  The effects of second-neighbor interactions in Kekule patterned graphene
electronic properties are studied starting from a tight-binding Hamiltonian.
Thereafter, a low-energy effective Hamiltonian is obtained by projecting the
high energy bands at the Gamma point into the subspace defined by the Kekule
wave vector. The spectrum of the low energy Hamiltonian is in excellent
agreement with the one obtained from a numerical diagonalization of the full
tight-binding Hamiltonian. The main effect of the second-neighbour interaction
is that a set of bands gains an effective mass and a shift in energy, thus
lifting the degeneracy of the conduction bands at the Dirac point. This band
structure is akin to a spin-one Dirac cone, a result expected for honeycomb
lattices with a distinction between one third of the atoms in one sublattice.
Finally, we present a study of Kekule patterned graphene nanoribbons. This
shows that the previous effects are enhanced as the width decreases. Moreover,
edge states become dispersive, as expected due to second neighbors interaction,
but here the Kek-Y bond texture results in an hybridization of both edge
states. The present study shows the importance of second neighbors in realistic
models of Kekule patterned graphene, specially at surfaces.
