We investigate a variational method for ill-posed problems, which embeds the fractional power of the standard graph Laplacian operator in the regularization term. We explore the dependence of the regularizer on a preliminary approximation of the solution, which is obtained using various existing reconstruction methods \(\varPsi \) from the literature. As a result, the regularization term is both dependent on and adaptive to the observed data, noise, and the choice of the fractional exponent. We present a selected numerical example problem on 2D computerized tomography, for which we consider various reconstruction techniques \(\varPsi \) , including Filtered Back Projection, Total Variation, and a trained deep neural network. Incorporating the fractional power of the graph Laplacian operator into the regularization term significantly enhances the quality of the approximated solutions for each method \(\varPsi \) . Additionally, we show that our proposal behaves as a regularization method and is also stable with respect to variations in the noise level.