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arXiv · 2309.05171

Bounds on Kemeny's constant of a graph and the Nordhaus-Gaddum problem

Sooyeong Kim·Neal Madras·Ada Chan·Mark Kempton·Stephen Kirkland·Adam Knudson
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Abstract

We study Nordhaus-Gaddum problems for Kemeny's constant $\mathcal{K}(G)$ of a connected graph $G$. We prove bounds on $\min\{\mathcal{K}(G),\mathcal{K}(\overline{G})\}$ and the product $\mathcal{K}(G)\mathcal{K}(\overline{G})$ for various families of graphs. In particular, we show that if the maximum degree of a graph $G$ on $n$ vertices is $n-O(1)$ or $n-Ω(n)$, then $\min\{\mathcal{K}(G),\mathcal{K}(\overline{G})\}$ is at most $O(n)$.

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Sooyeong Kim, Neal Madras, Ada Chan, Mark Kempton, Stephen Kirkland, Adam Knudson. 2023-09-11. Bounds on Kemeny's constant of a graph and the Nordhaus-Gaddum problem. https://arxiv.org/abs/2309.05171

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