On the Second Largest Scrambling Index of Primitive Matrices

Abstract

The scrambling index of an \(n \times n\) primitive matrix \(A\) is the smallest positive integer \(k\) such that \(A^k(A^T)^k > 0\), where \(A^T\) denotes the transpose of \(A\). In 2009, M. Akelbek and S. Kirkland gave an upper bound on the scrambling index of an \(n \times n\) primitive matrix \(M\) in terms of its order \(n\), and they also characterized the primitive matrices that achieve the upper bound. In this paper, we characterize primitive matrices which achieve the second largest scrambling index in terms of its order. Meanwhile, we show that there exists a gap in the scrambling index set of primitive matrices.