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 $R\bowtie I:=\{(r,r+i)\mathrm{\mid }r\in R,i\in I\}$ 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 $\bar{\mathrm{\Gamma }}(R\bowtie I)$ and $\mathrm{\Gamma }(R\bowtie I)$ coincide, to characterize when $\bar{\mathrm{\Gamma }}(R\bowtie I)$ is complete, and to compute the diameter and the girth of $\bar{\mathrm{\Gamma }}(R\bowtie I)$.