  The aim of this paper is to obtain new inequalities for a large family of
topological indices restricted to unicyclic graphs and to characterize the set
of extremal unicyclic graphs with respect to them. This family includes
variable first Zagreb, variable sum exdeg, multiplicative second Zagreb and
Narumi-Katayama indices. Our main results provide upper and lower bounds for
these topological indices on unicyclic graphs, fixing or not the maximum degree
or the number of pendant vertices.
