  This paper presents bounds for the variation of the spectral radius
$\lambda(G)$ of a graph $G$ after some perturbations or local vertex/edge
modifications of $G$. The perturbations considered here are the connection of a
new vertex with, say, $g$ vertices of $G$, the addition of a pendant edge (the
previous case with $g=1$) and the addition of an edge. The method proposed here
is based on continuous perturbations and the study of their differential
inequalities associated. Within rather economical information (namely, the
degrees of the vertices involved in the perturbation), the best possible
inequalities are obtained. In addition, the cases when equalities are attained
are characterized. The asymptotic behavior of the bounds obtained is also
discussed. For instance, if $G$ is a connected graph and $G_u$ denotes the
graph obtained from $G$ by adding a pendant edge at vertex $u$ with degree
$\delta_u$, then, $$ \lambda(G_u)\le
\lambda(G)+\frac{\delta_u}{\lambda^3(G)}+\textrm{o}(\frac{1}{\lambda^3(G)}). $$
