On the sandpile group of the cone of a graph

In this article, we study the sandpile group of the cone of a graph. After introducing the concept of uniform homomorphism of graphs we prove that every surjective uniform homomorphism of graphs induces an injective homomorphism between their sandpile groups. Also, we establish a relationship between the sandpile group of the cone of the cartesian product of graphs and the sandpile group of the cone of their factors. As an application of these results we obtain an explicit description of a set of generators of the sandpile group of the cone of the hypercube.