Abstract
The 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 spectrum for every \(n\). The proof proceeds by calculating an explicit eigenbasis. We conjecture that \(SR(d,n)\) is integral for all \(d\) and \(n\), and present evidence in support of this conjecture. For \(n<\left( {\begin{array}{c}d\\ 2\end{array}}\right) \), the evidence indicates that the smallest eigenvalue of the adjacency matrix is \(-n\), and that the corresponding eigenspace has dimension given by the Mahonian numbers, which enumerate permutations by number of inversions.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
