In QSAR and QSPR studies the most commonly used topological index was proposed by chemist Milan Randić in 1975 called Randić branching index or path-one molecular connectivity index, 1χ and it has many applications. In the extension of connectivity indices, in early 1990s, chemist Milan Randic´ introduced Variable Randić index defined as ∑v1v2∈E(G) ((dv1 + θ*)(dv2 + θ*))−1/2, where θ* is a non-negative real number and dv1 is the degree of vertex V1 in G. The main objective of the present study is to prove the conjecture proposed in [19]. In this study, we will show that the Pn (path graph) has the maximum Variable connectivity index among the collection of trees whose order is n, where n ≥ 4.

Molecular fragments Variable connectivity index (mfVCI) is proposed as a Variable molecular descriptor. Having in mind that the molecular structure unit exerts strong effect on the boiling point, molecular fragments could be defined as the atoms or functional groups having different characteristics due to different chemical bonding. Each molecular fragment is regarded as a vertex of topological diagram endued with Variable weights to substitute for topological matrix diagonal. The quantitative structure-property relationship (QSPR) model, obtained by mfVCI, shows its desirable robustness and predictivity.

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In the study of QSPR/QSAR, topological indices such as Zagreb index, Randic index, atom-bond connectivity index are exploited to estimate the bioactivity of chemical compounds. Inspired by many degree based topological indices, we propose here a new topological index, called the Atom Bond connectivity temperature index ABCT(G) of a molecular graph G, which shows good correlation with entropy, acentric factor, enthalpy of vaporization and standard enthalpy of vaporization of an octane isomers. In this paper we compute the Atom Bond connectivity temperature index ABCT(G) of line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC_4 C_8 [p,q].