Generalization or Improvement of Cheeger inequality on Graphs Let $G=(V,E)$ be an undirected graph with vertex set $V$ and edge set $E$. Let $A$ denote the adjacency matrix of $G$ and $D$ denote the diagonal matrix such that $D_{i,i}$ equals to the degree $d_i$ of vertex $i$. Then the Laplacian of $G$ is defined to be $L:=I-D^{-1/2}AD^{-1/2}$, which has $n$ nonnegative real eigenvalues, say, $0=\lambda_1\leq \lambda_2\leq \cdots\leq \lambda_n$. 
Now for any vertex subset $S\subseteq V$, let $e(S,\bar{S})$ denote the number of edges between $S$ and its complement $\bar{S}$. Let $vol(S):=\sum_{v\in S}deg_v$ be the sum of degrees of vertices in $S$. Then the conductance (or Cheeger constant) $h_G$ of graph $G$ is defined to be 
$$h_G:=\min_{S\subseteq V}\frac{e(S,\bar{S})}{\min \{vol(S), vol({\bar{S}})\}}.$$ 
The Cheeger inequality relates the second smallest eigenvalue $\lambda_2$ of $L$ to the conductance $h_G$ as follows:
$$2h_G\geq\lambda_2\geq \frac{h_G^2}{2}.$$
The above inequality is known to be tight.  For example, the left side of the inequlity is tight on the $d$-dimensional cube and the right side is tight on the $n$-vertex cycle.  Thus, we do not hope to improve the inequality that applies to every graph.
My question is: 

Is there any result which gives that for some special class of graphs, the Cheeger inequality has an improved form, say, $2h_G^{1.2}\geq\lambda_2\geq \frac{h_G^{1.5}}{2}$ for any $G\in\mathcal{C}$, where $\mathcal{C}$ is a set of graphs. 

Ideally, we would hope that there is a nice  tradeoff between the generality of the class $\mathcal{C}$ and the tightness of the Cheeger-type inequality. In another word, $\mathcal{C}$ contains a wide class of graphs and the upper bound is close to the lower bound.
 A: There is a lot known on the relations between the Cheeger constant $h_{G}$ and $\lambda_{2}$ and more general on the  whole spectrum of the normalized Laplacian for the following families of graphs:


*

*Random Trees (branching processes) Lyons and Peres book is an excellent reference.

*Random Geometric graphs, Penrose's book is an good reference.

*As previously mentioned the $G_{n,p}$ random model of Erdos and Renyi.

*Random regular graphs where you can find information in the fundamentals papers of McKay.

*Regular planar tessellations of hyperbolic space.
I hope it helps!
A: One place to look might be random graphs $G(n,p)$.  This might either give you a wide class of graphs for which an improvement holds, or else show you a limit to what you might hope for.
For Cheeger constant see:
MR2371054 (2008j:05316) Benjamini, Itai; Haber, Simi; Krivelevich, Michael; Lubetzky, Eyal The isoperimetric constant of the random graph process. Random Structures Algorithms 32 (2008), no. 1, 101–114.
For spectral gap see:
MR0637828 (83e:15010) Füredi, Z.; Komlós, J. The eigenvalues of random symmetric matrices.
Combinatorica 1 (1981), no. 3, 233–241.
A: There other classes of graphs such as generalizations of the $G(n,p)$ model with a given degree distribution. See for example Chung and Vu's paper on the spectrum of these graphs. In particular, they analyzed the case when the graph has a power law degree distribution.
