The paper investigates the hidden relationships among speech samples by applying graph tools. Specifically, we first estimate an applicable graph topology for unstructured speech signals, which can map speech signals into the vertex domain successfully and construct as Speech graph signals (SGSs). On the basis, we define a new graph Fourier transform for SGSs, which can investigate its related graph Fourier analysis. Moreover, we propose a new Graph structure spectral subtraction (GSSS) method for speech enhancement under different noisy environments. Simulation results show that the performance of the GSSS method can be significantly improved than the classical Basic spectral subtraction (BSS) method in terms of the average Segmental signal-to-noise ratio (SSNR), Perceptual evaluation of speech quality (PESQ) and the computational complexity.