# Centered Polygonal Lacunary Graphs: A Graph Theoretic Approach to p-Sequences of Centered Polygonal Lacunary Functions

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Notation and Background Results

#### 2.1. Lacunary Sequences and Lacunary Functions

#### 2.2. Centered Polygonal and Triangular Numbers

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

#### 2.3. p-Sequences

#### 2.4. Fiber Bundle Representation of Centered Polygonal Lacunary Sequences

**Theorem**

**4.**

## 3. Construction of Two-Dimensional Base Space Graphs

#### 3.1. The Construction of the Two-Dimensional Base Space Graphs

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Construction**

**1.**

- 1.
- Add $2p$ division points, ${D}_{p}$, to ${S}^{1}$.
- 2.
- Add a vertex at point 0 on ${S}^{1}$. This is referred to as vertex 0. The antipode to vertex 0 is referred to as vertex π.
- 3.
- Add a terminal edge to vertex 0.
- 4.
- Perform unit increase progression through ${D}_{p}$ for $2p-1$ iterations. This is done going counterclockwise from vertex 0. This builds the set of vertices, ${V}_{p}$. Each step in the progression creates an edge that terminates to either a new or an existing vertex and builds the set of edges, ${E}_{p}$.
- 5.
- Add a terminal edge to vertex π. This completes the graph.

#### 3.2. Information Carried in Base Space Graphs

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Theorem**

**6.**

**Proof.**

- creates a new vertex, in which case a new term is represented; or
- lands on an existing vertex, in which case represents a degeneracy (multiplicity) of a term that already exists.

#### 3.3. Rectangular Construction for Odd p

**Construction**

**2.**

- 1.
- Add $2p$ division points to ${S}^{1}$.
- 2.
- Add the 0 and π vertices and connecting edge. This is referred to as the 0-rectangle.
- 3.
- Simultaneously perform the first iteration of the unit increase progression counterclockwise from the 0 vertex and from the π vertex.
- 4.
- Add connecting edges from vertex $\frac{\pi}{p}$ to vertex π and $\pi +\frac{\pi}{p}$ to vertex 0. This forms the 1-rectangle.
- 5.
- The next iterations of the unit increase progression forms the 2-rectangle, etc. The number of iterations of the unit step progression is $\frac{p+1}{2}$. The final rectangle formed is called themaximal rectangle.
- 6.
- Add a terminal edge to vertices 0 and π. This completes the graph.

**Theorem**

**7.**

**Proof.**

**Corollary**

**8.**

**Corollary**

**9.**

**Corollary**

**10.**

## 4. Three-Dimensional Base Space Graphs

**Construction**

**3.**

- 1.
- Create a lattice on the cylinder ${S}^{1}\times {\mathbb{R}}^{+}$ using ${D}_{p}$ and the positive integers.
- 2.
- Create the three-dimensional graph from the corresponding two-dimensional graph by placing the jth vertex at its corresponding division point and value equal to half of its degree (equal to ${m}_{j}\in {M}_{p}$).
- 3.
- Add a terminal edge to vertex 0 and π (usually not shown in a rendering of the graph). This completes the graph.

## 5. Prime Decomposition

**Construction**

**4.**

- 1.
- The prime power polynomials are generated via,$$\begin{array}{cc}\hfill \mathcal{P}\left({2}^{m}\right)& ={2}^{m}\hfill \\ \hfill \mathcal{P}\left({q}^{n}\right)& =1+\sum _{j=1}^{n}{(-1)}^{j+1}{q}^{j}\phantom{\rule{1.em}{0ex}}n\phantom{\rule{4.pt}{0ex}}odd\hfill \\ \hfill \mathcal{P}\left({q}^{n}\right)& =2+\sum _{j=1}^{n}{(-1)}^{j}{q}^{j}\phantom{\rule{1.em}{0ex}}n\phantom{\rule{4.pt}{0ex}}even.\hfill \end{array}$$
- 2.
- Find the prime decomposition of p and let r equal the number of distinct odd primes present in the decomposition.
- 3.
- For a given p, multiply the prime power polynomials represented in the prime decomposition of p and divide by a factor of ${2}^{r-1}$ to obtain the overall polynomial.$$\mathcal{P}\left(p\right)=\mathcal{P}({2}^{m}{q}_{1}^{{n}_{1}}{q}_{2}^{{n}_{2}}\cdots {q}_{r}^{{n}_{r}})=\frac{1}{{2}^{r-1}}\mathcal{P}\left({2}^{m}\right)\mathcal{P}\left({q}_{1}^{{n}_{1}}\right)\mathcal{P}\left({q}_{2}^{{n}_{2}}\right)\cdots \mathcal{P}\left({q}_{r}^{{n}_{r}}\right).$$
- 4.
- Evaluating $\mathcal{P}\left(p\right)$ for a given p gives the number of vertices in ${\mathcal{G}}_{p}$.$$|{\mathcal{G}}_{p}|=|{V}_{p}|=\mathcal{P}\left(p\right).$$

