Skip to Main content Skip to Navigation
Conference papers

On the Spectra of Simplicial Rook Graphs

Abstract : The $\textit{simplicial rook graph}$ $SR(d,n)$ is the graph whose vertices are the lattice points in the $n$th dilate of the standard simplex in $\mathbb{R}^d$, with two vertices adjacent if they differ in exactly two coordinates. We prove that the adjacency and Laplacian matrices of $SR(3,n)$ have integral spectra for every $n$. We conjecture that $SR(d,n)$ is integral for all $d$ and $n$, and give a geometric construction of almost all eigenvectors in terms of characteristic vectors of lattice permutohedra. For $n \leq \binom{d}{2}$, we give an explicit construction of smallest-weight eigenvectors in terms of rook placements on Ferrers diagrams. The number of these eigenvectors appears to satisfy a Mahonian distribution.
Document type :
Conference papers
Complete list of metadata

Cited literature [14 references]  Display  Hide  Download
Contributor : Alain Monteil <>
Submitted on : Tuesday, November 17, 2015 - 10:21:00 AM
Last modification on : Thursday, April 4, 2019 - 11:52:02 AM
Long-term archiving on: : Friday, April 28, 2017 - 1:38:16 PM


Publisher files allowed on an open archive


  • HAL Id : hal-01229747, version 1



Jeremy L. Martin, Jennifer D. Wagner. On the Spectra of Simplicial Rook Graphs. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.373-386. ⟨hal-01229747⟩



Record views


Files downloads