  Phase transitions in the Hubbard model and ionic Hubbard model at
half-filling on the honeycomb lattice are investigated in the strong coupling
perturbation theory which corresponds to an expansion in powers of the hopping
$t$ around the atomic limit. Within this formulation we find analytic
expressions for the single-particle spectrum, whereby the calculation of the
insulating gap is reduced to a simple root finding problem. This enables high
precision determination of the insulating gap that does not require any
extrapolation procedure. The critical value of Mott transition on the honeycomb
lattice is obtained to be $U_c\approx 2.38 t$. Studying the ionic Hubbard model
at the lowest order, we find two insulating states, one with Mott character at
large $U$ and another with single-particle gap character at large ionic
potential, $\Delta$. The present approach gives a critical gapless state at
$U=2\Delta$ at lowest order. By systematically improving on the perturbation
expansion, the density of states around this critical gapless phase reduces.
