Algebraic connectivity of directed graphs

We consider a generalization of Fiedler's notion of algebraic connectivity to directed graphs. We show that several properties of Fiedler's definition remain valid for directed graphs and present properties peculiar to directed graphs. We prove inequalities relating the algebraic connectivity to quantities such as the bisection width, maximum directed cut and the isoperimetric number. Finally, we illustrate an application to the synchronization in networks of coupled chaotic systems.