Akelbek, Mahmud and Kirkland, Steve
Coefficients of ergodicity and the scrambling index.
Linear Algebra and its Applications, 430 (4).
For a primitive stochastic matrix S, upper bounds on the second
largest modulus of an eigenvalue of S are very important, because
they determine the asymptotic rate of convergence of the sequence
of powers of the corresponding matrix. In this paper, we introduce
the definition of the scrambling index for a primitive digraph. The
scrambling index of a primitive digraph D 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 directed walks of
length k; it is denoted by k(D).We investigate the scrambling index
for primitive digraphs, and give an upper bound on the scrambling
index of a primitive digraph in terms of the order and the girth of
the digraph. By doing so we provide an attainable upper bound on
the second largest modulus of eigenvalues of a primitive matrix that
make use of the scrambling index.
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