  In this paper, we introduce the uniform algebraic rank of a divisor class on
a finite graph. We show that it lies between Caporaso's algebraic rank and the
combinatorial rank of Baker and Norine. We prove the Riemann-Roch theorem for
the uniform algebraic rank, and show that both the algebraic and the uniform
algebraic rank are realized on effective divisors. As an application, we use
the uniform algebraic rank to show that Clifford representatives always exist.
We conclude with an explicit description of such Clifford representatives for a
large class of graphs.
