I will start describing some basics of the graph Laplacian eigenvectors of a given graph and their properties. In particular, I will describe the peculiar phase transition/localization phenomena of such eigenvectors observed on a certain type of graphs (e.g., dendritic trees of neurons). Then, I will describe how to construct wavelet packets on a given graph including the Haar-Walsh basis dictionary using the graph Laplacian eigenvectors. As an application of such basis dictionaries, I will discuss efficient approximation of functions given on graphs. This is a joint work with Jeff Irion, Yuji Nakatsukasa, and Ernest Woei.