  We study curves consisting of unions of projective lines whose intersections
are given by graphs. Under suitable hypotheses on the graph, these so-called
\emph{graph curves} can be embedded in projective space as line arrangements.
We discuss property $N_p$ for these embeddings and are able to produce products
of linear forms that generate the ideal in certain cases. We also briefly
discuss questions regarding the higher-dimensional subspace arrangements
obtained by taking the secant varieties of graph curves.
