In mathematics and physics, a quantum graph is a linear, network-shaped structure of vertices connected by bonds (or edges) with a differential or pseudo-differential operator acting on functions defined on the bonds. Such systems were first studied by Linus Pauling as models of free electrons in organic molecules in the 1930s. They arise in a variety of mathematical contexts, e.g. as model systems in quantum chaos, in the study of waveguides, in photonic crystals and in Anderson localization, or as limit on shrinking thin wires. Quantum graphs have become prominent models in mesoscopic physics used to obtain a theoretical understanding of nanotechnology. Another, more simple notion of quantum graphs was introduced by Freedman et al.
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.