<p>Based on the current situation, in which many numerical discretization on uniform meshes for the one-dimensional integral fractional Laplacian operator suffer from a severe order reduction, we will propose a novel finite difference scheme on nonuniform meshes to handle boundary singularities, and the resulting scheme is applied to solve fractional Poisson equation. The hypersingular integral form of integral fractional Laplacian is considered, and it can be approximated by composite linear interpolation quadrature. We give the strictly diagonally dominant of the coefficient matrix and its estimation of condition number. Utilizing the regularity of true solution and the property of the coefficient matrix, we give a rigorous analysis of our method on graded meshes in which show the relationship between optimal convergence order and the grading parameter and fractional index. Compared to uniform meshes, our method is more accurate on non-smooth solutions, such as benchmark fractional Laplacian problem. Numerical experiments validate the convergence and effectiveness of our method.</p>