Counting Independent Sets in Graphs of Hyperplane Arrangements

In this paper, we count the number of independent sets of a type of graph $G(\mathcal{A},q)$ associated to some hyperplane arrangement $\mathcal{A}$, which is a generalization of the construction of graphical arrangements. We show that when the parameters of $\mathcal{A}$ satisfy certain conditions, the number of independent sets of the disjoint union $G(\mathcal{A},q_1)\cup \cdots \cup G(\mathcal{A},q_s)$ depends only on the coefficients of $\mathcal{A}$ and the total number of vertices ${\sum }_i{q}_i$ when $q_i$’s are powers of large enough prime numbers. In addition it is independent of the coefficients as long as $\mathcal{A}$ is central and the coefficients are multiplicatively independent.