  We study the electronic structure and the phase diagram of non-interacting
fermions confined to hexagonal optical lattices. In the first part, we compare
the properties of Dirac points arising in the eigenspectrum of either honeycomb
or triangular lattices. Numerical results are complemented by analytical
equations for weak and strong confinements. In the second part we discuss the
phase diagram and the evolution of Dirac points in honeycomb lattices applying
a tight-binding description with arbitrary nearest-neighbor hoppings. With
increasing asymmetry between the hoppings the Dirac points approach each other.
At a critical asymmetry the Dirac points merge to open an energy gap, thus
changing the topology of the eigenspectrum. We analyze the trajectory of the
Dirac points and study the density of states in the different phases.
Manifestations of the phase transition in the temperature dependence of the
specific heat and in the structure factor are discussed.
