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