  We investigate electronic transport in single layer graphene coupled to
superconducting electrodes. In these Josephson transistors, we observe
significant suppression in the critical current Ic and large variation in the
product IcRn in comparison to theoretic predictions in the ballistic limit. We
show that the depression of Ic can be explained by premature switching in
underdamped Josephson junctions described within the resistively and
capacitively shunted junction (RCSJ) model. By considering the effect of
premature switching and dissipation, the calculated gate dependence of product
IcRn agrees with experimental data. Our discovery underscores the crucial role
of thermal fluctuations in electronic transport in graphene Josephson
transistors.
