By numerical and empirical means a dynamic transition of a complex network was found from a hierarchical scale-free Minimal Spanning Tree (MST ), representing the stock market before the recent worldwide financial crash, to a superstar-like MST decorated by a scale-free hierarchy of trees representing the market's state for the period containing the crash. Subsequently, a transition from this latter (perheps)an unstable state to hierarchical scale-free MST decorated by several star-like trees is observed after the worldwide financial crash. In the present work we applied the MST technique, as a particularly useful canonical tool of a graph theory, to show the transitions on the (German) Frankfurt Stock Exchange (FSE). Analogous results we obtained earlier for the Warsaw Stock Exchange . Our results can serve as  an empirical foundation for the theory of dynamic structural and topological phase transitions on financial markets.
We find numerical and empirical evidence for dynamical, structural and topological phase transitions on the (German) Frankfurt Stock Exchange (FSE) in the temporal vicinity of the worldwide financial crash. Using the Minimal Spanning Tree (MST) technique, a particularly useful canonical tool of the graph theory, two transitions of the topology of a complex network representing FSE were found. First transition is from a hierarchical scale-free MST representing the stock market before the recent worldwide financial crash, to a superstar-like MST decorated by a scale-free hierarchy of trees representing the market's state for the period containing the crash. Subsequently, a transition is observed from this transient, (meta)stable state of the crash, to a hierarchical scale-free MST decorated by several star-like trees after the worldwide financial crash. The phase transitions observed are analogous to the ones we obtained earlier for the Warsaw Stock Exchange and more pronounced than those found by Onnela-Chakraborti-Kaski-Kert\'esz for S P 500 index in the vicinity of Black Monday (October 19, 1987) and also in the vicinity of January 1, 1998. Our results provide an empirical foundation for the future theory of dynamical, structural and topological phase transitions on financial markets.
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