# Enumeration of Graphs and the Characteristic Polynomial of the Hyperplane Arrangements $\mathcal{J}_n$

## TL;DR

This paper derives a complete formula for the characteristic polynomial of specific hyperplane arrangements by linking them to graph enumeration, particularly bipartite graphs, using generating functions.

## Contribution

It introduces a novel method connecting hyperplane arrangements with graph enumeration, providing an explicit formula for their characteristic polynomial.

## Key findings

- Derived a formula for the characteristic polynomial of $\
- hyperplane arrangements.
- Connected hyperplane arrangements to bipartite graph enumeration using generating functions.

## Abstract

We give a complete formula for the characteristic polynomial of hyperplane arrangements $\mathcal J_n$ consisting of the hyperplanes $x_i+x_j=1$, $x_k=0$, $x_l=1$, $ 1\leq i, j, k, l\leq n$. The formula is obtained by associating hyperplane arrangements with graphs, and then enumerating central graphs via generating functions for the number of bipartite graphs of given order, size and number of connected components.