  Ballistic transport through a collection of quantum billiards in undoped
graphene is studied analytically within the conformal mapping technique. The
billiards show pseudodiffusive behavior, with the conductance equal to that of
a classical conductor characterized by the conductivity $\sigma_0=4e^2/\pi h$,
and the Fano factor $F=1/3$. By shrinking at least one of the billiard
openings, we observe a tunneling behavior, where the conductance shows a
power-law decay with the system size, and the shot noise is Poissonian (F=1).
In the crossover region between tunneling and pseudodiffusive regimes, the
conductance $G\approx (1-F)\times se^2/h$. The degeneracy $s=8$ for the Corbino
disk, which preserves the full symmetry of the Dirac equation, $s=4$ for
billiards bounded with smooth edges which break the symplectic symmetry, and
$s=2$ when abrupt edges lead to strong intervalley scattering. An alternative,
analytical or numerical technique, is utilized for each of the billiards to
confirm the applicability of the conformal mapping for various boundary
conditions.
