Sandpile Group For Cones Over Trees

Dorian Smith
PhD Student
University of Minnesota Twin Cities

The sandpile group $K(G)$ of a graph $G$ is a finite abelian group, isomorphic to the cokernel of the reduced graph Laplacian of $G.$ We study $K(G)$ when $G = Cone(T)$. The graph $Cone(T)$ is obtained from a tree $T$ on $n$ vertices by attaching a new cone vertex attached to all other vertices. For two such families of graphs, we will describe $K(G)$ exactly: the fan graphs $Cone(P_n)$ where $P_n$ is a path, and the thagomizer graph $Cone(S_n)$ where $S_n$ is the star-shaped tree. The motivation is that these two families turn out to be extreme cases among $Cone(T)$ for all trees $T$ on $n$ vertices.