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The realization of computational circuits in the nanometer scale leads to a higher speed, lower power consumption, and less occupied area on the chip surface in electronic systems. Two of the most important basic logic gates are three-input exclusive-OR (XOR3) and three-input majority (MAJ3) gates which are the two main functions used in designing computational circuits like full-adder (FA) cells. In this article, a new device (molecular transistor) is presented using the naphthalene molecule. The effects of the input terminal variation of the proposed transistor on its output have been studied. According to the results obtained from the simulation, by applying different voltages to the gate-terminal of the transistor (Vg), the negative differential resistance (NDR) behaviors are observed at different intervals of the source-drain voltage (Vsd). Due to the interesting characteristics of the proposed transistor, a full set of three-input basic logic gates is designed and presented. In these gates, a single transistor is utilized and by applying different bias voltages, it can generate different outputs suitable to the intended functions. Besides, an FA cell is designed using the proposed three-input logic gates. The transistor, the gates, and the FA are simulated using nonequilibrium Green’s function (NEGF) and density functional theory (DFT) to verify their functionality. [ABSTRACT FROM AUTHOR]