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{\bf C. Dalf\'{o}, M.A. Fiol and E. Garriga }
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{\bf A Differential Approach for Bounding the Index of Graphs under Perturbations}
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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,
$$
\textstyle
\lambda(G_u)\le \lambda(G)+\frac{\delta_u}{\lambda^3(G)}+\textrm{o}\left(\frac{1}{\lambda^3(G)}\right).
$$
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