On the Randić index

The Randić index of an organic molecule whose molecular graph is G is defined as the sum of (d(u)d(v))^−1/2 over all pairs of adjacent vertices of G, where d(u) is the degree of the vertex u in G. In Discrete Mathematics 257, 29–38 by Delorme et al. gave a best-possible lower bound on the Randić index of a triangle-free graph G with given minimum degree δ(G). In the paper, we first point out a mistake in the proof of their result (Theorem 2 of [2002]), and then we will show that the result holds when δ(G) ≥ 2.AMS subject classification: 05C18     

Temperature dependence of the Kováts retention index. The entropy index

According to a novel equation, the temperature dependence of the Kováts retention index, dI/dT is proportional to the difference of the Kováts retention index, I, and the new entropy index, I(S), defined similarly as the retention index, but based on solvation entropy instead of the free energy of solvations. The new relationship was tested with the experimental retention and thermodynamic data published by Kováts and coworkers for 32 compounds on 6 different stationary phases. Very good correlations (r>0.99) were observed for dI/dT versus (I-I(S)) and dI/dT versus deltaDeltaC(p), the molar heat capacity difference of the solute and the hypothetical n-alkane, which has the same retention index as the solute. Deviations in the dI/dT versus deltaDeltaC(p) relationship were observed only for alcohols, suggesting a different solvation mechanism for alcohols as compared with other compounds.     

An improved molecular connectivity index

Through modification of the delta values of the molecular connectivity indexes, and connecting the quantum chemistry with topology method effectively, the molecular connectivity indexes are converted into quantum-topology indexes. The modified indexes not only keep all information obtained from the original molecular connectivity method but also have their own virtue in application, and at the same time make up some disadvantages of the quantum and molecular connectivity methods.     

Randić index and lexicographic order

Let T be a tree and consider the Randi index (T)= ), where v _i –v _j runs over all edges of T and (v _i ) denotes the degree of the vertex v _i . Using counting arguments we show that the Randi index, is monotone increasing over the well (lexicographic order) ordered sequence of trees with unique branched vertex.