  We consider a metric graph consisting of two edges, one of which has length
$\varepsilon$ which we send to zero. On this graph we study the resolvent and
spectrum of the Laplacian subject to a general vertex condition at the
connecting vertex. Despite the singular nature of the perturbation (by a short
edge), we find that the resolvent depends analytically on the parameter
$\varepsilon$. In contrast, the negative eigenvalues escape to minus infinity
at rates that could be fractional, namely, $\varepsilon^0$,
$\varepsilon^{-2/3}$ or $\varepsilon^{-1}$. These rates take place when the
corresponding eigenfunction localizes, respectively, only on the long edge, on
both edges, or only on the short edge.
