The Negative Cycle Vectors of Signed Complete Graphs

TL;DR

This paper investigates the structure of negative cycle vectors in signed complete graphs, proving that the set of all such vectors spans the entire space of possible vectors, revealing a comprehensive understanding of their algebraic properties.

Contribution

It establishes that the affine subspace generated by negative cycle vectors in signed complete graphs is the entire space ^{n-2}, a significant generalization in graph theory.

Findings

The negative cycle vectors span ^{n-2}.

The affine subspace generated by these vectors is all of ^{n-2}.

The result applies to all signed complete graphs.

Abstract

A signed graph is a graph where the edges are assigned labels of either "$+$" or "$-$". The sign of a cycle in the graph is the product of the signs of its edges. We equip each signed complete graph with a vector whose entries are the number of negative $k$-cycles for $k\in\{3,\dots,n\}$. These vectors generate an affine subspace of $\mathbb{R}^{n-2}$. We prove that this subspace is all of $\mathbb{R}^{n-2}$.