On the critical group of matrices

TL;DR

This paper generalizes the concept of the critical group from graphs to matrices over commutative rings, providing methods to compute these groups for various graph-related matrices and their generalizations.

Contribution

It introduces a generalized framework for critical groups of matrices over rings and computes these groups for specific graph constructions and their matrix analogs.

Findings

Diagonal matrices equivalent to generalized Laplacians are found.

Critical groups of m-cones of l-duplications of graphs are calculated.

Results include critical groups of bipartite complete graphs with modifications.

Abstract

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 a commutative ring with identity. In this article we find diagonal matrices that are equivalent to some matrices that generalize the reduced Laplacian matrix of the path, the cycle, and the complete graph over an arbitrary commutative ring with identity. We are mainly interested in those cases when the base ring is the ring of integers and some subrings of matrices. Using these equivalent diagonal matrices we calculate the critical group of the m-cones of the l-duplications of the path, the cycle, and the complete graph. Also, as byproduct, we calculate the critical group of another matrices, as the m-cones of the l-duplication of the bipartite complete graph with m vertices in each partition, the bipartite complete graph with 2m vertices minus a matching.