1. January 2016 Spectra of graphs and geometric representations László Lovász Hungarian Academy of Sciences Eötvös Loránd University lovasz@cs.elte.hu

2. January 2016

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

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

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

6. 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

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

8. 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

9. M  U: rubber bands

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

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

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

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

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

15. January 2016 Braced stresses

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

17. 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)

18. 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

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

20. 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

21. 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

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

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

24. S+S+ S-S- Corank bound January 2016

25. 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

26. 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!

27. 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

28. 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

29. 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

30. 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

31. 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

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

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

34. January 2016