On the Sandpile group of the cone of a graph

TL;DR

This paper explores the structure of sandpile groups of graph cones, introduces uniform homomorphisms, and provides explicit generators for the sandpile group of hypercube cones.

Contribution

It offers a partial description of sandpile groups for cones of Cartesian product graphs and introduces uniform homomorphisms with injective properties.

Findings

Describes the sandpile group of the cone of a Cartesian product of graphs.

Introduces the concept of uniform homomorphism of graphs.

Provides explicit generators for the sandpile group of hypercube cones.

Abstract

In this article, we give a partial description of the sandpile group of the cone of the cartesian product of graphs in function of the sandpile group of the cone of their factors. Also, we introduce the concept of uniform homomorphism of graphs and prove that every surjective uniform homomorphism of graphs induces an injective homomorphism between their sandpile groups. As an application of these result we obtain an explicit description of a set of generators of the sandpile group of the cone of the hypercube of dimension d.