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Discrete fracture network (DFN) models are a powerful alternative to continuum models for subsurface flow and transport simulations because they explicitly include fracture geometry and network topology, thereby allowing for better characterization of the latter's influence on flow and transport through fractured media. Recent advances in high performance computing have opened the door for flow and transport simulations in large explicit three-dimensional DFN, but this increase in model fidelity and system size comes at a huge computational cost because of the large number of mesh elements required to represent thousands of fractures (with sizes that can range several orders of magnitude, from millimeter to kilometer). In most subsurface applications, fracture characteristics are only known statistically and numerous realizations are needed to bound uncertainty in flow and transport in the system, thereby exacerbating the computational burden. Graphs provide a simple and elegant way to characterize, query, and interrogate fracture network connectivity. We propose a DFN model reduction framework where various graph representations are used in conjunction with high-fidelity DFN simulations to increase computational efficiency while retaining accuracy of key quantities of interest. The appropriate choice of a graph representation, namely, which attributes of the DFN are to be represented as nodes and which ones as edges connecting those nodes, depends on the relevant scientific questions. We demonstrate that the proposed DFN model reduction framework provides an efficient means for DFN modeling through both system reduction of the DFN using graph-based properties and combining DFN and graph-based flow and transport simulations.