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Zeons, Orthozeons, and Graph Colorings

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Abstract

Nilpotent adjacency matrix methods have proven useful for counting self-avoiding walks (paths, trails, cycles, and circuits) in finite graphs. In the current work, these methods are extended for the first time to problems related to graph colorings. Nilpotent-algebraic formulations of graph coloring problems include necessary and sufficient conditions for k-colorability, enumeration (counting) of heterogeneous and homogeneous paths, trails, cycles, and circuits in colored graphs, and a matrix-based greedy coloring algorithm. Introduced here also are the orthozeons and their application to counting monochromatic self-avoiding walks in colored graphs. The algebraic formalism easily lends itself to symbolic computations, and Mathematica-computed examples are presented throughout.

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Authors and Affiliations

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

    G. Stacey Staples & Tiffany Stellhorn

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  1. G. Stacey Staples
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  2. Tiffany Stellhorn
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Correspondence to G. Stacey Staples.

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Communicated by Eckhard Hitzer

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Staples, G.S., Stellhorn, T. Zeons, Orthozeons, and Graph Colorings. Adv. Appl. Clifford Algebras 27, 1825–1845 (2017). https://doi.org/10.1007/s00006-016-0742-2

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