A ring-shaped carbon allotrope was recently synthesized for the first time, reinvigorating theoretical interest in this class of molecules. The dual $pi$ structure of these molecules allows for the possibility of novel electronic properties. In this work we use reduced density matrix theory to study the electronic structure and conductivity of cyclo[18]carbon and its boron nitride analogue, Btextsubscript{9}Ntextsubscript{9}. The variational 2RDM method replicates the experimental polyynic geometry of cyclo[18]carbon. We use a current-constrained 1-electron reduced density matrix (1-RDM) theory with Hartree-Fock molecular orbitals and energies to compute the molecular conductance in two cases: (1) conductance in the plane of the molecule and (2) conductance around the molecular ring as potentially driven by a magnetic field through the molecules center. In-plane conductance is greater than conductance around the ring, but cyclo[18]carbon is slightly more conductive than Btextsubscript{9}Ntextsubscript{9} for both in-the-plane and in-the-ring conduction. The computed conductance per molecular orbital provides insight into how the orbitals---their energies and densities---drive the conduction.
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