On Extremal Connected Graphs with Respect to the Forgotten Topological Index

In the study of the structure-dependency of the total π-electron from 1972, it is pointed out that it depends on the sums^∑v∈V^d(v)2 and^∑v∈V^d(v)3, where d(v) is the degree of a vertex v in the underlying molecular graph G. The first sum^∑v∈V^d(v)2, which was named the first Zagreb index, is one of the most investigated graph-based topological index. The second sum^∑v∈V^d(v)3 has been almost completely neglected except for its potential involvement in the study of the first Zagreb index and the zeroth-order general Randić index. This second sum was named the forgotten index or F-index in short. In this paper, we investigate some properties of the F-index with respect to some new graph transformations, which are used to characterize all extremal graphs having the maximum F-index in the set G_n,m of all connected graphs of order n and size m.