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On the Clean Graph of Commutative Artinian Rings

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  • R. Singh
  • M. Habibi
  • S. C. Patekar

Abstract

For a commutative Artinian ring R with unity, the clean graph ClR is a graph with vertices in the form of an ordered pair e,u, where e is an idempotent and u is a unit of ring R, respectively. Two distinct vertices e,u and f,v are adjacent in ClR if and only if ef=fe=0 or uv=vu=1. In this study, we consider Cl2R as the subgraph of ClR induced by e is a non−zero idempotent element of R. We show that Cl2R contains the Hamiltonian cycle. Also, we compute the graphic sequence, matching number, vertex cover number, edge cover number, vertex connectivity, and edge connectivity of Cl2R. As an application, we compute the Wiener and the first Zagreb indices of Cl2R. Moreover, we give the formula for the number of self invertible elements of Zn.

Suggested Citation

  • R. Singh & M. Habibi & S. C. Patekar, 2025. "On the Clean Graph of Commutative Artinian Rings," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2025, pages 1-14, November.
    • Handle: RePEc:hin:jijmms:8677973
    DOI: 10.1155/ijmm/8677973
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