Digraphs with Hermitian spectral radius below 2 and their cospectrality with paths

It is well-known that the paths are determined by the spectrum of the adjacency matrix. For digraphs, every digraph whose underlying graph is a tree is cospectral to its underlying graph with respect to the Hermitian adjacency matrix (H-cospectral). Thus every (simple) digraph whose underlying graph is isomorphic to Pn is H-cospectral to Pn. Interestingly, there are others. This paper finds digraphs that are H-cospectral with the path graph Pn and whose underlying graphs are nonisomorphic, when n is odd, and finds that such graphs do not exist when n is even. In order to prove this result, all digraphs whose Hermitian spectral radius is smaller than 2 are determined.