  In the field of graph signal processing (GSP), directed graphs present a
particular challenge for the "standard approaches" of GSP to due to their
asymmetric nature. The presence of negative- or complex-weight directed edges,
a graphical structure used in fields such as neuroscience, critical
infrastructure, and robot coordination, further complicates the issue. Recent
results generalized the total variation of a graph signal to that of directed
variation as a motivating principle for developing a graphical Fourier
transform (GFT). Here, we extend these techniques to concepts of signal
variation appropriate for indefinite and complex-valued graphs and use them to
define a GFT for these classes of graph. Simulation results on random graphs
are presented, as well as a case study of a portion of the fruit fly
connectome.
