the Cheeger bound is a bound of the second largest eigenvalue of the transition matrix of a finite-state, discrete-time, reversible stationary Markov chain. It can be seen as a special case of Cheeger inequalities in expander graphs.
Let $X$ be a finite set and let $K(x,y)$ be the transition probability for a reversible Markov chain on $X$. Assume this chain has stationary distribution $\pi$. $$ 1 - 2 \Phi \leq \lambda_2 \leq 1 - \frac{\Phi^2}{2}. $$
where $\Phi$ is the edge expansion of the graph for the random walk corresponding to the MC. Note that a homogeneous DTMC can be seen as a random walk on a directed graph with its vertex set being the state space of the MC.
When $G$ is d-regular, there is a relationship between edge expansion $h(G)$ and the spectral gap $d - \lambda_2$ of $G$. An inequality due to Tanner and independently Alon and Milman[7] states that $$ \frac{1}{2}(d - \lambda_2) \le h(G) \le \sqrt{2d(d - \lambda_2)}\,.$$
I cannot understand why the MC version is a special case of the graph version. Specifically, how can a reversible finite-state MC be seen as a regular graph? In particular, the stationary/reversible/limiting distribution of a reversible finite-state MC is not necessarily uniform.
I suspect the articles miss something? Maybe someone happen to know more general versions of Cheeger's inequality, which can have the two versions here as special cases?
Thanks and regards!