In the past decade, several multi-resolution representation theories for\ngraph signals have been proposed. Bipartite filter-banks stand out as the most\nnatural extension of time domain filter-banks, in part because perfect\nreconstruction, orthogonality and bi-orthogonality conditions in the graph\nspectral domain resemble those for traditional filter-banks. Therefore, many of\nthe well known orthogonal and bi-orthogonal designs can be easily adapted for\ngraph signals. A major limitation is that this framework can only be applied to\nthe normalized Laplacian of bipartite graphs. In this paper we extend this\ntheory to arbitrary graphs and positive semi-definite variation operators. Our\napproach is based on a different definition of the graph Fourier transform\n(GFT), where orthogonality is defined with the respect to the Q inner product.\nWe construct GFTs satisfying a spectral folding property, which allows us to\neasily construct orthogonal and bi-orthogonal perfect reconstruction\nfilter-banks. We illustrate signal representation and computational efficiency\nof our filter-banks on 3D point clouds with hundreds of thousands of points.\n