Unit graphs of group rings

In this paper, we studied the unit graph $\mathrm{{\rm Y}}(RG)$ 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 $\mathrm{{\rm Y}}(RG)$ 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 $\mathrm{{\rm Y}}(RG)$ 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.