Let $G$ be a finite, connected and simple graph. The critical group $K(G)$, or equivalently the sandpile group, is the twist group constructed by taking the cokernel of the graph Laplacian $\operatorname{cok}(L)$. A family of graphs with relatively simple acyclic critical groups, with the ultimate goal of understanding whether multiple divisors, i.e., formal linear combinations of the vertices of $G$, produce $K(G)$. to investigate. These graphs, called hinge graphs, can be intuitively understood by taking multiple primitive shapes and “gluing” them together by a single shared edge and corresponding two shared vertices. The construction of explicit critical groups when all primitive shapes are identical. Furthermore, we prove the order of the three special divisors. To prove the structure, we generalize many of the preceding results. The structure of the critical group in the generalized case can be translated into a problem in number theory, but the proof is left for future work.
