Unit graphs of group rings

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.