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<p>This work presents the main known results on some numerical combinatorial invariants of matrices and graphs, namely, the exponent, scrambling index, and chainable index. The notions of algebraically chainable matrices and algebraically chainable index are introduced, and their properties are studied. In particular, it is shown that the algebraically chainable index is bounded above by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and takes all integer values from 1 up to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Bibliography: 11 titles.</p>

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ALGEBRAICALLY CHAINABLE MATRICES AND THEIR PROPERTIES

  • 期刊论文

  • 发表日期：

    2026年5月25日

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  • A. E. Guterman,
  • E. R. Shafeev

摘要

This work presents the main known results on some numerical combinatorial invariants of matrices and graphs, namely, the exponent, scrambling index, and chainable index. The notions of algebraically chainable matrices and algebraically chainable index are introduced, and their properties are studied. In particular, it is shown that the algebraically chainable index is bounded above by \(n-1\) n - 1 and takes all integer values from 1 up to \(n-1\) n - 1 . Bibliography: 11 titles.