Abstract
We introduce a family of transforms that extends graph- and matroid-theoretic duality, and includes trinities and so on. Associated with each such transform are λ -minor operations, which extend deletion and contraction in graphs. We establish how the transforms interact with our generalised minors, extending the classical matroid-theoretic relationship between duality and minors: \({(M/e)^* =M^* \backslash e}\). Composition of the transforms is shown to correspond to complex multiplication of appropriate parameters. A new generalisation of the MacWilliams identity is given, using these transforms in place of ordinary duality. We also relate the weight enumerator of a binary linear code at a real argument < –1 to the transform, with parameter on the unit circle, of a close relative of the indicator function of the dual code. This result extends to arbitrary binary codes. The results on weight enumerators can also be recast in terms of the partition function of the Ising model from statistical mechanics. Most of our work is done at the level of binary functions \({f : 2^E \rightarrow \mathbb{C}}\), which include matroids as a special case. The specialisation to graphs is obtained by letting f be the indicator function of the cutset space of a graph.
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Part of the work of this paper was done while the author was a Visiting Fellow at the Isaac Newton Institute for Mathematical Sciences, Cambridge, U.K., Jan.–Feb. 2008. An earlier version of the paper was Isaac Newton Institute Preprint No. NI09015-CSM.
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Farr, G.E. Transforms and Minors for Binary Functions. Ann. Comb. 17, 477–493 (2013). https://doi.org/10.1007/s00026-013-0178-5
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DOI: https://doi.org/10.1007/s00026-013-0178-5