  Many systems comprising entities in interactions can be represented as
graphs, whose structure gives significant insights about how these systems
work. Network theory has undergone further developments, in particular in
relation to detection of communities in graphs, to catch this structure.
Recently, an approach has been proposed to transform a graph into a collection
of signals: Using a multidimensional scaling technique on a distance matrix
representing relations between vertices of the graph, points in a Euclidean
space are obtained and interpreted as signals, indexed by the vertices. In this
article, we propose several extensions to this approach, developing a framework
to study graph structures using signal processing tools. We first extend the
current methodology, enabling us to highlight connections between properties of
signals and graph structures, such as communities, regularity or randomness, as
well as combinations of those. A robust inverse transformation method is next
described, taking into account possible changes in the signals compared to
original ones. This technique uses, in addition to the relationships between
the points in the Euclidean space, the energy of each signal, coding the
different scales of the graph structure. These contributions open up new
perspectives in the study of graphs, by enabling processing of graphs through
the processing of the corresponding collection of signals, using reliable tools
from signal processing. A technique of denoising of a graph by filtering of the
corresponding signals is then described, suggesting considerable potential of
the approach.
