On certain hyperplane arrangements and colored graphs

TL;DR

This paper establishes a bijective relationship between 3-colored graphs and specific hyperplane arrangements, enabling the computation of their characteristic polynomials through graph centrality properties.

Contribution

It introduces a novel correspondence linking 3-colored graphs with hyperplane arrangements and expresses characteristic polynomials in terms of graph centrality.

Findings

Characteristic polynomial computed for n=2,3

Centrality of graphs corresponds to hyperplane arrangement properties

New combinatorial method for analyzing hyperplane arrangements

Abstract

We exhibit a one-to-one correspondence between $3$-colored graphs and subarrangements of certain hyperplane arrangements denoted $\mathcal J_n$, $n \in \mathbb N$. We define the notion of centrality of $3$-colored graphs which corresponds to the centrality of hyperplane arrangements. Via the correspondence, the characteristic polynomial $\chi_{\mathcal J_n}$ of $\mathcal J_n$ can be expressed in terms of the number of central $3$-colored graphs, and we compute $\chi_{\mathcal J_n}$ for $n = 2, 3$.