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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).