  This paper aims at revisiting Graph Convolutional Neural Networks by bridging
the gap between spectral and spatial design of graph convolutions. We
theoretically demonstrate some equivalence of the graph convolution process
regardless it is designed in the spatial or the spectral domain. The obtained
general framework allows to lead a spectral analysis of the most popular
ConvGNNs, explaining their performance and showing their limits. Moreover, the
proposed framework is used to design new convolutions in spectral domain with a
custom frequency profile while applying them in the spatial domain. We also
propose a generalization of the depthwise separable convolution framework for
graph convolutional networks, what allows to decrease the total number of
trainable parameters by keeping the capacity of the model. To the best of our
knowledge, such a framework has never been used in the GNNs literature. Our
proposals are evaluated on both transductive and inductive graph learning
problems. Obtained results show the relevance of the proposed method and
provide one of the first experimental evidence of transferability of spectral
filter coefficients from one graph to another. Our source codes are publicly
available at: https://github.com/balcilar/Spectral-Designed-Graph-Convolutions
