Metric Graphs
A metric graph is a graph consisting of a set of vertices and a set of edges where each edge has been associated with an interval so that is the coordinate on the interval, the vertex corresponds to and to 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 on the graph, is the shortest distance between them where distance is measured along the edges of the graph. It is possible to embed a metric graph in the plane, or space, if the lengths are chosen appropriately.
Open graphs: in the combinatorial graph model edges always join pairs of vertices however in a quantum graph one may also consider semi-infinite edges. These are edges associated with the interval attached to a single vertex at . A graph with one or more such open edges is referred to as an open graph.
Read more about this topic: Quantum Graph