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) sum of all eigenvalues equals the trace :. sum of all eigenvalues is zero

Eigenvalues of a Graph A = Images: wolframalpha and wikipedia

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

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

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

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

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

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

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

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

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

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

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

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>.

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

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