The Critical Group of a Line Graph |
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| Authors: | Andrew Berget Andrew Manion Molly Maxwell Aaron Potechin Victor Reiner |
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| Institution: | 1. Department of Mathematics, University of California, Davis, CA, 95616, USA
2. Department of Mathematics, Princeton University, Princeton, NJ, 08540-1000, USA
3. Department of Mathematics and Computer Science, Colorado College, 14 E. Cache la Poudre St., Colorado Springs, CO, 80903, USA
4. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA
5. School of Mathematics, University of Minnesota, 206 Church St. S.E., Minneapolis, MN, 55455, USA
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| Abstract: | 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 graphs containing no cut-edge. Here the number of independent cycles in the graph, which is known to bound the number of generators for the critical group of the graph, is shown also to bound the number of generators for the critical group of its line graph.
- The second gives, for each prime p, a constraint on the p-primary structure of the critical group, based on the largest power of p dividing all sums of degrees of two adjacent vertices.
- The third deals with connected graphs whose line graph is regular. Here known results relating the number of spanning trees of the graph and of its line graph are sharpened to exact sequences which relate their critical groups.
The first two results interact extremely well with the third. For example, they imply that in a regular nonbipartite graph, the critical group of the graph and that of its line graph determine each other uniquely in a simple fashion. |
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