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Structure and stability of the gold cluster ions of which skeleton are synthesized as a complex was analyzed using the Hückel method based on graph theory. Hückel Energy (HE) and Topological Resonance Energy (TRE) were determined for neutral, positive ion, and negative ion clusters, where all the isomers of the gold cluster up to octamer were considered. Since some graphically designed isomers include bonds that can not be realized in three-dimensional space, the screening was carried out by a molecular force field calculation with LAMMPS (lammps.sandia.gov/.). Among the isomers thus obtained, both HE and TRE were most stable when the tetramer was Au_4^{2+} with a tetrahedral structure, and with the hexamer, Au_6^{2+} with two tetrahedrons sharing one side. The complexes with these structures have actually been synthesized. On the other hand, there is no reported synthetic example of the most stable cluster of octamer Au8 with TRE and HE.