期刊
LINEAR ALGEBRA AND ITS APPLICATIONS
卷 430, 期 4, 页码 1111-1130出版社
ELSEVIER SCIENCE INC
DOI: 10.1016/j.laa.2008.10.007
关键词
Scrambling index; Primitive digraph; Coefficient of ergodicity
资金
- Natural Sciences and Engineering Research Council of Canada [OGP0138251]
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. Published by Elsevier Inc.
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