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Eigenvalues of a Graph

Eigenvalues of a Graph. Scott Grayson. Adjacency Matrix. source: http://www.stoimen.com/blog/2012/08/31/computer-algorithms-graphs-and-their-representation/. Properties of an Adjancency Matrix. Symmetric n eigenvalues corresponding to n eigenvectors Zero Trace (sum of the diagonal)

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Eigenvalues of a Graph

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  1. Eigenvalues of a Graph Scott Grayson

  2. Adjacency Matrix source: http://www.stoimen.com/blog/2012/08/31/computer-algorithms-graphs-and-their-representation/

  3. Properties of an Adjancency Matrix • Symmetric • n eigenvalues corresponding to n eigenvectors • Zero Trace (sum of the diagonal) • sum of all eigenvalues equals the trace • :. sum of all eigenvalues is zero

  4. Eigenvalues of a Graph A = Images: wolframalpha and wikipedia

  5. Eigenvalues of a Graph A = To find eigenvalues, solve for k: det( A - k*I ) = 0*where I is the Identity Images: wolframalpha and wikipedia

  6. Eigenvalues of a Graph A = Characteristic polynomial: To find eigenvalues, solve for k: det( A - k*I ) = 0*where I is the Identity Eigenvalues: k = -1, -1, 2 Images: wolframalpha and wikipedia

  7. More on EigenValues of A • The term “spectra” is used to describe the eigenvalues, eigenvectors and characteristic polynomial of the graph • Non isomorphic graphs with the same spectra are called “co-spectral” • Co-spectral Trees are common

  8. Co Spectral Trees Example • These trees are non-isomorphic, but co-spectral. • Characteristic polynomial: • “As n -> infinity, almost no trees are uniquely determines by their spectra” Images: “Introduction to Graph Theory” by West

  9. Laplacian Matrix • L = D - A • L is the Laplacian matrix • A is the adjacency matrix • D is the degree matrix • diagonal matrix containing the degree of each vertex Image: Wikipedia

  10. Properties of the Laplacian Spectrum • Eigenvalues will range between zero and 2 • The smallest eigenvalue of L is zero • If G is connected, the eigenvalue zero has multiplicity 1 • if multiplicity > 1 this tells us how many connected components the graph has • If the largest eigenvalue is 2, G has a bipartite component

  11. Part of a Lecture by Luca Trevisan http://youtu.be/iu6EX9Xt3gA?t=7m53s

  12. Applications • Minimization for other graph problems • ex. coloring • Examining connectivity in networks • Google PageRank algorithm • Recommendations (music, movies friends)

  13. PageRank • Developed in 1996 by Larry Page and Sergey Brin at Stanford • old method: “text ranking” • PageRank attempts to model a person randomly clicking links • Viewed as an eigenvalue problem • Adjacency matrix for links between web pages • Values between 0 and 1

  14. PageRank • Requires multiple passes • recursive • some links are more important than others • Damping factor • about 85% of links are self links R = PageRank vector M = adjacency matrix d = damping factor N = number of websites

  15. History • 1980 “Spectra of Graphs” by Cvetković, Doob, and Sachs • 2nd edition in 1988 • 3rd edition in 1995 • Some other research came from the quantum chemistry field

  16. References • Brouwer, Andries E., and Willem H. Haemers. "The Spectra of Graphs." N.p., n.d. Web. 2 Apr. 2014. <http://www.win.tue.nl/~aeb/2WF02/spectra.pdf>. • Chung, Fan. "Eigenvalues and the Laplacian of a Graph." N.p., n.d. Web. <http://www.math.ucsd.edu/~fan/research/cb/ch1.pdf>. • Fox, Jacob. "Spectral Graph Theory." N.p., n.d. Web. <http://math.mit.edu/~fox/MAT307-lecture18.pdf>. • "Lecture #3: PageRank Algorithm - The Mathematics of Google Search." PageRank Algorithm. N.p., n.d. Web. 02 Apr. 2014. <http://www.math.cornell.edu/~mec/Winter2009/RalucaRemus/Lecture3/lecture3.html>. • Lovasz, Laszlo. "Eigenvalues of Graphs." N.p., n.d. Web. 2 Apr. 2014. <http://www.cs.elte.hu/~lovasz/eigenvals-x.pdf>. • Spielman, Daniel. "The Laplacian." N.p., n.d. Web. <http://www.cs.yale.edu/homes/spielman/561/2009/lect02-09.pdf>. • West, Douglas Brent. Introduction to Graph Theory. Upper Saddle River, NJ: Prentice Hall, 2001. Print. • Wilf, H. S. "Eigenvalues of a Graph and Its Chromatic Number." N.p., n.d. Web. <http://www.math.upenn.edu/~wilf/website/Eigenvalues%20of%20a%20graph.pdf>.

  17. HW 1 Find the eigenvalues of the Laplacian of this graph: Image: Wikipedia

  18. HW 2 Prove or disprove: • If k vertices have identical neighborhoods. Then zero is an eigenvalue with multiplicity at least k-1 * this question refers to the eigenvalues of the adjacency matrix. Not Laplacian

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