  A unitary shift operator (GSO) for signals on a graph is introduced, which
exhibits the desired property of energy preservation over both backward and
forward graph shifts. For rigour, the graph differential operator is also
derived in an analytical form. The commutativity relation of the shift operator
with the Fourier transform is next explored in conjunction with the proposed
GSO to introduce a graph discrete Fourier transform (GDFT) which, unlike
existing approaches, ensures the orthogonality of GDFT bases and admits a
natural frequency-domain interpretation. The proposed GDFT is shown to allow
for a coherent definition of the graph discrete Hilbert transform (GDHT) and
the graph analytic signal. The advantages of the proposed GSO are demonstrated
through illustrative examples.
