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arXiv · 2606.12751

Nordhaus-Gaddum upper bounds for graph connectivity parameters

Mark Kempton·Xavier Zaitzeff·Sibi Muthuprakash
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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.

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Mark Kempton, Xavier Zaitzeff, Sibi Muthuprakash. 2026-06-10. Nordhaus-Gaddum upper bounds for graph connectivity parameters. https://arxiv.org/abs/2606.12751

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