Extremal benzenoid systems for two modified versions of the Randić index

Let G=(V,E) be a molecular graph, in which the vertices of V represent atoms and the edges of E the bonds between pairs of atoms. One of the earliest and most widely studied degree-based molecular descriptor, the Randić index of G, is defined as R(G)=∑uv∈E(dudv)−12, where du denotes the degree of u ∈ V. Bollobás and Erdős (1998) generalized this index by replacing −12 with any fixed real number. To facilitate the enumeration of these indices, motivated by earlier work of Dvořák et al. (2011) and Knor et al. (2015) introduced two other indices, that provide lower and upper bounds, respectively: Rα′(G)=∑uv∈Emin{duα,dvα} and Rα″(G)=∑uv∈Emax{duα,dvα}. In this paper, we give expressions for computing Rα′ and Rα″ of benzenoid systems and phenylenes, as well as a relation between Rα′ and Rα″ of a phenylene and its corresponding hexagonal squeeze. We also determine the extremal values of Rα′ and Rα″ in benzenoid systems with h hexagons for different intervals for the value of α.