Note on Indices of Convergence of Digraphs

Abstract

Let $k(D)$ be the index of convergence of a digraph $D$ of order $n\ge 8$. 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

$$k(D)\le \{\begin{array}{ll}\frac{1}{2}(n^2–8n+24)& \text{if }n\text{ is even},\\ \frac{1}{2}(n^2–10n+35)& \text{if }n\text{ is odd}.\end{array}$$

The cases of equality are also characterized.