Difference graphs of S-units

Let $K$ be an algebraic number field, $S$ a finite set of places of $K$ containing all infinite ones, $O_S$ the ring of $S$-integers in $K$ and $O_S^*$ the group of $S$-units. We suppose throughout that $O_S^*$ is infinite and that $S$ contains all infinite places of $K$. For any finite ordered subset $A=\{\alpha_1,\dots,\alpha_k\}$ of $O_S$, we denote by ${\mathcal G}_S(A)$ the graph whose vertices are $\alpha_1,\dots,\alpha_k$ and whose edges are the (unordered) pairs $\{\alpha_i,\alpha_j\}$ for which $$ \alpha_i-\alpha_j\in O_S^*. $$ Such difference graphs have been used in the study of irreducibility of polynomials, systems of linear equations in $S$-units and decomposable form equations. We present results on the finite, simple graphs which can be represented in this way, on the number of different $S$-representations of a graph, and on the graphs which can be represented for every $S$. This is joint research with Kalman Gyory and Lajos Hajdu.