Note on Indices of Convergence of Digraphs

Abstract

Let \(k(D)\) be the index of convergence of a digraph \(D\) of order \(n \geq 8\). It is proved that if \(D\) is not strong with only minimally strong components and the greatest common divisor of the cycle lengths of \(D\) is at least two, then

\[k(D) \leq \begin{cases}
\frac{1}{2}(n^2 – 8n + 24) & \text{if } n \text{ is even}, \\
\frac{1}{2}(n^2 – 10n + 35) & \text{if } n \text{ is odd}.
\end{cases}\]

The cases of equality are also characterized.