  In an earlier work we predicted the existence of a neutral triplet collective
mode in undoped graphene and graphite [Phys. Rev. Lett. {\bf 89} (2002) 16402].
In this work we study a phenomenological Hamiltonian describing the interaction
of tight-binding electrons on honeycomb lattice with such a dispersive neutral
triplet boson. Our Hamiltonian is a generalization of the Holstein polaron
problem to the case of triplet bosons with non-trivial dispersion all over the
Brillouin zone. This collective mode constitutes an important excitation branch
which can contribute to the decay rate of the electronic excitations. The
presence of such collective mode, modifies the spectral properties of electrons
in graphite and undoped graphene. In particular such collective mode, as will
be shown in this paper, can account for some part of the missing decay rate in
a time-domain measurement done on graphite.
