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Quantum graphs with varying edges lengths are constructed. We deal with two types of operators on the graph edges: the Schrodinger and the Dirac operators. The dynamics of quantum particles for these two cases is compared. From the physical point of view, the Schrodinger dynamics corresponds to nonrelativistic particle and the Dirac dynamics to the relativistic one.

Geometric graph composed from a ring and a segment is considered. We deal with the Schrodinger (quantum) and the wave (classical) operators at the edges and the Kirchhoff conditions at the internal vertex. The lengths of the graph edges varies in time. Time evolution of a wave packet is studied in classical and quantum cases.

We consider a metric graph under the assumption that lengths of its edges vary in time. The dynamics of waves of different nature on the graph is compared. We investigate classical waves governed by wave equation, non-relativistic quantum (Schrodinger) particles and relativistic (Dirac) particles. Star like graph and graph with a loop are considered.