Co-maximal Graph, its Planarity and Domination Number

Abstract

Let R be a finite commutative ring with identity. The co-maximal graph Γ(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, Γ2(R) is the subgraph of Γ(R) induced by non-unit elements and [Formula: see text] where J(R) is Jacobson radical. In this paper, we characterize the rings for which the graphs Γ(R) and L(Γ(R)) are planar. Also, we characterize rings for which [Formula: see text], Γ(R) and L(Γ(R)) are outerplanar along with some domination parameters on co-maximal graph.