Which Exterior Powers are Balanced?

TL;DR

This paper investigates the conditions under which the exterior powers of signed graphs are balanced, providing necessary and sufficient criteria based on the structure and cycle signs of the original graph.

Contribution

It introduces a novel framework for analyzing the balance of exterior powers of signed graphs and establishes precise conditions for their balance.

Findings

Balanced exterior powers depend on the original graph's cycle signs.

For k=1,..,n-2, balance depends on paths or cycles with specific sign patterns.

For k=n-1, all even cycles are positive and odd cycles are negative.

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.