2007 Algebraic Graph Theory 

Standing: (Left to Right) Dino Lorenzini, George Vulov, Brandon
Samples, Michael Berglund, Jennifer Muskovin 
One usually associates to a graph
G on n vertices two (n x n)matrices, the adjacency matrix A and the
Laplacian matrix L. Both A and L have a set of eigenvalues, and a Smith normal
form over the integers. Much has been written on the relationships between
the eigenvalues and the combinatorics/topology of the graph. Equivalent to the Smith normal
form of a graph is a finite abelian group that has 'appeared' independently
in several different fields, and is known under several names, such as the
component group, the critical group, or the sandpile group. This interesting
group is the main motivation for studying the Smith normal form of the
Laplacian. Its order is the number of spanning trees of the graph. In our VIGRE group, each
participant selects one or more problems to work on dealing with the Smith
normal form of the Laplacian. Often, students work together trying to solve a
problem as a team. Our weekly meetings are in the style of a seminar in which
a participant presents to the group. He or she may present background
material, interesting problems for solving, or proofs of original work. 
Links: 

2006 Maple
files created by Michael Guy 



August 29 
Dino
Lorenzini 
Background
on Laplacians of Graphs 
September
5 
Dino
Lorenzini 
Background
on Laplacians of Graphs 
September
12 
Grant Fiddyment 
Statistics
on Critical Group Structure 
September
19 

Paley
Graphs and Maple Computations 
September
26 
Michael
Berglund 
Eigenvalues
and Laplacians 
October 3

George
Vulov 
Algorithm
for the Smith 
October
10 

Paley
Graphs/Results/Conjectures 
October
17 
Brian Cook 
Generating
Graphs with a Cyclic Critical Group 
October
31 
Leopold
Matamba 
The
ChipFiring Game and the Critical Group 
November 7 
Leopold Matamba 
The
ChipFiring Game and the Critical Group 
November
14 
Dino
Lorenzini 
General
Discussion 
November
28 

Discussion 
December
5 
Jennifer
Muskovin 
Paley
Graphs 
Webpage created by Valerie
Hower
Modified by Brian Cook.
Archive: Fall 2006 web page.