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Algebraic Graph Theory |
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Standing: (Left to Right) Dino Lorenzini,
Nathan Walters, Grant Fiddyment, Renee Zawistowski, Al LaPointe, Juhyung Yi |
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One usualy
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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Maple
files created by Michael Guy |
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August 30
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Dino Lorenzini |
Background
on Laplacians of Graphs |
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September
6 |
Dino Lorenzini |
Background
on Laplacians of Graphs |
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Al LaPointe |
The
Order of the Component Group |
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September
13 |
Dino Lorenzini |
Interesting
Problems |
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Aja Johnson |
Almost
All Graphs are Asymmetrical |
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September
20 |
Aja Johnson |
Almost
All Graphs are Asymmetrical --Part 2 |
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September
27 |
Dino Lorenzini |
Resources
for Finding Research Papers |
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Jerry Hower |
Subgraphs of K_n and Their Complements |
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October 4
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Michael
Guy and Grant Fiddyment |
Writing
Programs in Maple |
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October
11 |
Valerie Hower |
Line
Graphs and Matroid Polytopes |
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October
18 |
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Dicussion of our conjectures |
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November
1 |
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Discussion
of our results |
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November
8 |
Renee Zawistowski |
Diagonalizing Matrices over a PID |
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November
15 |
Nathan
Walters |
Payley Graphs |
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November
29 |
Jerry Hower |
An
Analogue of the Riemann-Roch Theorem for Graphs |
Webpage created by Valerie
Hower