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We find numerical and empirical evidence for dynamical, structural andtopological phase transitions on the (German) Frankfurt Stock Exchange (FSE) inthe temporal vicinity of the worldwide financial crash. Using the MinimalSpanning Tree (MST) technique, a particularly useful canonical tool of thegraph theory, two transitions of the topology of a complex network representingFSE were found. First transition is from a hierarchical scale-free MSTrepresenting the stock market before the recent worldwide financial crash, to asuperstar-like MST decorated by a scale-free hierarchy of trees representingthe market's state for the period containing the crash. Subsequently, atransition is observed from this transient, (meta)stable state of the crash, toa hierarchical scale-free MST decorated by several star-like trees after theworldwide financial crash. The phase transitions observed are analogous to theones we obtained earlier for the Warsaw Stock Exchange and more pronounced thanthose found by Onnela-Chakraborti-Kaski-Kert\'esz for S&P 500 index in thevicinity of Black Monday (October 19, 1987) and also in the vicinity of January1, 1998. Our results provide an empirical foundation for the future theory ofdynamical, structural and topological phase transitions on financial markets.

Structural and topological phase transitions on the German Stock Exchange