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Eigenvalues and geometric representations of graphs L á szl ó Lov á sz Microsoft Research

Eigenvalues and geometric representations of graphs L á szl ó Lov á sz Microsoft Research One Microsoft Way, Redmond, WA 98052 lovasz@microsoft.com. Adjacency matrix:. Laplacian:. Matrices associated with graphs. (all graphs connected). Adjacency matrix:. Laplacian:.

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Eigenvalues and geometric representations of graphs L á szl ó Lov á sz Microsoft Research

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  1. Eigenvalues and geometric representations of graphs László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052 lovasz@microsoft.com

  2. Adjacency matrix: Laplacian: Matrices associated with graphs (all graphs connected)

  3. Adjacency matrix: Laplacian:

  4. eigenvalues of adjacency matrix eigenvalues of Laplacian Eigenvalues and eigenvectors

  5. -1 1 1.481 1.193 0 0 1 1 1 1 -1 1.481 -1 1 -1 Eigenvalues and eigenvectors

  6. average degree maximum degree If G is regular of degree d, then The largest eigenvalue

  7. Wilf The largest eigenvalue chromatic number maximum clique

  8. not used! The largest eigenvalue

  9. G bipartite  1 -1 1 1 1 -1 1 1 1 -1 1 1 The smallest eigenvalue

  10. Proof uses only 0 entries: can replace 1’s by anything G k-colorable  Hoffman The smallest eigenvalue maximizing we get: Polynomial time computable!

  11. Computing semidefinite optimization problem

  12. Transition matrix (of random walk): Another matrix associated with graphs Adjacency matrix: Laplacian: (Not much difference if graph is regular.)

  13. Random walks How long does it take to get completely lost?

  14. Sampling by random walk S: large and complicated set (all lattice points in convex body all states of a physical system all matchings in a graph...) Want: uniformly distributed random element from S Applications: - statistics - simulation - counting - numerical integration - optimization - card shuffling...

  15. One general method for sampling: random walks (+rejection sampling, lifting,…) Want: sample from setV Construct regular connected non-bipartite graph with node set V Walk for Tsteps ???????????? mixing time Output the final node

  16. 5 4 5 4 2 3 3 2 1 1 Given: poset Step: - pick randomly label i<n; - interchange i and i+1 if possible Node: compatible linear order Example: random linear extension of partial order

  17. The second largest eigenvalue

  18. in random walk: in sequence of independent samples: conductance: Conductance frequency of stepping from S to V \S Edge-density in cut

  19. Jerrum - Sinclair Conductance and eigenvalue gap eigenvalues of transition matrix up to a constant factor

  20. Jerrum - Sinclair Conductance and eigenvalue gap eigenvalues of transition matrix

  21. G connected  l1has multiplicity1 eigenvector is all-positive Frobenius-Perron What about the eigenvectors? eigenvalues of A

  22. Van der Holst are connected. What about the eigenvectors? eigenvalues of A

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