A metric graph is a graph consisting of a set ${\displaystyle V}$  of vertices and a set ${\displaystyle E}$  of edges where each edge ${\displaystyle e=(v_{1},v_{2})\in E}$  has been associated with an interval ${\displaystyle [0,L_{e}]}$  so that ${\displaystyle x_{e}}$  is the coordinate on the interval, the vertex ${\displaystyle v_{1}}$  corresponds to ${\displaystyle x_{e}=0}$  and ${\displaystyle v_{2}}$  to ${\displaystyle x_{e}=L_{e}}$  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 ${\displaystyle x,y}$  on the graph, ${\displaystyle \rho (x,y)}$  is the shortest distance between them where distance is measured along the edges of the graph.

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 ${\displaystyle [0,\infty )}$  attached to a single vertex at ${\displaystyle x_{e}=0}$ . A graph with one or more such open edges is referred to as an open graph.

Quantum graphs are metric graphs equipped with a differential (or pseudo-differential) operator acting on functions on the graph. A function ${\displaystyle f}$  on a metric graph is defined as the ${\displaystyle |E|}$ -tuple of functions ${\displaystyle f_{e}(x_{e})}$  on the intervals. The Hilbert space of the graph is ${\displaystyle \bigoplus _{e\in E}L^{2}([0,L_{e}])}$  where the inner product of two functions is

${\displaystyle \langle f,g\rangle =\sum _{e\in E}\int _{0}^{L_{e}}f_{e}^{*}(x_{e})g_{e}(x_{e})\,dx_{e},}$ 

${\displaystyle L_{e}}$  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 ${\displaystyle -{\frac {{\textrm {d}}^{2}}{{\textrm {d}}x_{e}^{2}}}}$  where ${\displaystyle x_{e}}$  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 ${\displaystyle H^{2}}$  of functions on the edges of the graph and specifying matching conditions at the vertices.

The trivial example of matching conditions that make the operator self-adjoint are the Dirichlet boundary conditions, ${\displaystyle f_{e}(0)=f_{e}(L_{e})=0}$  for every edge. An eigenfunction on a finite edge may be written as

${\displaystyle f_{e}(x_{e})=\sin \left({\frac {n\pi x_{e}}{L_{e}}}\right)}$ 

for integer ${\displaystyle n}$ . If the graph is closed with no infinite edges and the lengths of the edges of the graph are rationally independent then an eigenfunction is supported on a single graph edge and the eigenvalues are ${\displaystyle {\frac {n^{2}\pi ^{2}}{L_{e}^{2}}}}$ . The Dirichlet conditions don't allow interaction between the intervals so the spectrum is the same as that of the set of disconnected edges.

More interesting self-adjoint matching conditions that allow interaction between edges are the Neumann or natural matching conditions. A function ${\displaystyle f}$  in the domain of the operator is continuous everywhere on the graph and the sum of the outgoing derivatives at a vertex is zero,

${\displaystyle \sum _{e\sim v}f'(v)=0\ ,}$ 

where ${\displaystyle f'(v)=f'(0)}$  if the vertex ${\displaystyle v}$  is at ${\displaystyle x=0}$  and ${\displaystyle f'(v)=-f'(L_{e})}$  if ${\displaystyle v}$  is at ${\displaystyle x=L_{e}}$ .

The properties of other operators on metric graphs have also been studied.

• These include the more general class of Schrödinger operators,

${\displaystyle \left(i{\frac {\textrm {d}}{{\textrm {d}}x_{e}}}+A_{e}(x_{e})\right)^{2}+V_{e}(x_{e})\ ,}$ 

where ${\displaystyle A_{e}}$  is a "magnetic vector potential" on the edge and ${\displaystyle V_{e}}$  is a scalar potential.

• Another example is the Dirac operator on a graph which is a matrix valued operator acting on vector valued functions that describe the quantum mechanics of particles with an intrinsic angular momentum of one half such as the electron.
• The Dirichlet-to-Neumann operator on a graph is a pseudo-differential operator that arises in the study of photonic crystals.

All self-adjoint matching conditions of the Laplace operator on a graph can be classified according to a scheme of Kostrykin and Schrader. In practice, it is often more convenient to adopt a formalism introduced by Kuchment, see,^[3] which automatically yields an operator in variational form.