**Corollary**

**11.**

#### The Prime Power Family of Graphs

**Conjecture**

**12.**

## 6. Antipodal Condensed Graphs

**Construction**

**5.**

- 1.
- Identify all antipodal pairs as $\left\{{v}_{j},{v}_{j+p}\right\}\to {\widehat{v}}_{j}$.
- 2.
- The collections of edges, ${e}_{ij}\in {E}_{p}$, that connect $\left\{{v}_{i},{v}_{i+p}\right\}$ to $\left\{{v}_{j},{v}_{j+p}\right\}$ are identified as a single edge, ${\widehat{e}}_{ij}\in {\widehat{E}}_{p}$.
- 3.
- The terminal edges are viewed as connecting vertex 0 with vertex π via the “vertex at infinity”. This collapses to a self-loop ${\widehat{e}}_{0}\in {E}_{p}$. This completes the construction.

#### 6.1. Primes

#### 6.2. Powers of Primes

## 7. Scaling Properties

#### 7.1. Sprays

**Definition**

**7.**

#### 7.2. Sprays, Renormalization, and Fractal Character

**Conjecture**

**13.**

**Conjecture**

**14.**

**Definition**

**8.**

**Definition**

**9.**

**Conjecture**

**15.**

## 8. Graph Theoretic Properties

#### 8.1. Cliques and Chromatic Number

**Conjecture**

**16.**

**Proposition**

**17.**

**Conjecture**

**18.**

#### 8.2. Cycles and the Cycle Spectrum

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Two examples of centered polygonal lacunary functions, ${f}_{18}\left(z\right)$ plotted for $|{f}_{18}\left(z\right)|\le 1$, where $g\left(n\right)={C}^{\left(k\right)}\left(n\right)$ (see Equation (3)). The left panel shows the case for $k=2$ while the right shows the case of $k=5$. The superimposed line segments represent the symmetry angles ($p=1$ through $p=25$). The terminus of the line segments at the natural boundary sits at a point where $|{f}_{18}\left(z\right)|$ does not converge, but the limit value is a bounded $4p$-cycle.

**Figure 2.**Two-dimensional base space graphs for $p=1$ through $p=16$ (read left to right, top to bottom). The grey circle represents ${S}^{1}$ and small grey dots (sometimes obscured by larger blue dots) represent $2p$ equally spaced positions ${S}^{1}$. The graph, ${\mathcal{G}}_{p}$, is shown in blue. The vertices of the graphs represent terms of the form ${e}^{\frac{j\pi i}{p}}$, $j\in \mathbb{Z}$. The edges (blue lines) represent steps in the unit increase progression (Definition 3).

**Figure 3.**The complete set of cross-sections for the case of $p=3$ (corresponding to the third graph in the top row of Figure 2). The leftmost cross-section is considered the base cross-section and is obtained directly from ${\mathcal{G}}_{3}$. The complete set of cross-sections is obtained from the base cross-section via unit increase progression starting on the first fiber to the right of the midpoint. Each cross-section directly gives the $4p$ numerical values in the centered polygonal lacunary sequence at the position on the natural boundary at angle ${\alpha}_{3}$.

**Figure 4.**Three-dimensional base space graphs for $p=1$ to $p=16$ (read left to right, top to bottom). These correspond directly to the graphs in Figure 2, now with the third dimension being the degree of the vertex. In addition to offering a different visual perspective of the p-sequences, the degrees of the vertices within a given graph are intimately related to the nature of p itself and to the values of the centered polygonal lacunary functions at the natural boundary (see text for discussion).

**Figure 5.**Three-dimensional base space graphs: ${\overline{\mathcal{G}}}_{59}$ (

**top left**); ${\overline{\mathcal{G}}}_{96}$ (

**top right**); ${\overline{\mathcal{G}}}_{323}$ (

**bottom left**); and ${\overline{\mathcal{G}}}_{646}$ (

**bottom right**).

**Figure 6.**${\overline{\mathcal{G}}}_{{3}^{3}}$ and corresponding slices (

**first**and

**second panels**), ${\overline{\mathcal{G}}}_{{7}^{2}}$ (

**third panel**), and ${\overline{\mathcal{G}}}_{{3}^{4}}$ (

**fourth panel**).

**Figure 7.**${\overline{\mathcal{G}}}_{1260}$ and ${\overline{\mathcal{G}}}_{315}$. The p values associated with these graphs differ by a factor of ${2}^{2}$ and hence have equal saturation values.

