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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".

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  • $\begingroup$ looking at the undirected graph $A^T A$ or $A^T+A$ instead of $A$ would be easier ? $\endgroup$ – reuns Jan 22 '16 at 18:10
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These two papers should be helpful for you.

[1] J. G. Restrepo, E. Ott, and B. R. Hunt, Phys. Rev. Lett. 97, 094102 (2006)

[2] A. Milanese, J. Sun, and T. Nishikawa, Phys. Rev. E 81, 046112, (2010)

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