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Fifth-generation (5G) cellular networks are being developed to meet the ever-growing data traffic across mobile devices and their applications. The core of 5G cellular networks is leveraging wider and higher frequencies available at millimeter wave frequency (mmWave) bands, thus providing very high data rates for mobile devices. Multi-input multi-output (MIMO) is an essential technology for overcoming the high propagation loss at mmWave communications. In non-orthogonal multiple access (NOMA), multiple cellular user equipments (CUEs) communicate over the same time-frequency resources using a multiplexed power domain. In device-to-device (D2D) communications, two D2D user equipments (DUEs) communicate without passing through the base station. In the underlaying scenario, DUEs reuse the frequency resources allocated to CUEs for spectrum utilization but DUEs cause interferences for cellular and D2D communications. Integrating D2D communications with other 5G technologies has great potential for spectral efficiency improvement. Unfortunately, interference management and resource allocation are becoming increasingly challenging due to aggressive frequency reuse. In this paper, D2D communications at mmWave underlaying MIMO-NOMA cellular network system model is developed. Consequently, a novel resource allocation for D2D communications underlaying MIMO-NOMA cellular network is proposed. A resource allocation optimization problem is formulated for spectral efficiency maximization. To solve this NP-hard problem, the problem is decomposed into three subproblems: interference-aware graph-based user clustering, MIMO-NOMA beamforming design, and optimized power allocation based on particle swarm optimization. Simulation results demonstrate that the proposed algorithm for D2D communications at mmWave underlaying MIMO-NOMA cellular network delivers a greater spectral efficiency compared to the conventional D2D communications that operate underlay MIMO-orthogonal multiple access cellular networks.