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In this study, a bilayer zigzag graphyne nanoribbon with RKKY interaction is described using a ferrimagnetic mixed-spin (2, 5/2) Ising model. The compensation behaviors and hysteresis characteristics of the system generated by various physical parameters, such as crystal fields of sublattices, interlayer exchange coupling and the intralayer exchange coupling are explored using Monte Carlo simulation. In the research results, we found that among the 30 possible spin configurations that may occur in the ground state of the system, only 21 spin configurations are stable. At the same time, under certain parameters, the system exhibit interesting behaviors such as compensation phenomena and multi-loop hysteresis. This can provide a theoretical reference for multi-state magnetic recording. • Ising model and Monte Carlo method were used to study the bilayer zigzag graphyne nanoribbon. • The effect of RKKY interaction on the magnetic properties of the graphyne was investigated. • The ground-state phase diagram, magnetization, magnetic susceptibility were obtained. • Phase diagrams and hysteresis loop of the system were presented. • Interesting behaviors of compensation temperatures and multi-loop hysteresis were found under certain parameters. [ABSTRACT FROM AUTHOR]