This paper investigates graph spectral approaches to the problem of point pattern matching. Specifically, we concentrate on the issue of how to effectively use graph spectral properties to characterize point patterns in the presence of positional jitter and outliers. A novel local spectral descriptor is proposed to represent the attribute domain of feature points. For a point in a given point-set, weight graphs are constructed on its neighboring points and then their normalized Laplacian matrices are computed. According to the known spectral radius of the normalized Laplacian matrix, the distribution of the eigenvalues of these normalized Laplacian matrices is summarized as a histogram to form a descriptor. The proposed spectral descriptor is finally combined with the approximate distance order for recovering correspondences between point-sets. Extensive experiments demonstrate the effectiveness of the proposed approach and its superiority to the existing methods.