Deepa Sinha, Anita Kumari Rao, Bijan Davvaz

Let *R* be a finite commutative ring with identity. The co-maximal graph \(\varGamma (R)\) is a graph with vertices as elements of *R*, where two distinct vertices *a* and *b* are adjacent if and only if \(Ra+Rb = R\). Also, \(\varGamma _{2}(R)\) is the subgraph of \(\varGamma (R)\) induced by non-unit elements and \(\varGamma _{2}^{\prime }(R) = \varGamma _{2}(R){\setminus } J(R)\) where *J*(*R*) is Jacobson radical. In this paper, we characterize the rings for which the graphs \(\varGamma _{2}^{\prime }(R)\) and \(L(\varGamma _{2}^{\prime }(R))\) are planar. Also, we characterize rings for which \(\varGamma _{2}^{\prime }(R)\) and \(L(\varGamma _{2}^{\prime }(R))\) are split graphs, and obtain nullity of co-maximal graph for local rings and non-local rings along with domination number on co-maximal graph.