**Figure 8.**Behavior of the saturation of prime powers, ${q}^{m}$ for $q=3$ (circle); $q=5$ (square); $q=7$ (diamond); $q=11$ (triangle); and $q=13$,(inverted triangle). The vertical axis is s and the horizontal axis is m.

**Figure 9.**A plot of the entries of the adjacency matrix for several graphs. The special nature of primes and the inherent scaling properties are particularly noticeable. The

**top**shows the typical differences in the adjacency matrices for primes and non-primes by collecting ${\mathcal{G}}_{51}$, ${\mathcal{G}}_{52}$, ${\mathcal{G}}_{53}$, and ${\mathcal{G}}_{54}$. The

**second**row shows ${\mathcal{G}}_{{51}^{2}}$, ${\mathcal{G}}_{{52}^{2}}$, ${\mathcal{G}}_{{53}^{2}}$, and ${\mathcal{G}}_{{54}^{2}}$. The

**third**row shows ${\mathcal{G}}_{{5}^{2}}$, ${\mathcal{G}}_{{5}^{3}}$, ${\mathcal{G}}_{{5}^{4}}$, and ${\mathcal{G}}_{{5}^{5}}$. The

**bottom**row show several examples of other primes: ${\mathcal{G}}_{79}$, ${\mathcal{G}}_{179}$, ${\mathcal{G}}_{251}$, and ${\mathcal{G}}_{887}$.

**Figure 10.**Saturation values for the first 25,000 integer values of p. Visible striations appear because of the asymptotic “attractor” values of for number whose prime decomposition consist of high redundancy.

**Figure 11.**Renderings of ${\mathcal{G}}_{5}$. The

**left**panel is the two-dimensional base space graph. The grey dots represent the members of ${D}_{p}$ that are not vertices. The grey circle simple guides the eye to the embedding space, ${S}^{1}$. The

**middle**panel is a different representation of the graph that vertically juxtaposes antipodal vertices. The

**right**rendering is a reduction of the graph by associating antipodal points together as a single vertex. The resulting graphs is ${\mathcal{A}}_{5}$. This always reduces the number of vertices by half.

**Figure 13.**Antipodal condensed graphs, ${\mathcal{A}}_{p}$ for $p=2r$ for $r=3$–51 (odds). Graphs corresponding to $p=2$ times a prime are distinctively linearly in shape. Those corresponding to $p=2$ times a prime square also hold their distinctive flower shape.

**Figure 14.**The nested nature of the primes. ${\mathcal{A}}_{p}$ for $p=2\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}q$ (

**top row**), $p={2}^{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}q$ (

**middle row**) and $p={2}^{3}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}q$ (

**bottom row**) where $q=3,5,7$. In each case, the graph for a prime has a subgraph equal to the previous prime. This continues for higher primes.

**Figure 15.**Illustration of the twining of vertices that results when p is multiplied by 2. ${\mathcal{A}}_{25}$ is compared to ${\mathcal{A}}_{2\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}25}$ The middle two panels show the two copies of ${\mathcal{A}}_{25}$ contained in ${\mathcal{A}}_{2\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}25}$. The last panel shows the remaining edges. Twin vertices do not share an edge.

**Figure 16.**${\mathcal{A}}_{p}$ for p the squares of first 12 primes. The descriptor language for these types of graphs would be that they are $\frac{q-1}{2}$-petal flowers.

**Figure 17.**The ${\mathcal{A}}_{{5}^{n}}$ family of graphs. The

**top row**shows the n odd subfamily for $n=1,3,5$ and the

**bottom row**shows the n even subfamily for $n=2,4,6$.

**Figure 18.**A graph multiplication table for the first five odd primes, ${\mathcal{A}}_{p}$, where $p={q}_{i}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{q}_{j}$. The “product” is a web graph with $\frac{{q}_{i}-1}{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{q}_{j}-1}{2}$ faces (${q}_{i}\ne {q}_{j}$). The faces consist of one triangle and the remainder are quadrilaterals. The example of ${\mathcal{A}}_{77}$ is shown on the right with the faces labeled.

**Figure 19.**${\mathcal{A}}_{{5}^{4}}$ illustrates the concept of spray. The

**left panel**shows the entire graph while the

**right panel**is a blow-up of one of the petals. Five first-order sprays are connected to the ultimate vertex and each penultimate vertex is terminated by two sprays. Because $q=5$, the first-order sprays have five paths each and each path consists of five edges and four intermediate degree 3 vertices. Application of the renormalization scheme discussed in the text would yield ${\mathcal{A}}_{{5}^{2}}$, which is shown in Figure 17.

