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:
http://gmichaelguy.com/vigregraphs/  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 
Brandon Samples 
Paley Graphs and Maple Computations 
September 26 
Michael Berglund 
Eigenvalues and Laplacians 
October 3 
George Vulov 
Algorithm for the Smith Normal Form 
October 10 
Brandon Samples 
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 