  We discuss the problem of unique determination of the finite free discrete
Schr\"{o}dinger operator from its spectrum, also known as Ambarzumian problem,
with various boundary conditions, namely any real constant boundary condition
at zero and Floquet boundary conditions of any angle. Then we prove the
following Ambarzumian-type mixed inverse spectral problem: Diagonal entries
except the first and second ones and a set of two consecutive eigenvalues
uniquely determine the finite free discrete Schr\"{o}dinger operator.
