  When attempting to develop wavelet transforms for graphs and networks, some
researchers have used graph Laplacian eigenvalues and eigenvectors in place of
the frequencies and complex exponentials in the Fourier theory for regular
lattices in the Euclidean domains. This viewpoint, however, has a fundamental
flaw: on a general graph, the Laplacian eigenvalues cannot be interpreted as
the frequencies of the corresponding eigenvectors. In this paper, we discuss
this important problem further and propose a new method to organize those
eigenvectors by defining and measuring "natural" distances between eigenvectors
using the Ramified Optimal Transport Theory followed by embedding them into a
low-dimensional Euclidean domain. We demonstrate its effectiveness using a
synthetic graph as well as a dendritic tree of a retinal ganglion cell of a
mouse.
