Assuming that the crosses in the figure below are unknowns in a vertex-centered finite difference scheme in an Adaptive Mesh, how can I calculate the double derivate (Laplacian) at the Red x ? The Red x has only Eastern, Western and Northern neighbours but not a Southern neighbour. Can I take the average of just 3 points which are its immediate neighbours ?
You don't typically form a finite difference stencil involving the equation at these "hanging nodes", but set the value of this vertex to the mean value of the two adjacent vertices along the edge.
(See for example lecture 16 here: http://www.math.tamu.edu/~bangerth/videos.html).