End-regularity of zero-divisor graphs of group rings

A graph Γ is said to be End-regular if its endomorphism monoid End(Γ) is regular. D. Lu and T. Wu [25 Lu, D., Wu, T. (2008). On endomorphism-regularity of zero-divisor graphs. Discrete Math. 308:4811–4815.[Crossref], [Web of Science ®] , [Google Scholar]] posed an open problem: Given a ring R, when does the zero-divisor graph Γ(R) have a regular endomorphism monoid? and they solved the problem for R a commutative ring with at least one nontrivial idempotent. In this paper, we solve this problem for zero-divisor graphs of group rings.