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 $\overline{\Gamma}(R\Join I)$.