Making an adjacency matrix positive semidefinite I would like to ask how can we transform an adjacency matrix of a graph into a positive semidefinite matrix. Of course, we could set self loops, but I do not know of any result indicating how we can find on which nodes to set self-loops and how many of those.
Your question is too imprecise.  What does it mean to "transform a matrix"?  Are you asking whether there is an algorithm that takes as input a symmetric matrix and finds the minimal total weight that can be added to diagonal entries so as to make the new matrix positive semi-definite?  

If this is your question, the answer is yes, since it is an SDP.  If you restrict to adding integer values to diagonal elements ("self loops"), then I do not know the complexity of finding the minimum number of self loops needed.  

You might want to look into Gershgorin's circle theorem if you are trying to do this by hand.