Synchronous neural oscillations within the beta frequency range are observed across the parkinsonian basal ganglia network, including within the subthalamic nucleus (STN) - globus pallidus (GPe) subcircuit. The emergence of pathological synchrony in Parkinson's disease is often attributed to changes in neural properties or connection strength, and less often to the network topology, i.e. the structural arrangement of connections between neurons. This study investigates the relationship between network structure and neural synchrony in a model of the STN-GPe circuit comprised of conductance-based spiking neurons. Changes in net synaptic input were controlled for through a synaptic scaling rule, which facilitated separation of the effects of network structure from net synaptic input. Five topologies were examined as structures for the STN-GPe circuit: Watts-Strogatz, preferential attachment, spatial, stochastic block, k-regular random. Beta band synchrony generally increased as the number of connections increased, however the exact relationship was topology specific. Varying the wiring pattern while maintaining a constant number of connections caused network synchrony to be enhanced or suppressed, demonstrating the ability of purely structural changes to alter synchrony. This relationship was well-captured by the algebraic connectivity of the network, the second smallest eigenvalue of the network's Laplacian matrix. The structure-synchrony relationship was further investigated in a network model designed to emulate the action selection role of the STN-GPe circuit. It was found that increasing the number of connections and/or the overlap of action selection channels could lead to a rapid transition to synchrony, which was also predicted by the algebraic connectivity.