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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 |

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).