Let $A$ be a square $(0,1)$-matrix. Then $A$ is a Hall matrix provided it has a nonzero permanent. The Hall exponent of $A$ is the smallest positive integer $k$, if such exists, such that $A^k$ is a Hall matrix. The Hall exponent has received considerable attention, and we both review and expand on some of its properties. Viewing $A$ as the adjacency matrix of a digraph, we prove several properties of the Hall exponents of line digraphs with some emphasis on line digraphs of tournament (matrices).
The study on limit points of eigenvalues of undirected graphs was initiated by A. J. Hoffman in 1972. Now we extend the study to digraphs. We prove: 1. Every real number is a limit point of eigenvalues of graphs. Every complex number is a limit point of eigenvalues of digraphs. 2. For a digraph $D$, the set of limit points of eigenvalues of iterated subdivision digraphs of $D$ is the unit circle in the complex plane if and only if $D$ has a directed cycle. 3. Every limit point of eigenvalues of a set $\mathcal {D}$ of digraphs (graphs) is a limit point of eigenvalues of a set $\ddot{\mathcal {D}}$ of bipartite digraphs (graphs), where $\ddot{\mathcal {D}}$ consists of the double covers of the members in $\mathcal {D}$. 4. Every limit point of eigenvalues of a set $\mathcal {D}$ of digraphs is a limit point of eigenvalues of line digraphs of the digraphs in $\mathcal {D}$. 5. If $M$ is a limit point of the largest eigenvalues of graphs, then $-M$ is a limit point of the smallest eigenvalues of graphs.