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On Some Properties of Co-maximal Graphs of Commutative Rings
National Academy Science Letters  (IF0.788),  Pub Date : 2021-02-11, DOI: 10.1007/s40009-020-01030-x
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.