Extremal unicyclic and bicyclic graphs with respect to the F-index

In the study of structure-dependency of the total π-electron energy in 1972, it was shown that this energy depends on the degree based sum ∑_v∈Vdeg_G(v)² and ∑_v∈Vdeg_G(v)³, where deg(v) is the degree of a vertex of under consideration molecular graph G. Later on, the sum ∑_v∈Vdeg_G(v)² was named as first Zagreb index. An extensive research work has been done on the Zagreb index and its variants and generalizations, and it become the most investigated topological index in chemical graph theory. But the sum ∑_v∈Vdeg_G(v)³ has been never further investigated, except only it was used in finding some of the general first Zagreb index and the zeroth-order general Randić index. Recently, this sum was renamed Forgotten index or F-index and it was shown to have an exceptional applicative potential. In this paper, we determine the extremal graphs with respect to the F-index among the classes of connected unicyclic and bicyclic graphs.