Summary: Any directed graph \(G\) with \(N\) vertices and \(J\) edges has an associated line-graph \(L(G)\) where the \(J\) edges form the vertices of \(L(G)\). We show that the non-zero eigenvalues of the adjacency matrices are the same for all graphs of such a family \(L^n(G)\). We give necessary and sufficient conditions for a line-graph to be quantisable and demonstrate that the spectra of associated quantum propagators follow the predictions of random matrices under very general conditions. Line-graphs may therefore serve as models to study the semiclassical limit (of large matrix size) of a quantum dynamics on graphs with fixed classical behaviour.