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Neutrosophic cubic graph (NCG) belonging to FG family has good capabilities when facing problems that cannot be expressed by FGs. When an element membership is not clear, neutrality is a good option that can be well supported by a NCG. Hence, in this paper, some types of edge irregular neutrosophic cubic graphs (EI-NCGs) such as neighborly edge totally irregular (NETI), strongly edge irregular (SEI), and strongly edge totally irregular (SETI) are introduced. A comparative study between NEI-NCGs and NETI-NCGs is done. Finally, an application of neutrosophic cubic digraph to find the most effective person in a school has been presented.