A Rewriting Approach to Graph Invariants
Diagrammatic calculation is a powerful tool that gets near indispensable when one tries to manage some of the newer algebraic structures that have been popping up in the last couple of decades. Concretely, it generalises the underlying structure of expressions to being general graphs, where traditional algebraic notation only supports path- or treelike expressions. This paper demonstrates how to apply the author's Generic Diamond Lemma in diagrammatic calculations, by solving through elementary rewriting techniques the problem of classifying all multigraph invariants satisfying a linear contract—delete recursion. (As expected, this leads one to rediscover the Tutte polynomial, along with some more degenerate invariants.) In addition, a concept of “semigraph” is defined which formalises the concept of a graph-theoretical “gadget”.
KeywordsMonoidal Category Internal Edge Diagrammatic Calculation External Edge Spin Network
Unable to display preview. Download preview PDF.
- 5.Predrag Cvitanović: Group Theory — Tracks, LieÕs, and Exceptional Groups. Web book (modification of March 17, 2004), http://chaosbook.org/GroupTheory/
- 6.Reinhard Diestel: Graph Theory (Graduate Texts in Mathematics 173), Springer, Berlin, 1997; ISBN 0-387-98211-6Google Scholar
- 7.Lars Hellström: A Generic Framework for Diamond Lemmas, 2007; arXiv:0712.1142v1 [math.RA]Google Scholar
- 8.Gerardus't Hooft and Martinus J. G. Veltman: Diagrammar, CERN, Geneva, 1973Google Scholar
- 10.Seth A. Major: Spin Network Primer, (1999); arXiv:gr-qc/9905020v2Google Scholar
- 11.Alan D. Sokal: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, arXiv:math.CO/0503607v1Google Scholar