L e may be infinite in the case of an open edge. The simplest example of an operator on a metric graph is the Laplace operator. The operator on an edge is − − d 2 d x e 2 where x e is the coordinate on the edge. To make the operator self-adjoint a suitable domain must be specified. This is typically achieved by taking the Sobolev space H 2 of functions on the edges of the graph and specifying matching conditions at the vertices.