# How to perturb an adjacency matrix in order to have the highest increase in spectral radius?

Let's suppose I have a generic directed graph $$G$$ and it's adjacency matrix $$A$$. I can add an arc wherever I want in the graph. (i.e. perturb the matrix $$A$$ changing a single $$0$$ into a $$1$$). Where should I put that one to have the highest increase in the biggest eigenvector as possible?

I suppose that the answer is "where you can connect the two largest strongly connected components".

• looking at the undirected graph $A^T A$ or $A^T+A$ instead of $A$ would be easier ? – reuns Jan 22 '16 at 18:10