The topological characters of all fullerene isomers from C30 to C70 with respect to their number of shared pentagon bonds have been investigated. Systematic calculations were also carried out at the B3LYP/6-31G* level. It is found that the most stable structure has the smallest number of shared pentagon bonds, and the relative energies of a given size show an increasing tendency to rise with N55 number which denotes the number of C-C bonds shared by pentagon. The topological criterion that has extremely low computational cost can be used as filtering principle to prescreen the thermodynamically viable fullerenes and predict the relative stability of isomers. The topological characters of all fullerene isomers from C30 to C70 with respect to their number of shared pentagon bonds have been investigated. Systematic calculations were also carried out at the B3LYP / 6-31G * level. It is found that the most stable structure has the the smallest number of shared pentagon bonds, and the relative energies of a given size show an increasing tendency to rise with N55 number which shows the number of CC bonds shared by pentagon. The topological criterion that has extremely low computational cost can be used as filtering principle to prescreen the thermodynamically viable fullerenes and predict the relative stability of isomers.