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For a graph $G$ on $n$ vertices with normalized Laplacian eigenvalues $0 =\lambda_1(G) \leq \lambda_2(G) \leq \cdots \leq \lambda_n(G)$ and graphcomplement $G^c$, we prove that  \begin{equation*}  \max\{\lambda_2(G),\lambda_2(G^c)\}\geq \frac{2}{n^2}.  \end{equation*} We do this by way of lower bounding $\max\{i(G), i(G^c)\}$and $\max\{h(G), h(G^c)\}$ where $i(G)$ and $h(G)$ denote the isoperimetricnumber and Cheeger constant of $G$, respectively.

A Nordhaus-Gaddum type problem for the normalized Laplacian spectrum and graph Cheeger constant