By applying the elastic scattering Green's function theory in combination with t he hybrid density function theory, the electronic transport properties of molecu lar junctions constructed by the six-membered heterocyclic molecules, pyridazine -2,5-dithiol, pyrazine-2,5-dithiol, and pyrimidine-2,5-dithiol, have been studie d. For the heterocyclic molecule pyridine, the influence of the terminal atoms o n the current-voltage characteristics has also been obtained. We have accurately determined the coupling constant between the molecule and electrodes by using t he frontier molecular orbital theory and the perturbation method. The numerical results show that pyridazine-2,5-dithiol has higher current and conductance, whi le the conductance of pyrimidine-2,5-dithiol is very small when the bias voltage is lower than 4V. The conductivity of the molecule pyridine with selenium atom as the terminal atom is higher than that with oxygen atom or sulfur atom.