Small-tau expansion for the form factor of glued quantum star graphs

We compute the small-tau expansion up to the third order for the form factor of two glued quantum star graphs with Neumann boundary conditions, by taking into account only the most backscattering orbits. We thus show that the glueing has no effect if the number of glueing edges is negligible compared to the number of edges of the graph, whereas it has an effect on the $\tau^2$ term when the numbers of glueing and non glueing edges are of the same order.