Eigenvalues and critical groups of Adinkras

Abstract

Adinkras are signed graphs used to study supersymmetry in physics. We provide an introduction to these objects, and study the properties of their signed adjacency and signed Laplacian matrices. These matrices each have exactly two distinct eigenvalues (of equal multiplicity), making Adinkras closely related to the notions of strongly regular graphs. We also study the critical groups of Adinkras, and in particular determine their odd components. A novel technique of independent interest is used which considers modules over polynomial rings in order to study critical groups as Z-modules.