Let deg(v) denote the degree (= number of first neighbors) of the vertex v of a molecular graph G. Let e be an edge of G, connecting the vertices u and v. Then a weight w(e)=[deg(u) deg(v)]^L is associated to the edge e, where L is a suitably chosen exponent. The CONNECTIVITY INDEX of G is defined as the sum of w(e) over all edges of the graph G.

The first value put forward for the exponent L was L=+1 (Gutman et al., 1972, 1975), but the most famous choice is that of Randic (1975), namely L=-0.5. In his seminal 1975 paper Randic considered two options: L=-0.5 and L=-1, declaring both as plausible.

Eventually, the Randic (L=-0.5) connectivity index (and its generalizations) became the most successfully and most often applied topological indices in QSPR and QSAR studies. Yet, until recently no convincing argument in favor of the choice L=-0.5 was offered.

We have "attacked" the problem of the exponent in the definition of the connectivity index from two directions.

1. We showed (Gutman et al., 2000) that a few distinguished mathematical properties of the connectivity index holds only for the choice L=-0.5.
2. We show that the choice L=-0.5 makes the connectivity index a suitable measure of branching of the carbon-atom skeleton of organic molecules, being thus appropriate for QSPR and QSAR purposes. This, however, is not the case with L=-1. When diminishing L from -0.5 to -1 a "breakdown" of the connectivity index happens. The respective critical value of L depends on the number n of C atoms, increases with increasing n and for n up to 20 is around -0.9.

I. Gutman, N. Trinajstic, Chem. Phys. Lett. 17 (1972) 535.

I. Gutman, B. Ruscic, N.Trinajstic, F. Wilcox, J. Chem. Phys. 62 (1975) 3399.

M. Randic, J. Am. Chem. Soc. 97 (1975) 6609.

I. Gutman, O. Araujo, J.Rada, ACH Models Chem. 137 (2000) 653.