I'm reading Fan Chung's Spectral Graph Theory, and I'm now in chapter 2. There, Chung proves Cheeger's inequality, which is that $2h_G \geq \lambda_1 > h_G^2/2$ for any graph $G$. To me, this inequality raises the question of a physical interpretation of $\lambda_1$ and I'm wondering if such an interpretation exists. Below, I'll explain this terminology and ask my question a little more precisely.
Let $G$ be a graph with vertex set $V$, with $\#V =n$. The Laplacian of $G$ is the $n\times n$ matrix $\mathcal{L}$ whose $(u,v)^{\text{th}}$ entry is $\text{deg}(v)$ if $u=v$, $-(\text{deg}(u)\text{deg}(v))^{-1/2}$ if $u$ and $v$ are adjacent vertices, and $0$ otherwise. The eigenvalues of $\mathcal{L}$ are called the eigenvalues of $G$ and are denoted by $\lambda_0 \leq \lambda_1 \leq \dots \leq \lambda_{n-1}$. It's not too hard to show that $\lambda_0$ is always $0$, so we normally refer to $\lambda_1$ as the minimal eigenvalue of $G$.
If $S\subset V$ (with $S\neq V$) then we define $\overline{S} := V-S$ and $E(S,\overline{S})$ to be the set of edges of $G$ with one vertex in $S$ and the other in $\overline{S}$. Also, let the volume of $S$ be defined as $\text{vol}(S) := \sum_{v \in S} \text{deg}(v)$. We then define $ h_G(S) := \frac{\# E(S,\overline{S})}{\text{min}(\text{vol}(S),\text{vol}(\overline{S}))}.$ The Cheeger constant of $G$ is defined to be $h_G := \min_{S \subset V} h_G(S)$.
It's clear from the definition that the Cheeger constant measures how much the graph "bottlenecks" somewhere (to borrow Wikipedia's apt description). Loosely speaking, if Cheeger's constant is small then there's a small set of edges that you can remove from the graph to disconnect it into two relatively large and relatively connected subgraphs.
Now, it doesn't seem like $\lambda_1$ (which equals the minimal Rayleigh quotient over the space of harmonic eigenfunctions) would have much of a physical interpretation. However, Cheeger's inequality traps $\lambda_1$ fairly close to $h_G$, so it is some loose measure of bottlenecking. Also, there are other little hints in the text I've read so far that $\lambda_1$ might be related to the graph's connectedness. For instance, she proves $\lambda_1 = 0$ if and only if $G$ is disconnected and that $\lambda_1 \geq 1/D\text{vol}(G)$, where $D$ is the length of the maximal-length path in $G$.
So I'm wondering if there is a meaningful physical interpretation of $\lambda_1$ that is more precise than "it's bounded near something that is meaningful."