Nordhaus-Gaddum upper bounds for graph connectivity parameters

Abstract

We examine upper bounds on Nordhaus-Gaddum type problems for parameters related to graph connectivity. Our main result is that for a graph $G$ on $n$ vertices where both $G$ and its complement $G^c$ are connected, then the sum of the algebraic connectivity of $G$ and the algebraic connectivity of $G^c$ cannot exceed $n-3$ (with finitely many exceptions with a small number of vertices). We obtain similar results for the isoperimetric number of a graph, and explore similar Nordhaus-Gaddum type questions for the Cheeger constant and the second eigenvalue of the normalized Laplacian matrix.