1. Eigenvalues of a Graph
Scott Grayson

2. Adjacency Matrix
source:

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 = 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

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

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 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

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 <
Chung, Fan. "Eigenvalues and the Laplacian of a Graph." N.p., n.d. Web. <
Fox, Jacob. "Spectral Graph Theory." N.p., n.d. Web. <
"Lecture #3: PageRank Algorithm - The Mathematics of Google Search." PageRank Algorithm. N.p., n.d. Web Apr <
Lovasz, Laszlo. "Eigenvalues of Graphs." N.p., n.d. Web. 2 Apr < x.pdf>.
Spielman, Daniel. "The Laplacian." N.p., n.d. Web. < 09.pdf>.
West, Douglas Brent. Introduction to Graph Theory. Upper Saddle River, NJ: Prentice Hall, Print.
Wilf, H. S. "Eigenvalues of a Graph and Its Chromatic Number." N.p., n.d. Web. <

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