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
o
•
n eigenvalues corresponding to n eigenvectors
Zero Trace (sum of the diagonal)
o
sum of all eigenvalues equals the trace
o
:. 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
=
Characteristic polynomial:
To find eigenvalues, solve for k:
det( A - k*I ) = 0
Eigenvalues:
k = -1, -1, 2
*where I is the Identity
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”
o Co-spectral Trees are common
Co Spectral Trees Example
•
•
These trees are non-isomorphic, but co-spectral.
o 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
o L is the Laplacian matrix
o A is the adjacency matrix
o 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
o
•
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
o
ex. coloring
Examining connectivity in networks
o
Google PageRank algorithm
o
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
o Values between 0 and 1
PageRank
R = PageRank vector
M = adjacency matrix
d = damping factor
N = number of websites
•
•
Requires multiple
passes
o
recursive
o
some links are more
important than others
Damping factor
o
about 85% of links are
self links
History
•
1980 “Spectra of Graphs” by Cvetković,
Doob, and Sachs
o
•
o
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/eigenvalsx.pdf>.
Spielman, Daniel. "The Laplacian." N.p., n.d. Web. <http://www.cs.yale.edu/homes/spielman/561/2009/lect0209.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.
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