Abstract
The symmetric figure of a molecule and the fact that the bond polarities are equal imply that the polarities of the bonds cancel each other out and the molecule would be nonpolar. Many molecules are nonpolar, but have polar bonds. A chemical bond is polar if the atoms on either end of its molecular diagram are different. Therefore, the notion of symmetry plays an important role in chemical as well as the mathematical study of molecular graphs. By means of graph pieces such as cut vertices, cut edges and bridges, it is possible to separate a large graph into smaller pieces which we can handle easily. Using this new combinatorial technique for graphs, we obtain several formulae for some important topological graph indices of symmetrical graphs by means of smaller ones.