  We investigate lattice effects on wave functions that are lattice analogues
of bosonic and fermionic Laughlin wave functions with number of particles per
flux $\nu=1/q$ in the Landau levels. These wave functions are defined
analytically on lattices with $\mu$ particles per lattice site, where $\mu$ may
be different than $\nu$. We give numerical evidence that these states have the
same topological properties as the corresponding continuum Laughlin states for
different values of $q$ and for different fillings $\mu$. These states define,
in particular, particle-hole symmetric lattice Fractional Quantum Hall states
when the lattice is half-filled. On the square lattice it is observed that for
$q\leq 4$ this particle-hole symmetric state displays the topological
properties of the continuum Laughlin state at filling fraction $\nu=1/q$, while
for larger $q$ there is a transition towards long-range ordered
anti-ferromagnets. This effect does not persist if the lattice is deformed from
a square to a triangular lattice, or on the Kagome lattice, in which case the
topological properties of the state are recovered. We then show that changing
the number of particles while keeping the expression of these wave functions
identical gives rise to edge states that have the same correlations in the bulk
as the reference lattice Laughlin states but a different density at the edge.
We derive an exact parent Hamiltonian for which all these edge states are
ground states with different number of particles. In addition this Hamiltonian
admits the reference lattice Laughlin state as its unique ground state of
filling factor $1/q$. Parent Hamiltonians are also derived for the lattice
Laughlin states at other fillings of the lattice, when $\mu\leq 1/q$ or
$\mu\geq 1-1/q$ and when $q=4$ also at half-filling.
