Zero-divisor graphs of amalgamations

Let $f\colon A\rightarrow B$ be a homomorphism of commutative rings and let $J$ be an ideal of $B$. The amalgamation of $A$ with $B$ along $J$ with respect to $f$ is the subring of $A\times B$ given by \[ A\bowtie ^{f}J:=\{(a,f(a)+j) \mid a\in A, j\in J\}. \] This paper investigates the zero-divisor graph of amalgamations. Our aim is to characterize when the graph is complete and compute its diameter and girth for various contexts of amalgamations. The new results recover well-known results on duplications, and yield new and original examples issued from amalgamations