  Tight binding models like the Hubbard Hamiltonian are most often explored in
the context of uniform intersite hopping $t$. The electron-electron
interactions, if sufficiently large compared to this translationally invariant
$t$, can give rise to ordered magnetic phases and Mott insulator transitions,
especially at commensurate filling. The more complex situation of non-uniform
$t$ has been studied within a number of situations, perhaps most prominently in
multi-band geometries where there is a natural distinction of hopping between
orbitals of different degree of overlap. In this paper we explore related
questions arising from the interplay of multiple kinetic energy scales and
electron-phonon interactions. Specifically, we use Determinant Quantum Monte
Carlo (DQMC) to solve the half-filled Holstein Hamiltonian on a `decorated
honeycomb lattice', consisting of hexagons with internal hopping $t$ coupled
together by $t^{\,\prime}$. This modulation of the hopping introduces a gap in
the Dirac spectrum and affects the nature of the topological phases. We
determine the range of $t/t^{\,\prime}$ values which support a charge density
wave (CDW) phase about the Dirac point of uniform hopping $t=t^{\,\prime}$, as
well as the critical transition temperature $T_c$. The QMC simulations are
compared with the results of Mean Field Theory (MFT).
