The Negative Cycle Vectors of Signed Complete Graphs

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 $\mathbb{R}^{n-2}$.