# The Dimension of the Negative Cycle Vectors of Signed Graphs

## TL;DR

This paper investigates the structure of vectors representing negative cycle counts in signed graphs, establishing that for certain classes of graphs, these vectors span the entire space of possible cycle length distributions.

## Contribution

It introduces a method using matchings with permutability to determine the dimension of the negative cycle vector space, proving it is full-dimensional for key graph classes.

## Key findings

- For complete graphs, the negative cycle vector space is full-dimensional.
- For complete bipartite graphs, the space also spans the entire possible dimension.
- Provides lower bounds on the dimension of the negative cycle vector space using combinatorial techniques.

## Abstract

A "signed graph" is a graph $\Gamma$ where the edges are assigned sign labels, either "$+$" or "$-$". The sign of a cycle is the product of the signs of its edges. Let $\mathrm{SpecC}(\Gamma)$ denote the list of lengths of cycles in $\Gamma$. We equip each signed graph with a vector whose entries are the numbers of negative $k$-cycles for $k\in\mathrm{SpecC}(\Gamma)$. These vectors generate a subspace of $\mathbb R^{\mathrm{SpecC}(\Gamma)}$. Using matchings with a strong permutability property, we provide lower bounds on the dimension of this space; in particular, we show for complete graphs, complete bipartite graphs, and a few other graphs that this space is all of $\mathbb R^{\mathrm{SpecC}(\Gamma)}$.