The spectrum of the Laplacian on graphs which have certain symmetry properties can be studied via a decomposition of the operator as a direct sum of one-dimensional operators which are simpler to analyze. In the case of metric graphs, such a decomposition was described by M. Solomyak and K. Naimark when the graphs are radial trees. In the discrete case, there is a result by J. Breuer and M. Keller treating more general graphs.

We present a decomposition in the metric case which is derived from the discrete one. By doing so, we extend the family of (metric) graphs dealt with to also include certain symmetric graphs that are not trees. In addition, our analysis describes an explicit relation between the discrete and continuous cases. This is a joint work with Jonathan Breuer.