Abstract
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.
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Acknowledgements
This work is supported by the Research Grant from DST [MTR/2018/000607] under Mathematical Research Impact Centric Support (MATRICS) for a period of 3-years (2019–2022) and University Grants Commission [Sr. No. 2061540883 Ref. No. \(21/06/2015(i)EU-V\)] of first and second authors respectively. We would like to thank Prof. Bijan Davvaz for helping us finalise the paper.
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Sinha, D., Rao, A.K. & Davvaz, B. On Some Properties of Co-maximal Graphs of Commutative Rings. Natl. Acad. Sci. Lett. 44, 437–442 (2021). https://doi.org/10.1007/s40009-020-01030-x
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DOI: https://doi.org/10.1007/s40009-020-01030-x