  Making use of a majorization technique for a suitable class of graphs, we
derive upper and lower bounds for some topological indices depending on the
degree sequence over all vertices, namely the first general Zagreb index and
the first multiplicative Zagreb index. Specifically, after characterizing
$c-$cyclic graphs $(0\leq c\leq 6)$ as those whose degree sequence belongs to
particular subsets of $\mathbb{R}^{n}$, we identify the maximal and minimal
vectors of these subsets with respect to the majorization order. This technique
allows us to determine lower and upper bounds of the above indices recovering
those existing in the literature as well obtaining new ones.
