Coefficients of ergodicity and the scrambling index (Q999793)

The scrambling index of a primitive digraph \(D\), denoted by \(k(D)\), is the smallest positive integer \(k\) such that for every pair of vertices \(u\) and \(v\), there is a vertex, say \(w\), such that one can get to \(w\) from \(u\) and \(v\) in \(D\) by directed walks of length \(k\). In the paper under review, the authors investigate the scrambling index for primitive digraphs, and give an upper bound on the scrambling index \(k(D)\) of a primitive digraph \(D\) in terms of the order and the girth of \(D\). They also provide an attainable upper bound on the second largest modulus of eigenvalues of a primitive matrix that make use of the scrambling index.