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Eigenvalues of Simplicial Rook Graphs
Algebra & Discrete Mathematics| Speaker: | Jeremy Martin, University of Kansas |
| Location: | 1147 MSB |
| Start time: | Mon, Nov 4 2013, 11:00AM |
Description
The simplicial rook graph SR(d,n) is the graph whose vertices are the
lattice points in the nth dilate of the standard simplex in R^d, with
two vertices adjacent if they differ in exactly two coordinates. We
prove that the adjacency and Laplacian matrices of SR(3,n) have integer
eigenvalues for every n, by calculating an explicit eigenbasis. In
fact, we conjecture that SR(d,n) is integral for all d and n, for which
we have lots of evidence but no proof as of yet. For some cases, the
distribution of eigenvalues is partially given by Mahonian numbers.
Throughout this project, we have made extensive use of Sage to discover
patterns and formulate conjectures. This is joint work with Jennifer
Wagner (Washburn University).