Sandpile Groups of Cayley Graphs of $\mathbb{F}_2^r$

Abstract

The sandpile group K(G) of a connected graph G, defined to be the torsion part of the cokernel of its Laplacian matrix L(G), is a subtle graph isomorphism invariant with combinatorial, algebraic, and geometric descriptions. Past work has focused on determining the sandpile group of the hypercube. In this project, we study the sandpile group for a more general collection of graphs, the Cayley graphs of the group F r 2 . While the Sylow-p component of such groups has been classified for p = 2, much less is known about the Sylow-2 component. In this paper, we use ring theory to prove a sharp upper bound for the largest Sylow-2 subgroup of these sandpile groups. In the case of the hypercube, we provide an exact formula for the largest n − 1 among its Sylow-2 cyclic factors. We also find the number of Sylow-2 cyclic factors for "generic" Cayley graphs. With these methods, we also classify the sandpile group for r = 2 and r = 3 in the "generic" case.

The Critical Groups of Adinkras Up to 2-Rank of Cayley Graphs

“…Let A be an N -colored Adinkra on v vertices. Let the 2-rank of the Laplacian of the underlying Cayley graph be v/2 − m; we have m 󰃍 0 by [14…”