Notes on simplicial rook graphs

TL;DR

This paper studies the spectral properties and symmetries of simplicial rook graphs, revealing their integral eigenvalues, relation to Johnson graphs, and automorphism groups, thus advancing understanding of their combinatorial structure.

Contribution

It establishes the integral eigenvalues of simplicial rook graphs, identifies their relation to Johnson graphs, and determines their automorphism groups, providing new insights into their structure.

Findings

Eigenvalues are all integers.

Smallest eigenvalue is max(-n, -binomial(m, 2)).

Significant spectral overlap with Johnson graphs.

Abstract

The simplicial rook graph ${\rm SR}(m,n)$ is the graph of which the vertices are the sequences of nonnegative integers of length $m$ summing to $n$, where two such sequences are adjacent when they differ in precisely two places. We show that ${\rm SR}(m,n)$ has integral eigenvalues, and smallest eigenvalue $s = \max (-n, -{m \choose 2})$, and that this graph has a large part of its spectrum in common with the Johnson graph $J(m+n-1,n)$. We determine the automorphism group and several other properties.