  Neutral excitations in a fractional quantum Hall droplet define the
incompressibility gap of the topological phase. In this work, we derived a set
of analytical results for the energy gap of the graviton modes with two-body
and three-body Hamiltonians in both the long-wavelength and thermodynamic
limit. These allow us to construct model Hamiltonians for the graviton modes in
different FQH phases, and to elucidate a hierarchical structure of conformal
Hilbert spaces (nullspaces of model Hamiltonians) with respect to the graviton
modes and their corresponding ground states. Using the analytical tools
developed, we perform numerical analysis with a particular focus on the
Laughlin $\nu = 1/5$ and the Gaffnian $\nu = 2/5$ phases. Our calculation shows
that for gapped phases, low-lying neutral excitations can undergo a "phase
transition" even when the ground state is invariant. We discuss the
compressibility of the Gaffnian phase, the possibility of multiple graviton
modes, and the transition from the graviton modes to the "hollow-core" modes,
as well as their experimental consequences.
