Let $d$ be a fixed number. By the Cheeger theory and theory of expanders, the second smallest eigenvalue of the Laplacian for a family of $d$-regular graphs is bounded bellow by a positive constant if and only if the isoperimetric number is bounded bellow by a positive constant.
Can this be proved for any family of regular graphs (which degree of regularity of its members can be different from each other)?
There is a normalized version of Cheeger's inequality which uses $\mathcal{L} = \frac{\mathbf{L}}d$ as the Laplacian for $d$-regular graphs. More generally for non-regular graphs, $\mathcal L = \mathbf{I} - \mathbf{D}^{-1/2} \mathbf{A} \mathbf{D}^{1/2}$ where $\mathbf{D}$ is the degree matrix.
As seen in Chung's Spectral Graph Theory (http://www.math.ucsd.edu/~fan/research/revised.html)
$$ \frac{h_G^2}2 \le \lambda_2 \le 2 h_G $$
where $h_G$ is the Cheeger constant and $\lambda_1$ is the second smallest eigenvalue of \mathcal{L}. This is what you are looking for.
