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On the Characterization of Regular Ring Lattices and their Relation with the Dirichlet Kernel

Marco Fabris

TL;DR

Basic properties of RRL graphs and the eigenvalues of the corresponding Laplacian and Randi\'{c} matrices are investigated and the Fiedler value of such a network topology is found analytically.

Abstract

Regular ring lattices (RRLs) are defined as peculiar undirected circulant graphs constructed from a cycle graph, wherein each node is connected to pairs of neighbors that are spaced progressively in terms of vertex degree. This kind of network topology is extensively adopted in several graph-based distributed scalable protocols and their spectral properties often play a central role in the determination of convergence rates for such algorithms. In this work, basic properties of RRL graphs and the eigenvalues of the corresponding Laplacian and Randić matrices are investigated. A deep characterization for the spectra of these matrices is given and their relation with the Dirichlet kernel is illustrated. Consequently, the Fiedler value of such a network topology is found analytically. With regard to RRLs, properties on the bounds for the spectral radius of the Laplacian matrix and the essential spectral radius of the Randić matrix are also provided, proposing interesting conjectures on the latter quantities.

On the Characterization of Regular Ring Lattices and their Relation with the Dirichlet Kernel

TL;DR

Basic properties of RRL graphs and the eigenvalues of the corresponding Laplacian and Randi\'{c} matrices are investigated and the Fiedler value of such a network topology is found analytically.

Abstract

Regular ring lattices (RRLs) are defined as peculiar undirected circulant graphs constructed from a cycle graph, wherein each node is connected to pairs of neighbors that are spaced progressively in terms of vertex degree. This kind of network topology is extensively adopted in several graph-based distributed scalable protocols and their spectral properties often play a central role in the determination of convergence rates for such algorithms. In this work, basic properties of RRL graphs and the eigenvalues of the corresponding Laplacian and Randić matrices are investigated. A deep characterization for the spectra of these matrices is given and their relation with the Dirichlet kernel is illustrated. Consequently, the Fiedler value of such a network topology is found analytically. With regard to RRLs, properties on the bounds for the spectral radius of the Laplacian matrix and the essential spectral radius of the Randić matrix are also provided, proposing interesting conjectures on the latter quantities.
Paper Structure (15 sections, 16 theorems, 30 equations, 2 figures, 3 tables)

This paper contains 15 sections, 16 theorems, 30 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Basic notions of graph theory
  5. Circulant matrices
  6. Definition and properties of the Dirichlet kernel
  7. Main results related to RRLs
  8. Definition and basic properties
  9. Spectral analysis
  10. Essential spectral radius analysis
  11. Further discussions and numerical examples
  12. Conjecture on a potential upper bound for $j^{\star}$
  13. Numerical examples for $4\leq N \leq 11$
  14. Conjecture on the values taken by $\sigma(\mathbf{R})$
  15. Conclusions and future directions

Key Result

Lemma 1

For an undirected graph $\mathcal{G}$, the sum of all its degrees equals twice the number of its edges, i.e. $\mathrm{vol}(\mathcal{G}) = 2M(\mathcal{G})$.

Figures (2)

  • Figure 1: All the three RRLs with $N=9$ vertices. A layer of edges is added for each increasing value of $m \in \{1,2,3\}$: (a) first layer in black, (b) second layer in green, (c) third layer in red.
  • Figure 2: General eigenvalue distribution of the Randić matrix spectrum $\Lambda(\mathbf{R})$ for the RRLs $C_{N}^{m}$ with $N=4,\ldots, 11$ and $m = 1,\ldots ,n-1 = \lfloor N/2\rfloor-1$.

Theorems & Definitions (39)

  • Lemma 1: Handshaking lemma Euler1741
  • Definition 1: Circulant matrix Gray2005
  • Theorem 1: Spectrum of circulant matrices Gray2005
  • Definition 2: Dirichlet kernel BruncknerBruncknerThomson1997
  • Theorem 2: Well-known properties of the Dirichlet kernel BruncknerBruncknerThomson1997Wiggins2007Kirkwood2018
  • Definition 3: RRL $C_{N}^{m}$
  • Remark 1
  • Proposition 1: Regularity and common degree of RRLs
  • proof
  • Proposition 2: Connectivity of RRLs
  • ...and 29 more