The Blume–Emery–Griffiths model with a random crystal field is studied in a random graph architecture, in which the average connectivity is a controllable parameter. The disordered average over the graph realizations is treated by replica symmetry formalism of order parameter functions. A self consistent equation for the distribution of local fields is derived, and numerically solved by a population dynamics algorithm. The results show that the average connectivity amounts to changes in the topology of the phase diagrams. Phase diagrams for representative values of the model parameters are compared with those obtained for fully connected mean field and renormalization group approaches.