Related papers: Critical groups of simplicial complexes

The critical group of a finite graph is an abelian group defined by the Smith normal form of the Laplacian. We determine the the critical groups of the Peisert graphs, a certain family of strongly regular graphs similar to, but different…

The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. This paper provides three basic structural results on the critical group of a line graph. The first deals with connected…

This paper makes some preliminary observations towards an extension of current work on graphs defined on groups to simplicial complexes. I define a variety of simplicial complexes on a group which are preserved by automorphisms of the…

Motivated by the appearance of embeddings in the theory of chip firing and the critical group of a graph, we introduce a version of the critical group (or sandpile group) for combinatorial maps, that is, for graphs embedded in orientable…

The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. For a graph G with a certain reflective symmetry, we generalize a result of Ciucu-Yan-Zhang factorizing the spanning tree…

The critical group of an abelian network is a finite abelian group that governs the behavior of the network on large inputs. It generalizes the sandpile group of a graph. We show that the critical group of an irreducible abelian network…

An arithmetical structure on a finite, connected graph without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there is a divisor of the sum of the integers at adjacent vertices,…

Given a graph G with a distinguished vertex s, the critical group of (G,s) is the cokernel of their reduced Laplacian matrix L(G,s). In this article we generalize the concept of the critical group to the cokernel of any matrix with entries…

In recent work, Benkart, Klivans, and Reiner defined the critical group of a faithful representation of a finite group $G$, which is analogous to the critical group of a graph. In this paper we study maps between critical groups induced by…

In this expository article intended to be accessible to undergraduate students we introduce a finite abelian group that can be associated to any finite connected graph. This group can be defined in an elementary combinatorial way in terms…

The critical group of a connected graph is closely related to the graph Laplacian, and is of high research value in combinatorics, algebraic geometry, statistical physics, and several other areas of mathematics. In this paper, we study the…

Chip firing provides a way to study the sandpile group (also known as the Jacobian) of a graph. We use a generalized version of chip firing to bound the number of invariant factors of the critical group of an arithmetical structure on a…

An arithmetical structure on a finite, connected graph without loops is an assignment of positive integers to the vertices that satisfies certain conditions. Associated to each of these is a finite abelian group known as its critical group.…

The kernel of the natural projection of a graph product of groups onto their direct product is called the Cartesian subgroup of the graph product. This construction generalises commutator subgroups of right-angled Coxeter and Artin groups.…

Critical ideals generalize the critical group, Smith group and the characteristic polynomials of the adjacency and Laplacian matrices of a graph. We give a complete characterization of the digraphs with at most one trivial critical ideal.…

Let $G$ be a finite, connected, simple graph. The critical group $K(G)$, also known as the sandpile group, is the torsion subgroup of the cokernel of the graph Laplacian $\operatorname{cok}(L)$. We investigate a family of graphs with…

This expository article presents a self-contained introduction to simplicial homology for finite simplicial complexes, emphasizing concrete computation and geometric intuition. Beginning with orientations of simplices and the construction…

The edge group of a simplicial complex is a well-known, combinatorial version of the fundamental group. It is a group associated to a simplicial complex that consists of equivalence classes of edge loops and that is isomorphic to the…

We introduce some determinantal ideals of the generalized Laplacian matrix associated to a digraph G, that we call critical ideals of G. Critical ideals generalize the critical group and the characteristic polynomials of the adjacency and…

The notion of Laplacian of a graph can be generalized to simplicial complexes and hypergraphs, and contains information on the topology of these structures. Even for a graph, the consideration of associated simplicial complexes is…