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Let Rbe 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 aand bare adjacent if and only if Ra+Rb=R. Also, Γ2(R)is the subgraph of Γ(R)induced by non-unit elements and Γ2′(R)=Γ2(R)\J(R)where J(R) is Jacobson radical. In this paper, we characterize the rings for which the graphs Γ2′(R)and L(Γ2′(R))are planar. Also, we characterize rings for which Γ2′(R)and L(Γ2′(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.

Let Rbe a commutative ring with unity. The co-maximal graph Γ(R)is the graph with vertex set Rand two vertices aand bare adjacent if Ra+Rb=R. In this paper, we characterize rings for which the co-maximal graph Γ(R)is a line graph of some graph Gand determine rings for which Γ(R)is Eulerian or Hamiltonian. We also find the diameter and girth of Γ(R).

AbstractLet be a commutative ring with unity. The co-maximal graph is the graph with vertex set and two vertices and are adjacent if . In this paper, we characterize rings for which the co-maximal graph is a line graph of some graph and determine rings for which is Eulerian or Hamiltonian. We also find the diameter and girth of .