The paper introduces a novel method for constructing wavelet transforms for functions defined on the vertices of an arbitrary finite weighted graph. The approach is based on defining scaling using the spectral decomposition of the discrete graph Laplacian, which serves as an analogue of the Fourier domain for graphs. Given a wavelet generating kernel \( g \) and a scale parameter \( t \), the scaled wavelet operator \( T_t^* = g(t \mathcal{L}) \) is defined, and the spectral graph wavelets are formed by localizing this operator to an indicator function. The procedure is shown to define an invertible transform under an admissibility condition on \( g \). The paper explores the localization properties of the wavelets in the limit of fine scales and presents a fast Chebyshev polynomial approximation algorithm for computing the transform, avoiding the need to diagonalize the Laplacian. The transform is demonstrated to have potential applications in various problem domains, including network analysis, machine learning, and pattern recognition.The paper introduces a novel method for constructing wavelet transforms for functions defined on the vertices of an arbitrary finite weighted graph. The approach is based on defining scaling using the spectral decomposition of the discrete graph Laplacian, which serves as an analogue of the Fourier domain for graphs. Given a wavelet generating kernel \( g \) and a scale parameter \( t \), the scaled wavelet operator \( T_t^* = g(t \mathcal{L}) \) is defined, and the spectral graph wavelets are formed by localizing this operator to an indicator function. The procedure is shown to define an invertible transform under an admissibility condition on \( g \). The paper explores the localization properties of the wavelets in the limit of fine scales and presents a fast Chebyshev polynomial approximation algorithm for computing the transform, avoiding the need to diagonalize the Laplacian. The transform is demonstrated to have potential applications in various problem domains, including network analysis, machine learning, and pattern recognition.