Which Exterior Powers are Balanced?

Abstract

A signed graph is a graph whose edges are given (-1,+1) weights. In such a graph, the sign of a cycle is the product of the signs of its edges. A signed graph is called balanced if its adjacency matrix is similar to the adjacency matrix of an unsigned graph via conjugation by a diagonal (-1,+1) matrix. For a signed graph $\Sigma$ on n vertices, its exterior k-th power, where k=1,..,n-1, is a graph $\bigwedge^{k} \Sigma$ whose adjacency matrix is given by \[ A({$\bigwedge^{k} {\Sigma}$}) = P^{\dagger} A(\Sigma^{\Box k}) P, \] where P is the projector onto the anti-symmetric subspace of the k-fold tensor product space $(\mathbb{C}^{n})^{\otimes k}$ and $\Sigma^{\Box k}$ is the k-fold Cartesian product of $\Sigma$ with itself. The exterior power creates a signed graph from any graph, even unsigned. We prove sufficient and necessary conditions so that $\bigwedge^{k} \Sigma$ is balanced. For k=1,..,n-2, the condition is that either $\Sigma$ is a signed path or $\Sigma$ is a signed cycle that is balanced for odd k or is unbalanced for even k; for k=n-1, the condition is that each even cycle in $\Sigma$ is positive and each odd cycle in $\Sigma$ is negative.