  We investigate a variational method for ill-posed problems, named
$\texttt{graphLa+}\Psi$, which embeds a graph Laplacian operator in the
regularization term. The novelty of this method lies in constructing the graph
Laplacian based on a preliminary approximation of the solution, which is
obtained using any existing reconstruction method $\Psi$ from the literature.
As a result, the regularization term is both dependent on and adaptive to the
observed data and noise. We demonstrate that $\texttt{graphLa+}\Psi$ is a
regularization method and rigorously establish both its convergence and
stability properties.
  We present selected numerical experiments in 2D computerized tomography,
wherein we integrate the $\texttt{graphLa+}\Psi$ method with various
reconstruction techniques $\Psi$, including Filter Back Projection
($\texttt{graphLa+FBP}$), standard Tikhonov ($\texttt{graphLa+Tik}$), Total
Variation ($\texttt{graphLa+TV}$), and a trained deep neural network
($\texttt{graphLa+Net}$). The $\texttt{graphLa+}\Psi$ approach significantly
enhances the quality of the approximated solutions for each method $\Psi$.
Notably, $\texttt{graphLa+Net}$ is outperforming, offering a robust and stable
application of deep neural networks in solving inverse problems.
