On the critical group of hinge graphs

TL;DR

This paper studies the structure of the critical (sandpile) group of hinge graphs, a family formed by gluing base shapes, providing explicit computations and generalizations for their critical groups.

Contribution

It introduces hinge graphs and derives explicit structures of their critical groups, including cases with varying base shape sizes, advancing understanding of divisors generating these groups.

Findings

Explicit critical group structure for identical base shapes

Order of three special divisors computed

Generalization to hinge graphs with variable base shape sizes

Abstract

Let $G$ be a finite, connected, simple graph. The critical group $K(G)$, also known as the sandpile group, is the torsion subgroup of the cokernel of the graph Laplacian $\operatorname{cok}(L)$. We investigate a family of graphs with relatively simple non-cyclic critical group with an end goal of understanding whether multiple divisors, i.e., formal linear combinations of vertices of $G$, generate $K(G)$. These graphs, referred to as hinge graphs, can be intuitively understood by taking multiple base shapes and ``gluing" them together by a single shared edge and two corresponding shared vertices. In the case where all base shapes are identical, we compute the explicit structure of the critical group. Additionally, we compute the order of three special divisors. We prove the structure of the critical group of hinge graphs when variance in the number of vertices of each base shape is allowed, generalizing many of the aforementioned results.