  On the Euclidean domains of classical signal processing, linking of signal
samples to the underlying coordinate structure is straightforward. While graph
adjacency matrices totally define the quantitative associations among the
underlying graph vertices, a major problem in graph signal processing is the
lack of explicit association of vertices with an underlying quantitative
coordinate structure. To make this link, we propose an operation, called the
vertex multiplication, which is defined for graphs and can operate on graph
signals. Vertex multiplication, which generalizes the coordinate multiplication
operation in time series signals, can be interpreted as an operator which
assigns a coordinate structure to a graph. By using the graph domain extension
of differentiation and graph Fourier transform (GFT), vertex multiplication is
defined such that it shows Fourier duality, which states that differentiation
and coordinate multiplication operations are duals of each other under Fourier
transformation (FT). The proposed definition is shown to reduce to coordinate
multiplication for graphs corresponding to time series. Numerical examples are
also presented.
