphs László Lovász Microsoft Research One Microsoft Way, Redmond, WA

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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?

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:

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

Stresses of tensegrity frameworks bars struts cables x y Equilibrium:

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

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

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

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

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

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

From polyhedra to circles horizon

From polyhedra to representation of the dual