  We investigate the Josephson current in a graphene
superconductor/normal/superconductor junction, where superconductivity is
induced by means of the proximity effect from external contacts. We take into
account the possibility of anisotropic pairing by also including singlet
nearest-neighbor interactions, and investigate how the transport properties are
affected by the symmetry of the superconducting order parameter. This
corresponds to an extension of the usual on-site interaction assumption, which
yields an isotropic s-wave order parameter near the Dirac points. Here, we
employ a full numerical solution as well as an analytical treatment, and show
how the proximity effect may induce exotic types of superconducting states near
the Dirac points, e.g. $p_x$- and $p_y$-wave pairing or a combination of s-wave
and $p+\i p$-wave pairing. We find that the Josephson current exhibits a
weakly-damped, oscillatory dependence on the length of the junction when the
graphene sheet is strongly doped. The analytical and numerical treatments are
found to agree well with each other in the s-wave case when calculating the
critical current and current-phase relationship. For the scenarios with
anisotropic superconducting pairing, there is a deviation between the two
treatments, especially for the effective $p_x$-wave order parameter near the
Dirac cones which features zero-energy states at the interfaces. This indicates
that a numerical, self-consistent approach becomes necessary when treating
anisotropic superconducting pairing in graphene.
