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Spinorial Formulations of Graph Problems

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Abstract

Beginning with an arbitrary finite graph, various spinor spaces are constructed within Clifford algebras of appropriate dimension. Properties of spinors within these spaces then reveal information about the structure of the graph. Spinor polynomials are introduced and the notions of degrees of polynomials and Fock subspace dimensions are tied together with matchings, cliques, independent sets, and cycle covers of arbitrary finite graphs. In particular, matchings, independent sets, cliques, cycle covers, and cycles of arbitrary length are all enumerated by dimensions of spinor subspaces, while sizes of maximal cliques and independent sets are revealed by degrees of spinor polynomials. The spinor adjacency operator is introduced and used to enumerate cycles of arbitrary length and to compute graph circumference and girth.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL, 62026-1653, USA

    Glenn M. Harris & G. Stacey Staples

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  1. Glenn M. Harris
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  2. G. Stacey Staples
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Correspondence to G. Stacey Staples.

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Harris, G.M., Staples, G.S. Spinorial Formulations of Graph Problems. Adv. Appl. Clifford Algebras 22, 59–77 (2012). https://doi.org/10.1007/s00006-011-0298-0

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  • DOI: https://doi.org/10.1007/s00006-011-0298-0

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