Can you hear the shape of a graph?
In June 2011, Bristol University took part in the Royal Society Summer Science Exhibition. Our exhibition investigated the mathematics behind `Quantum Graphs'. Below is some background on these fascinating objects. You can see a more detailed summary of what our exhibit actually entailed, together with the full list of people who contributed to it by clicking here (pdf).My role in all this: I was involved in the theoretical and design aspects of the exhibition. This involved deciding what material should be included in the leaflets, and how it should be phrased. I was also responsible for ensuring our team were well prepared to deal with the public, attending the Royal Society's Communication skills course with a colleague, we subsequently organised a day spent training the rest of the team on public engagement.
Just before the exhibition week I was also responsible for giving a few talks to fill in the rest of our team on theoretical aspects of Quantum Graphs. This included giving mathematical definitions of metric graphs, and quantum graphs, and to explicitly solve the (very easiest) case of a star graph (one vertex attached to several edges).
Below you can find some introductory material to quantum graphs, aimed at someone new to the area:
A typical example of a `Quantum Graph' with 5 vertices and 4 edges.
The graph is vibrating at a particular resonant frequency
Why are they sometimes referred to as Quantum Graphs?
The special frequencies, or eigenvalues, of a graph are determined by solving the same governing equation which appears in time-independent quantum mechanics. In particular, the associated wavefunction (yellow line above) can be used to calculate the probability to find a particle (say an electron) at a particular place in the graph.
In the picture above, the points where the yellow and grey lines intersect is where the quantum particle is least likely to be found. These points are referred to as nodes. They divide the graph up into many distinct pieces called nodal domains. See if you can count how many nodal domains there are. I am a mathematician (or a physicist). Where can I find out more?
This survey article by Peter Kuchment is a good place to start, describing the origins of quantum graphs and their applications. The concept of a metric graph is defined, and then the Quantum Graph is introduced as a formal mathematical object.
A graph is simply a configuration of points ("vertices") connected together by lines ("edges"). In our exhibition, we were interested in studying wave phenomena on these types of structures. These waves arise due to a phenomena known as resonance, which has big implications for everything from building bridges to the construction of your car radio.
Why do Mathematicians study them?
When a graph is vibrated (or plucked, like a guitar), it will only produce special standing-wave patterns if the vibration occurs at a very specific frequency. These frequencies are called eigenvalues (or harmonics to musicians). Mathematicians can calculate these frequencies only for very simple types of graphs. They are interested in the general question
How can a graph be reconstructed only by knowing its natural vibrational frequencies?
In particular, can one hear the shape of a graph? It turns out the answer to this question is in general no, and many examples of isospectral (that is, different graphs with the same set of frequencies, or sounds) are known to exist.
For example, the two graphs below are known to be isospectral partners:
