A general modeling of some vertex-degree based topological indices in benzenoid systems and phenylenes

Abstract

A graph invariant I ( G ) of a connected graph G = ( V , E ) contributed by the weights of all edges is defined as I ( G ) = ∑ c i j x i j with the summation over all edges, c i j is the weight of edges connecting vertices of degree i and j , x i j is the number of edges of G connecting vertices of degree i and j . It generalizes Randić index, Zagreb index, sum-connectivity index, G A 1 index, ABC index etc. In this paper, we first give the expressions for computing this invariant I ( G ) of benzenoid systems and phenylenes, and a relation between this invariant of a phenylene and its corresponding hexagonal squeeze, and then determine the extremal values of I ( G ) and extremal graphs in catacondensed benzenoid systems and phenylenes, and a unified approach to the extremal values and extremal graphs of Randić index, the general Randić index, Zagreb index, sum-connectivity index, the general sum-connectivity index, G A 1 index and ABC index in benzenoid systems and phenylenes.