  A Horizontal Visibility Graph (HVG) is a simple graph extracted from an
ordered sequence of real values, and this mapping has been used to provide a
combinatorial encryption of time series for the task of performing network
based time series analysis. While some properties of the spectrum of these
graphs --such as the largest eigenvalue of the adjacency matrix-- have been
routinely used as measures to characterise time series complexity, a theoretic
understanding of such properties is lacking. In this work we explore some
algebraic and spectral properties of these graphs associated to periodic and
chaotic time series. We focus on the family of Feigenbaum graphs, which are
HVGs constructed in correspondence with the trajectories of one-parameter
unimodal maps undergoing a period-doubling route to chaos (Feigenbaum
scenario). For the set of values of the map's parameter $\mu$ for which the
orbits are periodic with period $2^n$, Feigenbaum graphs are fully
characterised by two integers (n,k) and admit an algebraic structure. We
explore the spectral properties of these graphs for finite n and k, and among
other interesting patterns we find a scaling relation for the maximal
eigenvalue and we prove some bounds explaining it. We also provide numerical
and rigorous results on a few other properties including the determinant or the
number of spanning trees. In a second step, we explore the set of Feigenbaum
graphs obtained for the range of values of the map's parameter $\mu$ for which
the system displays chaos. We show that in this case, Feigenbaum graphs form an
ensemble for each value of $\mu$ and the system is typically weakly
self-averaging. Unexpectedly, we find that while the largest eigenvalue can
distinguish chaos from an iid process, it is not a good measure to quantify the
chaoticity of the process, and that the eigenvalue density does a better job.

Data Analysis, Statistics and Probability;Combinatorics;Spectral Theory;Chaotic Dynamics