  In this work we propose the construction of two-channel wavelet filterbanks
for analyzing functions defined on the vertices of any arbitrary finite
weighted undirected graph. These graph based functions are referred to as
graph-signals as we build a framework in which many concepts from the classical
signal processing domain, such as Fourier decomposition, signal filtering and
downsampling can be extended to graph domain. Especially, we observe a spectral
folding phenomenon in bipartite graphs which occurs during downsampling of
these graphs and produces aliasing in graph signals. This property of bipartite
graphs, allows us to design critically sampled two-channel filterbanks, and we
propose quadrature mirror filters (referred to as graph-QMF) for bipartite
graph which cancel aliasing and lead to perfect reconstruction. For arbitrary
graphs we present a bipartite subgraph decomposition which produces an
edge-disjoint collection of bipartite subgraphs. Graph-QMFs are then
constructed on each bipartite subgraph leading to "multi-dimensional" separable
wavelet filterbanks on graphs. Our proposed filterbanks are critically sampled
and we state necessary and sufficient conditions for orthogonality, aliasing
cancellation and perfect reconstruction. The filterbanks are realized by
Chebychev polynomial approximations.
