Solution and significance of structural representative of fullerene cages.

The number of hexagon-carbon ring with different polimerity index in the st ructural representation of fullerene cages C-n, x, y, z and u are solution for indeterminate equation HPI = (A(x)+B-y+C-z+D-u)/(x+y+z+u). This paper g ives a method finding solution for the equation and demonstrate two factors , we call them two rules, which stable fullerene cages have to satisfy: eve n carbon rule and sixty carbon rule. Indeterminate equation have many solut ion groups, each one corresponds to one kind of fullerene cage. They have v arious molecular configuration being isoresm. The larger n, the more soluti on group is, and the more variant of fullerene cage C-n would be.