# Applications of Cayley Graphs in Physics

I have been recently reading about Cayley graphs and character theory. It is evident that Cayley graphs are very useful tool in theoretical computer science. In physics, Cayley graphs seem do appear in the study of quantum walks. I wonder however, if they have been used anywhere else in physics, specially in the study of the spectral properties of physical systems. Any references will be helpful.

• I find this question very interesting, but I suspect, unfortunately, that there really aren't any such applications. I hope someone surprises me. – Alexander Gruber Apr 28 '14 at 8:55
• I can think of few simple physical systems, for example, a particle hopping on a line and a stabilizer Hamiltonian, where corresponding Cayley graph (the ring graph and the hypercube) is obvious. Understanding spectral properties of these Cayley graphs can help in the understanding of the properties of the corresponding physical system. The goal is to find a complex enough physical Hamiltonians to which the corresponding Cayley graph can reveal something non-trivial. – Abbas Apr 28 '14 at 10:47
• Maybe physics.stackexchange is a better place for this question. – Moishe Kohan Apr 28 '14 at 17:45
• See for instance this discussion: physics.stackexchange.com/questions/26895/… – Moishe Kohan Apr 28 '14 at 17:48

Given that you've already noticed the utility in theoretical comp. sci., you might want to look into the quantum computing literature. Typically, you find these come up in terms of special cases of Cayley graphs rather than general Cayley graphs. (I don't know if you are interested in the particular properties of a Cayley graph, however.)

One simple example is an Ising Hamiltonian for some qubits in quantum theory. Take, for instance, the simple Hamiltonian

$\displaystyle H = -\sum_i \sigma_x^{(i)}$.

Now, consider how this Hamiltonian maps particular qubits. Specifically, if $|b_i\rangle$ is a qubit-string, then

$-\sigma_x^{(j)} |b_i\rangle = |b_i \oplus 1_j\rangle$

where $1_j$ is the bit string with a 1 in the $j^{th}$ position and 0s elsewhere. Now, we see that

$\langle b_i | H | b_j \rangle = -\delta_{ij}$.

This Hamiltonian can then be represented as the adjacency matrix of an N-cube acting on the set of vertices identified with bit-strings of length N where $b_i$ and $b_j$ share an edge if and only if $|b_i \oplus b_j| =1$ where $|\cdot|$ represents the Hamming-distance. We could now go and modify this Hamiltonian or the basis it acts on (ie. change it from qubits to qutrits, dits, or whatever else you choose) to cover a broader range of Cayley graphs.