Algorithmic aspects of arithmetical structures

Arithmetical structures on graphs were first mentioned in \cite{Lorenzini89} by D. Lorenzini. Later in \cite{arithmetical} they were further studied on square non-negative integer matrices. In both cases, necessary and sufficient conditions for the finiteness of the set of arithmetical structures were given. Therefore, it is natural to ask for an algorithm that compute them. This article is divided in two parts. In the first part 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. We recall that arithmetical structures on a matrix $L$ are solutions of the polynomial Diophantine equation \[ f_L(X):=\det(\text{Diag}(X)-L)=0. \] In the second part, the ideas developed to solve the problem over matrices are generalized to a wider class of polynomials, which we call dominated. In particular the concept of arithmetical structure is generalized on this new setting. All this leads to an algorithm that computes arithmetical structures of dominated polynomials. Moreover, we show that any other integer solution of a dominated polynomial is bounded by a finite set and we explore further methods to obtain them.