Suppose you are interested in random walks on an extremely structured graph such as a hypercube graph or a cycle graph. If your graph happens to be the Cayley graph of an abelian group $G$, as in both of the above examples, then it is easy to describe the behavior of random walks on it because the eigenvectors of the adjacency matrix are precisely the characters of $G$ and the eigenvalues depend in a simple way on the characters; in other words, you should learn about the discrete Fourier transform.
Edit: Some elaboration. Let $G$ be a finite abelian group with $|G| = n$. A character of $G$ is a homomorphism $G \to \mathbb{C}$, and it is a basic fact of character theory that the characters form a basis of the space of functions $G \to \mathbb{C}$; this is the discrete Fourier transform. Now let $\mathbf{A}(G)$ be the adjacency matrix of a Cayley graph of $G$ using generators ${ s_1, ... s_k }$. The group $G$ acts on the space of functions $G \to \mathbb{C}$ by sending a function $f : G \to \mathbb{C}$ to $f(gx)$. Call this representation $\rho$; then (and this is the important connnecting observation) one may regard $\mathbf{A}(G)$ as the linear operator $\displaystyle \sum_{i=1}^{k} \rho(s_i)$.
Proposition: Let $\chi_j : G \to \mathbb{C}$ be a character of $G$. Then $\chi_j$ is an eigenvector of $\mathbf{A}(G)$ with eigenvalue $\displaystyle \sum_{i=1}^{k} \chi_j(s_i)$.chi_j(s_i)$, and these are all the eigenvectors.
Proof. Just observe that $\rho(s_i) \chi_j(g) = \chi_j(s_i g) = \chi_j(s_i) \chi_j(g)$. The fact that these exhaust the set of eigenvectors follows from the basic fact cited above.
For example, the cycle graph $C_n$ is the Cayley graph of the cyclic group $\mathbb{Z}/n\mathbb{Z}$ with generators ${ 1, -1 }$, so its eigenvectors are just the rows of the discrete Fourier transform matrix on $\mathbb{Z}/n\mathbb{Z}$ and its eigenvalues are $e^{ \frac{2\pi i k}{n} } + e^{- \frac{2\pi ik}{n} } = 2 \cos \frac{2\pi k}{n}$. (Note that I have implicitly identified the space of functions $G \to \mathbb{C}$ with the free vector space on the elements of $G$ in the usual way.)
