  We consider an arbitrary metric graph, to which we glue another graph with
edges of lengths proportional to $\varepsilon$, where $\varepsilon$ is a small
positive parameter. On such graph, we consider a general self-adjoint second
order differential operator $\mathcal{H}_\varepsilon$ with varying coefficients
subject to general vertex conditions; all coefficients in differential
expression and vertex conditions are supposed to be analytic in $\varepsilon$.
We introduce a special operator on a certain graph obtained by rescaling the
aforementioned small edges and assume that it has no embedded eigenvalues at
the threshold of its essential spectrum. Under such assumption, we show that
certain parts of the resolvent of $\mathcal{H}_\varepsilon$ are analytic in
$\varepsilon$. This allows us to represent the resolvent of
$\mathcal{H}_\varepsilon$ by an uniformly converging Taylor-like series and its
partial sums can be used for approximating the resolvent up to an arbitrary
power of $\varepsilon$. In particular, the zero-order approximation reproduces
recent convergence results by G. Berkolaiko, Yu. Latushkin, S. Sukhtaiev and by
C. Cacciapuoti, but we additionally show that next-to-leading terms in
$\varepsilon$-expansions of the coefficients in the differential expression and
vertex conditions can contribute to the limiting operator producing the Robin
part at the vertices, to which small edges are incident. We also discuss
possible generalizations of our model including both the cases of a more
general geometry of the small parts of the graph and a non-analytic
$\varepsilon$-dependence of the coefficients in the differential expression and
vertex conditions.
