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Graph signal processing uses the graph eigenvector basis to analyze signals.However, these graph eigenvectors are typically linearly ordered (by totalvariation), which may not be reasonable for many graph structures. There havebeen structure based similarity metrics proposed in the literature that bettercapture the geometry of graph eigenvectors. On the other hand, there has beenwork that attempts to generalize the concept of duality to graph signalprocessing. The (signal processing) the dual graph captures the relationshipbetween graph frequencies and is obtained by typically inverting the graphFourier transform operation. In this work, we investigate the connectionsbetween these two concepts. We propose a dualness measure of two graphs, whichquantifies how close the graphs are being (signal processing) duals of eachother. We show that this definition satisfies some desirable properties anddevelop an algorithm based on coordinate descent and perfect matching tocompute an approximation to dualness. By computing the dualness measure, weobserve that for structured graphs, the similarity metric-based techniques givea better dual graph (and vice-versa for ER graphs), suggesting a potentiallynovel approach unifying the two methods.

Investigating the relationship between graph eigenvector ordering and the signal processing dual