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 $P_n$ is $H$-cospectral to $P_n$. Interestingly, there are others. This paper finds digraphs that are $H$-cospectral with the path graph $P_n$ 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.