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 \(R\Join I:=\{(r,r+i)| r\in R, i\in I\}\) of \(R\times R\). 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 \(\overline{\Gamma}(R\Join I)\) and \(\Gamma(R\Join I)\) coincide, to characterize when \(\overline{\Gamma}(R\Join I)\) is complete, and to compute the diameter and the girth of \(\overline{\Gamma}(R\Join I)\).