  Spectral graph convolution, an important tool of data filtering on graphs,
relies on two essential decisions; selecting spectral bases for signal
transformation and parameterizing the kernel for frequency analysis. While
recent techniques mainly focus on standard Fourier transform and vector-valued
spectral functions, they fall short in flexibility to describe specific signal
distribution for each node, and expressivity of spectral function. In this
paper, we present a novel wavelet-based graph convolution network, namely
WaveGC, which integrates multi-resolution spectral bases and a matrix-valued
filter kernel. Theoretically, we establish that WaveGC can effectively capture
and decouple short-range and long-range information, providing superior
filtering flexibility, surpassing existing graph convolutional networks and
graph Transformers (GTs). To instantiate WaveGC, we introduce a novel technique
for learning general graph wavelets by separately combining odd and even terms
of Chebyshev polynomials. This approach strictly satisfies wavelet
admissibility criteria. Our numerical experiments showcase the capabilities of
the new network. By replacing the Transformer part in existing architectures
with WaveGC, we consistently observe improvements in both short-range and
long-range tasks. This underscores the effectiveness of the proposed model in
handling different scenarios. Our code is available at
https://github.com/liun-online/WaveGC.
