In this article, the efficiency of all the well-known valency-based molecular descriptors has been measured using certain comparative testing. Importantly, we use a computational technique for hexagonal systems to generate regression models to determine the correlation of valency-based molecular descriptors with the π-electronic energies for polycyclic aromatic hydrocarbons. The study implicates some favorable outcomes as the sum-connectivity index and certain Randić index have correlation coefficient of 0.9999. To ensure the applicability of our proposed study, we compute analytically explicit expressions for efficient valency-based molecular descriptors for certain infinite families of carbon nanotubes and carbon nanocones. Our results assist in correlating the π-electronic energies of underlying chemical structures of these nanotubes and nanocones.

In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n, $n-1$ , and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, a method is presented to locate fault-tolerant resolving sets by using classical resolving sets in graphs. The second part of the paper applies the proposed method to three infinite families of regular graphs and locates certain fault-tolerant resolving sets. By accumulating the obtained results with some known results in the literature, we present certain lower and upper bounds on the fault-tolerant metric dimension of these families of graphs. As a byproduct, it is shown that these families of graphs preserve a constant fault-tolerant resolvability structure.

Several chemical and medical experimentation reveals a dependence of physicochemical and biological properties of a compound on its molecular structure. Molecular/topological descriptors/indices retrieve this dependence by employing mathematical/statistical tools to generate quantitative structure property/activity relationship (QSAR/QSPR) models. QSAR/QSPR models are regression models which theoretically relate a physicochemical/biological property to a molecular descriptor. By converting a compound to a chemical graph, graph-theoretic topological indices use algorithmic graph theory to generate QSAR/QSPR models. Degree-related molecular indices have been well-known for correlating better with certain physicochemical characteristics of compounds. In this paper, we conduct a comparative analysis to put forward the best degree-based index for correlating with physicochemical characteristics of polycyclic aromatic hydrocarbons. Two different computational techniques have been put forward and then used to conduct our comparative analysis. Our comparative testing generates several favorable and unexpected conclusions as the well-known indices such as the augmented Zagreb index, the atom-bond connectivity index, the geometric arithmetic index deliver relatively poor performance. Moreover, exceptional performance of the generalized variants of Randić and sum-connectivity indices is somehow unexpected as they significantly outperform the well-studied and well-researched degree-based indices. This warrants to study further the general Randić and sum-connectivity indices for QSAR/QSPR models. Applications to certain families of carbon polyhex aromatic nanotubes are reported.