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Eigenvalues and Geometric Representations of Graphs: Insights from Laplacian and Adjacency Matrices

Dive into the world of graph theory and matrices associated with graphs in this comprehensive study. Learn about eigenvalues of adjacency and Laplacian matrices, average and maximum degrees, chromatic numbers, bipartite graphs, and more. Discover the importance of eigenvalues and eigenvectors in understanding graph structures, regularity, and properties. Explore applications such as statistics, simulation, optimization, and functional sampling through random walks and semidefinite optimization techniques. Uncover the connections between eigenvalues, conductance, and edge density in graphs, leading to insights on graph connectivity and spectral properties.

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Eigenvalues and Geometric Representations of Graphs: Insights from Laplacian and Adjacency Matrices

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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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