Centered Polygonal Lacunary Graphs: A Graph Theoretic Approach to p-Sequences of Centered Polygonal Lacunary Functions
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Department of Mathematics, Concordia College, Moorhead, MN 56562, USA
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Department of Chemistry, Concordia College, Moorhead, MN 56562, USA
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Author to whom correspondence should be addressed.
Mathematics 2019, 7(11), 1021; https://doi.org/10.3390/math7111021
Received: 14 October 2019 / Revised: 23 October 2019 / Accepted: 24 October 2019 / Published: 28 October 2019
(This article belongs to the Special Issue Advanced Mathematics for Physical Chemistry and Chemical Physics)
This work is on the nature and properties of graphs which arise in the study of centered polygonal lacunary functions. Such graphs carry both graph-theoretic properties and properties related to the so-called p-sequences found in the study of centered polygonal lacunary functions. p-sequences are special bounded, cyclic sequences that occur at the natural boundary of centered polygonal lacunary functions at integer fractions of the primary symmetry angle. Here, these graphs are studied for their inherent properties. A ground-up set of planar graph construction schemes can be used to build the numerical values in p-sequences. Further, an associated three-dimensional graph is developed to provide a complementary viewpoint of the p-sequences. Polynomials can be assigned to these graphs, which characterize several important features. A natural reduction of the graphs original to the study of centered polygonal lacunary functions are called antipodal condensed graphs. This type of graph provides much additional insight into p-sequences, especially in regard to the special role of primes. The new concept of sprays is introduced, which enables a clear view of the scaling properties of the underling centered polygonal lacunary systems that the graphs represent. Two complementary scaling schemes are discussed.
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Keywords:
lacunary function; gap function; centered polygonal numbers; triangular numbers; renormalization; scaling; fractal character; primes
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MDPI and ACS Style
Sullivan, K.; Rutherford, D.; Ulness, D.J. Centered Polygonal Lacunary Graphs: A Graph Theoretic Approach to p-Sequences of Centered Polygonal Lacunary Functions. Mathematics 2019, 7, 1021.
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