  We use numerical simulations of different dark energy cosmologies to
investigate the concentration-mass (c-M) relation in galaxy clusters. In
particular, we consider a reference Lambda cold dark matter (LambdaCDM) model,
two models with dynamical dark energy, viewed as a quintessence scalar field
[using a Ratra and Peebles (RP) and a supergravity (SUGRA) potential form], and
two extended quintessence models, one with positive and one with negative
coupling (EQp and EQn respectively), where the quintessence scalar field
interacts non-minimally with gravity (scalar-tensor theories). All the models
are normalized in order to match CMB data from Wilkinson Microwave Anisotropy
Probe 3 (WMAP3). [Abridged] We consider both the complete catalog of clusters
and groups and a subsample of relaxed objects. The c-M relation of our
reference LambdaCDM model is in good agreement with the results in literature,
and relaxed objects have a higher normalization and a shallower slope with
respect to the complete sample. For the different dark energy models, we find
that for LambdaCDM, RP and SUGRA the normalization of the c-M relation is
linked to the growth factor D+ and the power spectrum normalization sigma8,
with models having a higher value of sigma8 D+ having also a higher
normalization. This simple scheme is no longer valid for EQp and EQn because in
these models it is present a time dependent effective gravitational
interaction, whose redshift evolution depends on the sign of the coupling. This
leads to a decrease (increase) of the expected normalization in the EQp (EQn)
model. This result shows a direct manifestation of the coupling between gravity
and the quintessence scalar field characterizing EQ models that cannot be seen
at the background level but can be investigated in the non-linear regime.
