On the Randić index of quasi-tree graphs

In studying branching properties of alkanes, several numbering schemes for the edges of the associated hydrogen-suppressed graph were proposed based on the degrees of the end vertices of an edge [11]. To preserve rankings of certain molecules, some inequalities involving the weights of edges needed to be satisfied. Randić [11] stated that weighting all edges uv of the associated graph G by (d(u)d(v))−1/2 preserved these inequalities, where d(u) and d(v) are the degrees of u and v. The sum of weights over all edges of G, which is called the Randić index or molecular connectivity index or simply connectivity index of G and denoted by R(G), has been closely correlated with many chemical properties [8] and found to parallel the boiling point, Kovats constants, and a calculated surface. In addition, the Randić index appears to predict the boiling points of alkanes more closely, and only it takes into account the bonding or adjacency degree among carbons in alkanes (see [9]). It is said in [7] that Randić index “together with its generalizations it is certainly the molecular-graph-based structuredescriptor, that found the most numerous applications in organic chemistry,