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 Randi\'{c} 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 Randi\'{c} matrix are also provided, proposing interesting conjectures on the latter quantities.

翻译：正则环格（RRL）定义为由环图构造的特殊无向循环图，其中每个节点与按顶点度数逐步间隔的邻居对相连。这类网络拓扑被广泛应用于多种基于图的分布式可扩展协议中，其谱特性常对确定此类算法的收敛速度起核心作用。本文研究了RRL图的基本性质以及相应拉普拉斯矩阵和Randi\'{c}矩阵的特征值，给出了这些矩阵谱的深层刻画，并阐述了它们与狄利克雷核的关系。由此，解析求解了该网络拓扑的Fiedler值。针对RRL，还提供了拉普拉斯矩阵谱半径和Randi\'{c}矩阵本质谱半径的界值性质，并针对后者提出了有趣的猜想。