arXiv:2611.12951v1 Announce Type: new Abstract: Let $D$ be an Eulerian digraph of order $n$ and minimum degree $\delta$. We prove that the average eccentricity of $D$ satisfies $\avec(D) \leq \frac{3n}{\delta+1}$, and that $\avec(D) \leq \frac{2n}{\delta}+2$ if $D$ is bipartite. We show that both bounds are best possible apart from an additive constant and the $\delta+1$ versus $\delta+2$ gap (resp.\ the $\delta$ versus $\delta+1$ gap in the bipartite case), where the average eccentricity of the extremal examples shows that $\avec = \frac{3n}{\delta+2}-2$ in general and $\avec = \frac{2n}{\delta+1}-\frac{5}{2}$ in the bipartite case for $\delta \geq 2$. We further conjecture that these values are the true maxima up to an additive constant (for $\delta \geq 3$ in the bipartite case).
Continue to arXiv Mathematics →You are being forwarded in 6s — stay on this page