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

Abstract: 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 graph complement $G^c$, 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 isoperimetric number and Cheeger constant of $G$, respectively.