Graphs with smallest forgotten index

The forgotten topological index of a molecular graph $G$ is defined as $F(G)=sum_{vin V(G)}d^{3}(v)$, where $d(u)$ denotes the degree of vertex $u$ in $G$. The first through the sixth smallest forgotten indices among all trees, the first through the third smallest forgotten indices among all connected graph with cyclomatic number $gamma=1,2$, the first through the fourth for $gamma=3$, and the first and the second for $gamma=4,5$ are determined. These results are compared with those obtained for the first Zagreb index.