Although the one-dimensional non-conjugated alkane chain, which has an important influence on the electron transport process, does not possess the characteristics of electron-rich and electron-deficient, it often exists in single-molecule devices and biological molecules such as peptides and proteins. In order to understand the electron transport characteristics of alkane chain, a one-dimensional linear non-conjugate (CH₂)_n molecular junction model is designed in this study. Subsequently, we conduct the systematic study of the electronic transport behavior of (CH₂)_n (n = 1–12) molecular linear chain coupling to two graphene electrodes, based on the density functional theory and nonequilibrium Green’s function formalism. The results reveal that the structure and conductance of CH₂ chain are highly sensitive to the odevity of CH₂ unit. When the value of n is odd, the groups of CH₂ extend in a zigzag way from the left electrode to the right electrode in the plane of graphene, while the value of n is even, what is different is that the groups of CH₂ are arranged above and below the electrode plane. For this reason, the odd-even behavior of conductance occurs in the (CH₂)_n (n = 1–12) molecular chain. Furthermore, n is also an important factor to affect their transport properties (odd or even behavior of conductance). The longer the (CH₂)_n chain, the deeper the suppression in transmission spectrum and the lower the equilibrium conductance. What is more, the conductance decreases exponentially with the increase of molecular length, with a decay constant β of 0.67 and 0.60 for odd and even, respectively, which is in good agreement with the experimental research. Additionally, by analyzing their eigenchannels of odd and even (CH₂)_n molecular chain, we find that the coplanar σ electron with graphene electrode makes a major contribution to the electronic transport channel. The current-voltage curve of (CH₂)_n molecular chain exhibits nonlinearity, implying their semiconductor characteristics. The interesting mechanical and electronic transport properties are expected to conduce to further experimental synthesis, design and operation of the single molecular nanodevices.