Exotic eigenvalues and analytic resolvent for a graph with a shrinking edge

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.