  We provide a construction of the induced subgraphs of the zero-divisor graph
of $M_2(R)$ for the ring $R$ of algebraic integers of some number fields that
are neither complete nor connected, and study the structure of the induced
subgraphs explicitly. As an application, we prove that the automorphism group
of the zero-divisor graph of $M_2(R)$ is not a Jordan group.
