A metric graph is a graph consisting of a set V of vertices and a set E of edges where each edge e = ( v 1 , v 2 ) ∈ ∈ E has been associated with an interval [ 0 , L e ] so that x e is the coordinate on the interval, the vertex v 1 corresponds to x e = 0 and v 2 to x e = L e or vice versa. The choice of which vertex lies at zero is arbitrary with the alternative corresponding to a change of coordinate on the edge. The graph has a natural metric: for two points x , y on the graph, ρ ρ ( x , y ) is the shortest distance between them where distance is measured along the edges of the graph.