Algebraically chainable matrices and their properties

A. E. Guterman^a, E. R. Shafeev<sup>bc

a</sup> Bar-Ilan University, Ramat Gan
^b Lomonosov Moscow State University
^c Moscow Center for Fundamental and Applied Mathematics

Abstract: This work presents the main known results concerning numerical combinatorial invariants of matrices and graphs: 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 $n-1$ and can equal every integer from $1$ up to $n-1$.

Key words and phrases: chainable matrices, chainable index, nonnegative matrices.

UDC: 512.643

Received: 01.11.2025