The critical polynomial of a graph

Abstract

Let G be a connected graph on n vertices with adjacency matrix AG. Associated to G is a polynomial dG(x1,…,xn) of degree n in n variables, obtained as the determinant of the matrix MG(x1,…,xn), where MG=Diag(x1,…,xn)−AG. We investigate in this article the set VdG(r) of non-negative values taken by this polynomial when x1,…,xn≥r≥1. We show that VdG(1)=Z≥0. We show that for a large class of graphs one also has VdG(2)=Z≥0. When VdG(2)≠Z≥0, we show that for many graphs VdG(2) is dense in Z≥0. We give numerical evidence that in many cases, the complement of VdG(2) in Z≥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.