  Contemporary data is often supported by an irregular structure, which can be
conveniently captured by a graph. Accounting for this graph support is crucial
to analyze the data, leading to an area known as graph signal processing (GSP).
The two most important tools in GSP are the graph shift operator (GSO), which
is a sparse matrix accounting for the topology of the graph, and the graph
Fourier transform (GFT), which maps graph signals into a frequency domain
spanned by a number of graph-related Fourier-like basis vectors. This
alternative representation of a graph signal is denominated the graph frequency
signal. Several attempts have been undertaken in order to interpret the support
of this graph frequency signal, but they all resulted in a one-dimensional
interpretation. However, if the support of the original signal is captured by a
graph, why would the graph frequency signal have a simple one-dimensional
support? That is why, for the first time, we propose an irregular support for
the graph frequency signal, which we coin the dual graph. The dual GSO leads to
a better interpretation of the graph frequency signal and its domain, helps to
understand how the different graph frequencies are related and clustered,
enables the development of better graph filters and filter banks, and
facilitates the generalization of classical SP results to the graph domain.
