January 2016 Spectra of graphs and geometric representations László Lovász Hungarian Academy of Sciences Eötvös Loránd University

January 2016

Extreme graphs? Shannon capacity? Strong regularity lemma? Property testing? Combinatorial Nullstellensatz? Anti-Hadamard matrices?Optimization?Eigenvalues? Eigenvalues!

January 2016 The eigenvalue gap Laplacian: adjacent positions degrees Eigenvalues: 1  2 ...  n

January 2016 Graphs and the eigenvalue gap Gap between 1 and 2  expander graph Alon - Milman Alon 1 < 2  graph is connected

January 2016 G-matrix: G = (V,E): simple graph, V=[n] well-signed G-matrix: Graphs, matrices, geometric representations Want to understand: UM=0, M: G-matrix, U   dxn d=rank(U)=corank(M) really good G-matrix: well-signed, one negative eigenvalue

January 2016 U M = 0 nullspace representation M  U: nullspace representation unique up to linear transformation

cycle fixed to convex polygon edges replaced by rubber bands M  U: rubber bands G is 3-connected planar, fixed cycle a face  planar embedding Tutte

M  U: rubber bands

Energy: M  U: rubber bands Equilibrium: (j free node) stress matrix stress in rubber band or strength of rubber band

January 2016 M  U: rubber bands M ij : stress define stress M ij so that equilibrium condition holds at all nodes

January 2016 U  M: bar-and-joint structures M has corank 3 and is positive semidefinite. Connelly

January 2016 U  M: bar-and-joint structures uiui M ij

January 2016 Braced stresses U M = 0 nullspace representation M’M’ M U 0 U’ U’M’=0

January 2016 Braced stresses

PP*P* u v q p January 2016 U  M: canonical stress on 3-polytopes Canonical braced stress

PP*P* u v q p January 2016 U  M: canonical stress on 3-polytopes The canonical braced stress matrix has 1 negative and 3 zero eigenvalues. L (really good G-matrix)

January 2016 M  U: the Colin de Verdière number G : connected graph Roughly: multiplicity of second largest eigenvalue of adjacency matrix And: non-degeneracy condition on weightings Largest has multiplicity 1. But: maximize over weighting the edges and diagonal entries

M ii arbitrary Strong Arnold Property normalization M=(M ij ): well-signed G-matrix M has =1 negative eigenvalue January 2016 [  (G)-connected]

μ(G) is minor monotone deleting and contracting edges μ  k is polynomial time decidable for fixed k for μ>2, μ(G) is invariant under subdivision for μ>3, μ(G) is invariant under Δ-Y transformation January 2016 Colin de Verdière number Basic properties

μ(G)  1  G is a path μ(G)  3  G is a planar Colin de Verdière, using pde’s Van der Holst, elementary proof μ(G)  2  G is outerplanar January 2016 Colin de Verdière number Special values

are connected. discrete Courant Nodal Theorem January 2016 M: really good G-matrix Mx = 0 supp(x) minimal Van der Holst’s lemma

like convex polytopes? or… connected January 2016 Van der Holst’s lemma for nullspace representation

S+S+ S-S- Corank bound January 2016

The eigenvalue gap Gap between 1 and 2  expander graph Alon - Milman Alon 1 < 2  graph is connected 2 < 3  G[supp + (v 2 )], G[supp - (v 2 )] are connected van der Holst

January 2016 The eigenvalue gap Gap between 2 < 3  G[supp + (v 2 )], G[supp - (v 2 )] are expanders expander ? Use (v 2 ) i 2 as weights!

G 3-connected planar  nullspace representation, scaled to unit vectors, gives embedding in S 2 L-Schrijver G 3-connected planar  nullspace representation can be scaled to convex polytope L January 2016 M  U: Steinitz representations

μ ( G )  1  G is a path μ ( G )  3  G is a planar μ ( G )  2  G is outerplanar μ ( G )  4  G is linklessly embeddable in 3-space L - Schrijver January 2016 Colin de Verdière number Special values

G 4-connected linkless embed. nullspace representation gives linkless embedding in  3 ? G path  nullspace representation gives embedding in  1 properly normalized G 2-connected  nullspace representation gives outerplanarouterplanar embedding in  2 G 3-connected  nullspace representation gives planarplanar embedding in  2, and also Steinitz representation L-Schrijver; L January 2016

Computing G-matrices Input: A 2-connected graph G=(V,E). Output: Either an outerplanar embedding of G, or a really good G-matrix with corank 3. Special case: G 3-connected planar  Steinitz representation of G

January 2016 U  M: circulations h: circulation on edges ij with u i and u j not parallel i  u i   2 Every G-matrix arises this way

January 2016 M well-signed  h is a counterclockwise circulation M has one negative eigenvalue  ? U  M: circulations

January 2016 Shifting the origin u i : nullspace representation, |u i |=1 M: really good G-matrix with corank 2

January 2016