Disconnectivity graphs are used to characterize the potential energy surfaces of Lennard-Jones clusters containing 13, 19, 31, 38, 55, and 75 atoms. This set includes members which exhibit either one or two "funnels'' whose low-energy regions may be dominated by a single deep minimum or contain a number of competing structures. The graphs evolve in size due to these specific size effects and an exponential increase in the number of local minima with the number of atoms. To combat the vast number of minima we investigate the use of monotonic sequence basins as the fundamental topographical unit. Finally, we examine disconnectivity graphs for a transformed energy landscape to explain why the transformation provides a useful approach to the global optimization problem.