Sandpile Groups of Cones over Trees

Let $G=(V,E)$ be a loopless, undirected, connected graph. The sandpile group $K(G)$ is the finite abelian group isomorphic to the cokernel of the reduced graph Laplacian of $G$. Here we prove the exact group stucture of $K(G)$ for two types of graphs $G$: a path with a cone vertex attached, commonly referred to as an $n$-fan, and that of a star with a cone vertex attached. The motivation is that we hope that these two families will be the extreme cases, giving bounds on the structure of the sandpile groups for all other cones over trees on a fixed number of vertices.