Connectedness in Projection Graphs of Commutative Rings

S Teresa Arockiamary; C Meera; V Santhi

doi:10.17485/IJST/v16iSP3.icrtam293

Article

Connectedness in Projection Graphs of Commutative Rings

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Indian Journal of Science and Technology

DOI: 10.17485/IJST/v16iSP3.icrtam293

Year: 2023, Volume: 16, Issue: Special Issue 3, Pages: 55-63

Original Article

Connectedness in Projection Graphs of Commutative Rings

S Teresa Arockiamary¹, C Meera², V Santhi^3*

¹Assistant Professor, Department of Mathematics, Stella Maris College, University of Madras, Tamil Nadu, India
²Assistant Professor, Department of Mathematics, Bharathi Women’s College, University of Madras, Tamil Nadu, India
³Assistant Professor, Department of Mathematics, Presidency College, University of Madras, Tamil Nadu, India

*Corresponding Author
Email: [email protected]

Received Date:24 February 2023, Accepted Date:11 August 2023, Published Date:04 December 2023

This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

Background/Objectives: Connectedness plays an essential role in applications of graph theory. Connected graphs are used to represent social networks, transportation networks, and communication networks. In this study, various classes of commutative rings are explored to identify connected projection graphs. Methods: An investigation is carried out to exclude classes of rings whose projection graphs possess isolated vertices and connectedness is shown by establishing spanning subgraphs. Findings : Unipotent units and zero-divisors are found non-isolated, in which zero-divisors include idempotents and nilpotents. The element 2 is isolated if it is invertible. Necessary condition for the rings to have connected projection graphs is derived. Projection graphs of finite Boolean rings are connected and Hamiltonian. The local rings of integers modulo n with even characteristics are connected and contain spanning bistars. A criterion for certain class of nonlocal rings to have connected projection graphs is described. Novelty: Study on projection graphs in the perception of unipotent units is carried out, which is not done earlier in any other algebraic graph.

Keywords: Unipotent, Von Neumann regular ring, Boolean ring, Spanning subgraph, Bistar

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Copyright

© 2023 Arockiamary et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Published By Indian Society for Education and Environment (iSee)