Abstract

We will study various quantities related to connectivity of a graph. To this end, we look at Nordhaus-Gaddum type problems, which are problems where the same quantity is studied for a graph $G$ and its complement $G^c$ at the same time. 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. We also discuss some related Nordhaus-Gaddum questions.

Degree

College and Department

Computational, Mathematical, and Physical Sciences; Mathematics

Rights

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