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Möbius function for the set of acyclic directed backbone graphs

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Abstract

Consider a two-rooted graphG, the edges of which are directed in such a way that there are no cycles and every edge belongs to some self-avoiding walk from rootu to rootv which follows the direction of the edges. Au-v backbone ofG is a subgraph formed by taking the union of any subset of directed self-avoiding walks fromu tov. Let ℬ uv be the set of all such backbones ofG partially ordered by set-inclusion. We prove the conjecture of Bhatti and Essam that the Möbius function of this set is given, for acyclicb,b′∈ℬ uv withb⩽b′, byμ(b,b′)=(−1)c′-c, wherec andc′ are the respective cycle ranks ofb andb′. The significance of this result in percolation theory is reviewed together with previous results for other sets of subgraphs.

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Authors and Affiliations

  1. School of Mathematical Sciences, Queen Mary College, University of London, London, UK

    D. K. Arrowsmith

  2. Department of Mathematics, Royal Holloway & Bedford New College, University of London, Egham, Surrey, UK

    J. W. Essam

Authors
  1. D. K. Arrowsmith
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  2. J. W. Essam
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Arrowsmith, D.K., Essam, J.W. Möbius function for the set of acyclic directed backbone graphs. J Stat Phys 58, 553–574 (1990). https://doi.org/10.1007/BF01112762

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  • DOI: https://doi.org/10.1007/BF01112762

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