  The adsorption of an alkali-metal submonolayer on graphene occupying every
third hexagon of the honeycomb lattice in a commensurate
$(\sqrt{3}\times\sqrt{3})R30^\circ$ arrangement induces an energy gap in the
spectrum of graphene. To exemplify this type of band gap, we present \textit{ab
initio} density functional theory calculations of the electronic band structure
of C$_6$Li. An examination of the lattice geometry of the compound system shows
the possibility that the nearest-neighbor hopping amplitudes have alternating
values constructed in a Kekul\'e-type structure. The band structure of the
textured tight-binding model is calculated and shown to reproduce the expected
band gap as well as other characteristic degeneracy removals in the spectrum of
graphene induced by lithium adsorption. More generally we also deduce the
possibility of energy gap opening in periodic metal on graphene compounds
C$_x$M if $x$ is a multiple of 3.
