The aim of this paper is to obtain new sharp inequalities for a large family
of topological indices, including the first variable Zagreb index $M_1^\alpha$,
and to characterize the set of extremal graphs with respect to them. Our main
results provide lower bounds on this family of topological indices involving
just the minimum and the maximum degree of the graph. These inequalities are
new even for the first Zagreb, the inverse and the forgotten indices.Comment: 17 page