**Figure 20.**${\mathcal{A}}_{{5}^{6}}$ is an example of a graph with second-order sprays. The

**left panel**shows the entire graph while the

**middle**and

**right panels**are blow-ups of the area around the ultimate vertex. Five second-order sprays are connected to the ultimate vertex and each member of the spray is itself a (first-order) spray. Twenty-five first-order sprays connect the ultimate vertex to an antepenultimate vertex. These 25 first-order sprays are organized into five second-order sprays, each of which connects the ultimate vertex to the the penultimate vertex. Because $q=5$, the first-order sprays have five paths each and each path consists of five edges and four intermediate degree 3 vertices while the second-order sprays have five paths of first-order sprays. One iteration of the renormalization scheme discussed in the text would yield ${\mathcal{A}}_{{5}^{4}}$ which is shown in Figure 19. A second iteration of the renormalization scheme would yield ${\mathcal{A}}_{{5}^{2}}$, which is shown in Figure 17.

**Figure 21.**Maximum clique subgraphs for the cases of p equals two, three, four, and five distinct primes. The maximum clique subgraph is unique.

**Figure 22.**Maximum cliques for: ${\mathcal{A}}_{30}$ (

**top row**); ${\mathcal{A}}_{60}$ (

**middle row**); and ${\mathcal{A}}_{90}$ (

**bottom row**). The left panel in the middle row shows the complete graph isomorphic to the maximal cliques in this set of graphs. Note that the isomorphism is up to the self-loops.

**Figure 23.**The normalized cycle spectrum for products of two odd primes. The

**top two rows**are $p=3q$, where $q=3,5,7,11,13,17$. The

**third and fourth rows**are $p=5q$, where $q=5,7,11,13,17,19$. The final row is $p=5q$, where $q=7,11,13$.

**Figure 24.**The normalized cycle spectrum for products of powers of two and one odd prime. The cases of $p={2}^{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}3$ (

**top row**) and $p={2}^{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}5$ (

**bottom row**) where $m=2,3,4$.

Nature of p | s | Descriptor | Example | Remark |
---|---|---|---|---|

${2}^{m}$ | 1 | disk | Figure 4 | unbranched graph |

q | $\frac{q+1}{2q}$ | table two legs | Figure 5 | legs antipodal |

${2}^{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}q$ | $\frac{q+1}{2q}$ | table ${2}^{m+1}$ legs | Figure 5 | legs equally spaced |

${q}^{2}$ | $\frac{{q}^{2}-q+2}{2{q}^{2}}$ | crown two spikes | Figure 6 | spikes antipodal |

${2}^{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{q}^{2}$ | $\frac{{q}^{2}-q+2}{2{q}^{2}}$ | crown ${2}^{m+1}$ spikes | — | legs equally spaced |

q${}^{3}$ | $\frac{{q}^{3}-{q}^{2}+q+1}{2{q}^{3}}$ | double level crown | Figure 6 | spike positions at q |

${q}^{n}$ (odd) | $\frac{1+{\sum}_{j=1}^{n}{(-1)}^{j+1}{q}^{j}}{2{q}^{n}}$ | $\frac{\mathrm{odd}+1}{2}$ level crown | — | nested orders |

${q}^{n}$ (even) | $\frac{2+{\sum}_{j=1}^{n}{(-1)}^{j}{q}^{j}}{2{q}^{n}}$ | $\frac{\mathrm{even}}{2}$ level crown | — | nested orders |

${q}_{1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{q}_{2}$ | $\frac{{q}_{1}{q}_{2}+{q}_{1}+{q}_{2}+1}{2{q}_{1}{q}_{2}}$ | drum two legs | Figure 5 | legs antipodal |

${2}^{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{q}_{1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{q}_{2}$ | $\frac{{q}_{1}{q}_{2}+{q}_{1}+{q}_{2}+1}{2{q}_{1}{q}_{2}}$ | drum ${2}^{m+1}$ legs | Figure 5 | legs equally spaced |

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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.
https://doi.org/10.3390/math7111021

**AMA Style**

Sullivan K, Rutherford D, Ulness DJ.
Centered Polygonal Lacunary Graphs: A Graph Theoretic Approach to *p*-Sequences of Centered Polygonal Lacunary Functions. *Mathematics*. 2019; 7(11):1021.
https://doi.org/10.3390/math7111021

**Chicago/Turabian Style**

Sullivan, Keith, Drew Rutherford, and Darin J. Ulness.
2019. "Centered Polygonal Lacunary Graphs: A Graph Theoretic Approach to *p*-Sequences of Centered Polygonal Lacunary Functions" *Mathematics* 7, no. 11: 1021.
https://doi.org/10.3390/math7111021