In this paper, we characterize the finite commutative rings such that their comaximal graph (or zero-divisor graph) are ring graphs, and we also study the case where they are outerplanar.
"When the comaximal and zero-divisor graphs are ring graphs and outerplanar." Rocky Mountain J. Math. 44 (6) 1745 - 1761, 2014. https://doi.org/10.1216/RMJ-2014-44-6-1745