  Thus far, digraphs that are uniquely determined by their Hermitian spectra
have proven elusive. Instead, researchers have turned to spectral determination
of classes of switching equivalent digraphs, rather than individual digraphs.
In the present paper, we consider the traditional notion: a digraph (or mixed
graph) is said to be strongly determined by its Hermitian spectrum (abbreviated
SHDS) if it is isomorphic to each digraph to which it is cospectral. Convincing
numerical evidence to support the claim that this property is extremely rare is
provided. Nonetheless, the first infinite family of connected digraphs that is
SHDS is constructed. This family is obtained via the introduction of twin
vertices into a structure that is named negative tetrahedron. This special
digraph, that exhibits extreme spectral behavior, is contained in the
surprisingly small collection of all digraphs with exactly one negative
eigenvalue, which is determined as an intermediate result.
