This study introduces a parametrized family of representations for evolving graphs together with a benchmark function that is diagnostic of an important quality of a representation for graph evolution, its natural distribution of edge densities. The new benchmark function, the edge maximization function, is equivalent to the trivial onemax function for some representations and represents a difficult problem for others. The utility of the edge maximization function lies in the fact that the edge density distribution in a graph is a critical parameter for evolving graphs and so performance of a representation on edgemax is diagnostic of an important aspect of its behavior. Three cases of the edgemax problem are examined using six different parameterizations of the new representation. The representations presented here are generative and so need not have any particular length. For each problem case and parametrization of the representation two lengths of chromosome are examined, one that is just long enough to solve the benchmark problem and one that is 10% longer. The edgemax is found to be diagnostic of representation properties.