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2007 Algebraic Graph Theory |
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Standing: (Left to Right) Dino Lorenzini, George Vulov, Brandon
Samples, Michael Berglund, Jennifer Muskovin |
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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. |
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Links: |
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2006 Maple
files created by Michael Guy |
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August 29 |
Dino
Lorenzini |
Background
on Laplacians of Graphs |
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September
5 |
Dino
Lorenzini |
Background
on Laplacians of Graphs |
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September
12 |
Grant Fiddyment |
Statistics
on Critical Group Structure |
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September
19 |
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Paley
Graphs and Maple Computations |
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September
26 |
Michael
Berglund |
Eigenvalues
and Laplacians |
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October 3
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George
Vulov |
Algorithm
for the Smith |
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October
10 |
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Paley
Graphs/Results/Conjectures |
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October
17 |
Brian Cook |
Generating
Graphs with a Cyclic Critical Group |
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October
31 |
Leopold
Matamba |
The
Chip-Firing Game and the Critical Group |
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November 7 |
Leopold Matamba |
The
Chip-Firing Game and the Critical Group |
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November
14 |
Dino
Lorenzini |
General
Discussion |
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November
28 |
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Discussion |
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December
5 |
Jennifer
Muskovin |
Paley
Graphs |
Webpage created by Valerie
Hower
Modified by Brian Cook.
Archive: Fall 2006 web page.