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

A graph invariant I(G)I(G) of a connected graph G=(V,E)G=(V,E) contributed by the weights of all edges is defined as I(G)=∑cijxijI(G)=∑cijxij with the summation over all edges, cijcij is the weight of edges connecting vertices of degree ii and jj, xijxij is the number of edges of GG connecting vertices of degree ii and jj. It generalizes Randić index, Zagreb index, sum-connectivity index, GA1GA1 index, ABC index etc. In this paper, we first give the expressions for computing this invariant I(G)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)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, GA1GA1 index and ABC index in benzenoid systems and phenylenes.