2021/01/31 by Carlos E. Valencia, R. R. Villagrán · 1 citation
Mathematics · #math.NT #math.CO #msc:11D72 #msc:11Y50 #msc:11C20 #msc:15B48
paper · pdf · doi:10.1016/j.laa.2022.01.020
published as Linear Algebra and its Applications 640 (2022) 191-208 · 14 pages. Major changes, sections 4 and 5 was deleted. Section 4 is the base of the article "Arithmetical structures on dominated polynomials"
arxiv created 2022/01/12 · arxiv updated 2022/02/17
Arithmetical structures on graphs were first introduced in \citeLorenzini89. Later in \citearithmetical they were further studied in the setting of square non-negative integer matrices. In both cases, necessary and sufficient conditions for the finiteness of the set of arithmetical structures were given. More precisely, an arithmetical structure on a non-negative integer matrix L with zero diagonal is a pair (d,r)∈ ℕ+n× ℕ+n such that (\textrmDiag(d)-L)rt=0t and gcd(r1,…,rn)=1. Thus, arithmetical structures on L are solutions of the polynomial Diophantine equation fL(X):=det(Diag(X)-L)=0. Therefore, it is of interest to ask for an algorithm that compute them. We present an algorithm that computes arithmetical structures on a square integer non-negative matrix L with zero diagonal. In order to do this we introduce a new class of Z-matrices, which we call quasi M-matrices.