On the spectral reconstruction problem for digraphs

The idiosyncratic polynomial of a graph $G$ with adjacency matrix $A$ is the characteristic polynomial of the matrix $A + y(J-A-I)$, where $I$ is the identity matrix and $J$ is the all-ones matrix. It follows from a theorem of Hagos (2000) combined with an earlier result of Johnson and Newman (1980) that the idiosyncratic polynomial of a graph is reconstructible from the multiset of the idiosyncratic polynomial of its vertex-deleted subgraphs. For a digraph $G$ with adjacency matrix $A$, we define its idiosyncratic polynomial as the characteristic polynomial of the matrix $A + y(J-A-I)+zA^{T}$. By forbidding two fixed digraphs on three vertices as induced subdigraphs, we prove that the idiosyncratic polynomial of a digraph is reconstructible from the multiset of the idiosyncratic polynomial of its induced subdigraphs on three vertices. As an immediate consequence, the idiosyncratic polynomial of a tournament is reconstructible from the collection of its $3$-cycles. Another consequence is that all the transitive orientations of a comparability graph have the same idiosyncratic polynomial.