I generalize David Zureick-Brown and John Voight's work on log-canonical rings to graph curves. I use a paper of Noot as a starting point. I outline some of the difficulties in developing Max Noether-like and Petri-like theorems. I work out theorems for the generators of most well behaved graph curves. I also find a useful construction for hyperelliptic graph curves. Introduction - 1 – Noneffective canoincal divisors and bridges - 2 – Inductive Step - 3 – One point log divisors - 3 – Two point log divisors - 4 – Three point log divisors - 5 – Hyper Elliptic Curves - 5 Advisors/Committee Members: Raman, Parimala (Committee Member), Allison, Blake A (Committee Member), Zureick-Brown, David M (Thesis Advisor).