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

TL;DR

This paper investigates bounds on Kemeny's constant for a graph and its complement, providing new limits for various graph families and exploring the Nordhaus-Gaddum problem in this context.

Contribution

It establishes bounds on Kemeny's constant for graphs and their complements, advancing understanding of Nordhaus-Gaddum problems for this graph invariant.

Findings

Bounds on the minimum of Kemeny's constants for a graph and its complement.

Bounds on the product of Kemeny's constants for a graph and its complement.

Specific bounds for graphs with maximum degree close to n.

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 $n-\Omega(n)$, then $\min\{\mathcal{K}(G),\mathcal{K}(\overline{G})\}$ is at most $O(n)$.