Affiliations: Lomonosov Moscow State University, 119991, Moscow, Russia; Moscow Center for Continuous Mathematical Education, 119002, Moscow, Russia; Moscow Institute of Physics and Technology, 141701, Dolgoprudny, Russia, [email protected], [email protected]
Correspondence:
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Address for correspondence: Lomonosov Moscow State University, 119991, Moscow, Russia
Note: [1] The work is financially supported by the grant RSF 16-11-10075.
Abstract: The notion of scrambling index was firstly introduced by Akelbek and Kirkland in 2009. For a primitive digraph D, it is defined as the smallest positive integer k such that for every pair of vertices u and v of D there exist two directed paths of lengths k to a common vertex w. This notion turned out to be useful for several applications, e. g., to estimate eigenvalues of non-negative primitive stochastic matrices. In 2010 Huang and Liu with the background of a memoryless communication system generalized this notion to λ-tuples of vertices and named it λ-th upper scrambling index. These notions can be reformulated in terms of matrix theory. A standard way to generate matrices with the given λ-th upper scrambling index is to apply certain matrix transformations that preserve this index to the existing examples of matrices with known λ-th upper scrambling index. In this paper we completely characterize bijective linear maps preserving λ-th upper scrambling index 1 or 0.