Let ${\displaystyle v}$  be a vertex with ${\displaystyle d}$  edges emanating from it. For simplicity we choose the coordinates on the edges so that ${\displaystyle v}$  lies at ${\displaystyle x_{e}=0}$  for each edge meeting at ${\displaystyle v}$ . For a function ${\displaystyle f}$  on the graph let

${\displaystyle \mathbf {f} =(f_{e_{1}}(0),f_{e_{2}}(0),\dots ,f_{e_{d}}(0))^{T},\qquad \mathbf {f} '=(f'_{e_{1}}(0),f'_{e_{2}}(0),\dots ,f'_{e_{d}}(0))^{T}.}$ 

Matching conditions at ${\displaystyle v}$  can be specified by a pair of matrices ${\displaystyle A}$  and ${\displaystyle B}$  through the linear equation,

${\displaystyle A\mathbf {f} +B\mathbf {f} '=\mathbf {0} .}$ 

The matching conditions define a self-adjoint operator if ${\displaystyle (A,B)}$  has the maximal rank ${\displaystyle d}$  and ${\displaystyle AB^{*}=BA^{*}.}$ 

The spectrum of the Laplace operator on a finite graph can be conveniently described using a scattering matrix approach introduced by Kottos and Smilansky .^[4][5] The eigenvalue problem on an edge is,

${\displaystyle -{\frac {d^{2}}{dx_{e}^{2}}}f_{e}(x_{e})=k^{2}f_{e}(x_{e}).\,}$ 

So a solution on the edge can be written as a linear combination of plane waves.

${\displaystyle f_{e}(x_{e})=c_{e}{\textrm {e}}^{ikx_{e}}+{\hat {c}}_{e}{\textrm {e}}^{-ikx_{e}}.\,}$ 

where in a time-dependent Schrödinger equation ${\displaystyle c}$  is the coefficient of the outgoing plane wave at ${\displaystyle 0}$  and ${\displaystyle {\hat {c}}}$  coefficient of the incoming plane wave at ${\displaystyle 0}$ . The matching conditions at ${\displaystyle v}$  define a scattering matrix

${\displaystyle S(k)=-(A+ikB)^{-1}(A-ikB).\,}$ 

The scattering matrix relates the vectors of incoming and outgoing plane-wave coefficients at ${\displaystyle v}$ , ${\displaystyle \mathbf {c} =S(k){\hat {\mathbf {c} }}}$ . For self-adjoint matching conditions ${\displaystyle S}$  is unitary. An element of ${\displaystyle \sigma _{(uv)(vw)}}$  of ${\displaystyle S}$  is a complex transition amplitude from a directed edge ${\displaystyle (uv)}$  to the edge ${\displaystyle (vw)}$  which in general depends on ${\displaystyle k}$ . However, for a large class of matching conditions the S-matrix is independent of ${\displaystyle k}$ . With Neumann matching conditions for example

${\displaystyle A=\left({\begin{array}{ccccc}1&-1&0&0&\dots \\0&1&-1&0&\dots \\&&\ddots &\ddots &\\0&\dots &0&1&-1\\0&\dots &0&0&0\\\end{array}}\right),\quad B=\left({\begin{array}{cccc}0&0&\dots &0\\\vdots &\vdots &&\vdots \\0&0&\dots &0\\1&1&\dots &1\\\end{array}}\right).}$ 

Substituting in the equation for ${\displaystyle S}$  produces ${\displaystyle k}$ -independent transition amplitudes

${\displaystyle \sigma _{(uv)(vw)}={\frac {2}{d}}-\delta _{uw}.\,}$ 

where ${\displaystyle \delta _{uw}}$  is the Kronecker delta function that is one if ${\displaystyle u=w}$  and zero otherwise. From the transition amplitudes we may define a ${\displaystyle 2|E|\times 2|E|}$  matrix

${\displaystyle U_{(uv)(lm)}(k)=\delta _{vl}\sigma _{(uv)(vm)}(k){\textrm {e}}^{ikL_{(uv)}}.\,}$ 

${\displaystyle U}$  is called the bond scattering matrix and can be thought of as a quantum evolution operator on the graph. It is unitary and acts on the vector of ${\displaystyle 2|E|}$  plane-wave coefficients for the graph where ${\displaystyle c_{(uv)}}$  is the coefficient of the plane wave traveling from ${\displaystyle u}$  to ${\displaystyle v}$ . The phase ${\displaystyle {\textrm {e}}^{ikL_{(uv)}}}$  is the phase acquired by the plane wave when propagating from vertex ${\displaystyle u}$  to vertex ${\displaystyle v}$ .

