Unit graphs of group rings

Eman Alluqmani; Eman Alluqmani

doi:10.3934/math.2026116

AIMS Mathematics

2026, Volume 11, Issue 1: 2907-2934. doi: 10.3934/math.2026116

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Research article

Unit graphs of group rings

Eman Alluqmani ^,

Mathematics Department, Faculty of Sciences, Umm Al-Qura University, P. O. Box 14035, Makkah 21955, Saudi Arabia

Received: 28 November 2025 Revised: 12 January 2026 Accepted: 21 January 2026 Published: 29 January 2026
MSC : 05C25, 05C75, 16S34, 16U60, 20C05

In this paper, we studied the unit graph $ \Upsilon(RG) $ of a finite group ring $ RG $, where two vertices $ x, y\in RG $ were adjacent when $ x+y $ was a unit. We established algebraic criteria for connectivity, described all connected components, and determined the diameter of $ \Upsilon(RG) $ in terms of the semisimple quotient $ RG/J(RG) $. We classified completely when the unit graph was a cycle or a complete graph, and we determined the girth of $ \Upsilon(RG) $ from the field factorization of the quotient. The results revealed a tight correspondence between the additive structure generated by units and the global geometry of the unit graph.
- unit graph,
- group ring,
- connectivity,
- diameter,
- cycle classification,
- girth
Citation: Eman Alluqmani. Unit graphs of group rings[J]. AIMS Mathematics, 2026, 11(1): 2907-2934. doi: 10.3934/math.2026116

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Abstract

In this paper, we studied the unit graph $ \Upsilon(RG) $ of a finite group ring $ RG $, where two vertices $ x, y\in RG $ were adjacent when $ x+y $ was a unit. We established algebraic criteria for connectivity, described all connected components, and determined the diameter of $ \Upsilon(RG) $ in terms of the semisimple quotient $ RG/J(RG) $. We classified completely when the unit graph was a cycle or a complete graph, and we determined the girth of $ \Upsilon(RG) $ from the field factorization of the quotient. The results revealed a tight correspondence between the additive structure generated by units and the global geometry of the unit graph.

References

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