Counting points of slope varieties over finite fields

TL;DR

This paper establishes a bijection between the solutions of the slope variety of complete graphs over _2 and complement-reducible graphs, linking algebraic geometry with graph theory.

Contribution

It introduces a novel connection between the zeroes of slope varieties over _2 and cographs, providing a new perspective on graph representations.

Findings

Bijection between slope variety zeroes and cographs

Characterization of slope varieties over _2

Insights into algebraic and combinatorial graph properties

Abstract

The slope variety of a graph is an algebraic set whose points correspond to drawings of a graph. A complement-reducible graph (or cograph) is a graph without an induced four-vertex path. We construct a bijection between the zeroes of the slope variety of the complete graph on $n$ vertices over $\mathbb{F}_2$, and the complement-reducible graphs on $n$ vertices.