Quantization condition: An eigenfunction on the graph can be defined through its associated ${\displaystyle 2|E|}$  plane-wave coefficients. As the eigenfunction is stationary under the quantum evolution a quantization condition for the graph can be written using the evolution operator.

${\displaystyle |U(k)-I|=0.\,}$ 

Eigenvalues ${\displaystyle k_{j}}$  occur at values of ${\displaystyle k}$  where the matrix ${\displaystyle U(k)}$  has an eigenvalue one. We will order the spectrum with ${\displaystyle 0\leqslant k_{0}\leqslant k_{1}\leqslant \dots }$ .

The first trace formula for a graph was derived by Roth (1983). In 1997 Kottos and Smilansky used the quantization condition above to obtain the following trace formula for the Laplace operator on a graph when the transition amplitudes are independent of ${\displaystyle k}$ . The trace formula links the spectrum with periodic orbits on the graph.

${\displaystyle d(k):=\sum _{j=0}^{\infty }\delta (k-k_{j})={\frac {L}{\pi }}+{\frac {1}{\pi }}\sum _{p}{\frac {L_{p}}{r_{p}}}A_{p}\cos(kL_{p}).}$ 

${\displaystyle d(k)}$  is called the density of states. The right hand side of the trace formula is made up of two terms, the Weyl term ${\displaystyle {\frac {L}{\pi }}}$  is the mean separation of eigenvalues and the oscillating part is a sum over all periodic orbits ${\displaystyle p=(e_{1},e_{2},\dots ,e_{n})}$  on the graph. ${\displaystyle L_{p}=\sum _{e\in p}L_{e}}$  is the length of the orbit and ${\displaystyle L=\sum _{e\in E}L_{e}}$  is the total length of the graph. For an orbit generated by repeating a shorter primitive orbit, ${\displaystyle r_{p}}$  counts the number of repartitions. ${\displaystyle A_{p}=\sigma _{e_{1}e_{2}}\sigma _{e_{2}e_{3}}\dots \sigma _{e_{n}e_{1}}}$  is the product of the transition amplitudes at the vertices of the graph around the orbit.

Quantum graphs were first employed in the 1930s to model the spectrum of free electrons in organic molecules like Naphthalene, see figure. As a first approximation the atoms are taken to be vertices while the σ-electrons form bonds that fix a frame in the shape of the molecule on which the free electrons are confined.

A similar problem appears when considering quantum waveguides. These are mesoscopic systems - systems built with a width on the scale of nanometers. A quantum waveguide can be thought of as a fattened graph where the edges are thin tubes. The spectrum of the Laplace operator on this domain converges to the spectrum of the Laplace operator on the graph under certain conditions. Understanding mesoscopic systems plays an important role in the field of nanotechnology.

In 1997^[6] Kottos and Smilansky proposed quantum graphs as a model to study quantum chaos, the quantum mechanics of systems that are classically chaotic. Classical motion on the graph can be defined as a probabilistic Markov chain where the probability of scattering from edge ${\displaystyle e}$  to edge ${\displaystyle f}$  is given by the absolute value of the quantum transition amplitude squared, ${\displaystyle |\sigma _{ef}|^{2}}$ . For almost all finite connected quantum graphs the probabilistic dynamics is ergodic and mixing, in other words chaotic.

Quantum graphs embedded in two or three dimensions appear in the study of photonic crystals.^[7] In two dimensions a simple model of a photonic crystal consists of polygonal cells of a dense dielectric with narrow interfaces between the cells filled with air. Studying dielectric modes that stay mostly in the dielectric gives rise to a pseudo-differential operator on the graph that follows the narrow interfaces.

Periodic quantum graphs like the lattice in ${\displaystyle {\mathbb {R} }^{2}}$  are common models of periodic systems and quantum graphs have been applied to the study the phenomena of Anderson localization where localized states occur at the edge of spectral bands in the presence of disorder.

• Schild's Ladder, a novel dealing with a fictional quantum graph theory
• Feynman diagram