Counting Independent Sets in Graphs of Hyperplane Arrangements

TL;DR

This paper investigates counting independent sets in graphs derived from hyperplane arrangements, revealing that under certain conditions, the count depends only on arrangement coefficients and total vertices, not on specific coefficients.

Contribution

It introduces a method to count independent sets in hyperplane arrangement graphs, showing dependence only on arrangement coefficients and total vertices under specific conditions.

Findings

Number of independent sets depends only on arrangement coefficients and total vertices.

Counting formula is independent of coefficients for central arrangements with multiplicatively independent coefficients.

The results generalize graphical arrangements to hyperplane arrangements with specific parameter conditions.

Abstract

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.