The Mathematics of Chip-firing is a solid introduction and overview of the growing field of
chip-firing. It offers an appreciation for the richness and diversity of the subject. Chip-firing …

Let M denote the Laplacian matrix of a graph G. Associated with G is a finite group Φ (G),
obtained from the Smith normal form of M, and whose order is the number of spanning trees …

We consider four conjectures related to the largest eigenvalue of (the adjacency matrix of) a
graph (ie, to the index of the graph). Three of them have been formulated after some …

The Jacobian group (also known as the critical group or sandpile group) is an important
invariant of a finite, connected graph X; it is a finite abelian group whose cardinality is equal …

The sandpile group of a graph is a refinement of the number of spanning trees of the graph
and is closely connected with the graph Laplacian. In this paper, the structure of the sandpile …

A graph is Laplacian integral if the spectrum of its Laplacian matrix consists entirely of
integers. We consider the class of constructably Laplacian integral graphs–those graphs that …

Reiner proposed two conjectures about the structure of the critical group of the n-cube Qn. In
this paper we confirm them. Furthermore we describe its p-primary structure for all odd …

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 …

We generalize the theory of critical groups from graphs to simplicial complexes. Specifically,
given a simplicial complex, we define a family of abelian groups in terms of combinatorial …

Sandpile groups are a subtle graph isomorphism invariant, in the form of a finite abelian
group, whose cardinality is the number of spanning trees in the graph. We study their group …