The Scrambling Indices of Primitive Digraphs with Exactly Two Cycles

Abstract

In 2009, Akelbek and Kirkland introduced a useful parameter called the scrambling index of a primitive digraph \(D\), which is the smallest positive integer \(k\) such that for every pair of vertices \(u\) and \(v\), there is a vertex \(w\) such that we can get to \(w\) from \(u\) and \(v\) in \(D\) by walks of length \(k\). In this paper, we study and obtain the scrambling indices of all primitive digraphs with exactly two cycles.