2000/10/25 by David G. Wagner, Wagner, David G. · 1 citation
Mathematics · #05C25 #05C30 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Rings and Algebras (math.RA) #math.CO #math.RA #msc:05C25 #msc:05C30
paper · pdf · doi:10.48550/arxiv.math/0010241
23 pages
arxiv created 2000/10/25 · openalex publication_date 2000/10/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The critical group K(G) of a directed graph G=(V,E) is the cokernel of the transpose of the Laplacian matrix of G acting on the integer lattice ZV. For undirected graphs G, this has been considered by Bacher, de la Harpe, and Nagnibeda, and by Biggs. We prove several things, among which are: K(G/p) is a subgroup of K(G) when p is an equitable partition and G is strongly connected; for undirected graphs, the torsion subgroup of K(G) depends only on the graphic matroid of G; and, the `dollar game' of Biggs can be generalized to give a combinatorial interpretation for the elements of K(G), when G is strongly connected.