In the study of the structure-dependency of the total \( \pi \)-electron from 1972, it is pointed out that it depends on the sums \( \sum_{v\in V}d(v)^2 \) and \( \sum_{v\in V}d(v)^3 \), where \( d(v) \) is the degree of a vertex \( v \) in the underlying molecular graph \( G \). The first sum \( \sum_{v\in V}d(v)^2 \), which was named the first Zagreb index, is one of the most investigated graph-based topological index. The second sum \( \sum_{v\in V}d(v)^3 \) has been almost completely neglected except for its potential involvement in the study of the first Zagreb index and the zeroth-order general Randi\'{c} 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 \( \mathcal{G}_{n,m} \) of all connected graphs of order \( n \) and size \( m \).