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28 March 2024
 
  » arxiv » math.CO/0010241

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The critical group of a directed graph
David G. Wagner ;
Date 25 Oct 2000
Subject Combinatorics; Rings and Algebras MSC-class: 05C30, 05C25 | math.CO math.RA
AbstractThe 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 Z^V. 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.
Source arXiv, math.CO/0010241
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