The negative tetrahedron and the first infinite family of connected digraphs that are strongly determined by the Hermitian spectrum
It is very elusive to find digraphs that are uniquely determined by their Hermitian spectra. So, people usually turn to finding spectral determination of classes of switching equivalent digraphs, rather than individual digraphs. In the present paper, the authors return back to considering the traditional notion: the strongly determined digraph (or mixed graph) by its Hermitian spectrum (abbreviated SHDS), which is a digraph (or mixed graph) that is isomorphic to each digraph (or mixed graph) to which it is Hermitian cospectral. From numerical analysis (not a rigorous proof), they come to the conclusion that digraphs with this property are extremely rare. However, fortunately they can construct an infinite family of connected digraphs that is SHDS. 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.
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