The extended zero-divisor graph of the amalgamated duplication of a ring along an ideal

Let R be a commutative ring and I be an ideal of R. The amalgamated duplication of R along I is the subring R I:=\(r,r+i)| r∈ R, i∈ I\ of R× R. This paper investigates the extended zero-divisor graph of the amalgamated duplication of R along I. The purpose of this work is to study when Γ̄(R I) and Γ(R I) coincide, to characterize when Γ̄(R I) is complete, and to compute the diameter and the girth of Γ̄(R I).