  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.
