The critical polynomial of a graph

Let $G$ be a connected graph on $n$ vertices with adjacency matrix $A_G$. Associated to $G$ is a polynomial $d_G(x_1,\dots, x_n)$ of degree $n$ in $n$ variables, obtained as the determinant of the matrix $M_G(x_1,\dots,x_n)$, where $M_G={\rm Diag}(x_1,\dots,x_n)-A_G$. We investigate in this article the set $V_{d_G}(r)$ of non-negative values taken by this polynomial when $x_1, \dots, x_n \geq r \geq 1$. We show that $V_{d_G}(1) = {\mathbb Z}_{\geq 0}$. We show that for a large class of graphs one also has $V_{d_G}(2) = {\mathbb Z}_{\geq 0}$. When $V_{d_G}(2) \neq {\mathbb Z}_{\geq 0}$, we show that for many graphs $V_{d_G}(2)$ is dense in ${\mathbb Z}_{\geq 0}$. We give numerical evidence that in many cases, the complement of $V_{d_G}(2)$ in ${\mathbb Z}_{\geq 0}$ might in fact be finite. As a byproduct of our results, we show that every graph can be endowed with an arithmetical structure whose associated group is trivial.