GRAPH-THEORETICAL INVARIANT OF RANDIC REVISITED

The mathematical properties of the graph theoretical invariant of Randic for a series of regular graphs was studied, taking as examples the calculation of vertex and edge connectivity indices. Mathematical relations that explain the degeneracy of both indices for this type of graphs are obtained. The exponent -1/2 used in the graph invariant of Randic was identified as the main cause of this accidental degeneracy of studied topological indices. Several examples of this kind of degeneracy for edge and vertex connectivity indices in graphs representing alkanes and cycloalkanes are presented. Degeneracy in all studied graphs was eliminated by using an exponent of -1/3 in the graph invariant, given a mathematical basis for the empirical generalization of connectivity index carried out by Altenburg. The importance of using regular graphs to test novel graph invariants was also analyzed.