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Subgraph of generalized co-maximal graph of commutative rings

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Abstract

Let R be a commutative ring with 1. In Biswas et al. (Disc Math Algorithms Appl 11(1):1950013, 2019), we introduced a graph G(R) whose vertices are elements of R and two distinct vertices a, b are adjacent if and only if \(aR+bR=eR\) for some nonzero idempotent e in R. Let \(G'(R)\) be the subgraph of G(R) generated by the non-units of R. In this paper, we characterize those rings R for which the graph \(G'(R)\) is connected and Eulerian. Also we characterize those rings R for which genus of the graph \(G'(R)\) is \(\le 2\). Finally, we show that the graph \(G'(R)\) is a line graph of some graph if and only if R is either a regular ring or a local ring.

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Acknowledgements

The authors are thankful to the anonymous referees for making several useful comments and suggestions which have definitely enriched this paper.

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Authors and Affiliations

  1. Department of Mathematics, Jadavpur University, Kolkata, 700032, India

    B. Biswas & S. Kar

  2. Department of Pure Mathematics, University of Calcutta, Kolkata, 700019, India

    M. K. Sen

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  1. B. Biswas
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  2. S. Kar
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  3. M. K. Sen
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Biswas, B., Kar, S. & Sen, M.K. Subgraph of generalized co-maximal graph of commutative rings. Soft Comput 26, 1587–1596 (2022). https://doi.org/10.1007/s00500-022-06748-y

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  • DOI: https://doi.org/10.1007/s00500-022-06748-y

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