Which Exterior Powers are Balanced?

A signed graph is a graph whose edges are given ±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 matrix. For a signed graph Σ on n vertices, its exterior kth power, where k = 1, . . . , n−1, is a graph ∧k Σ whose adjacency matrix is given by A( ∧k Σ) = P † ∧A(Σ )P∧, where P∧ is the projector onto the anti-symmetric subspace of the k-fold tensor product space (Cn)⊗k and Σ k is the k-fold Cartesian product of Σ with itself. The exterior power creates a signed graph from any graph, even unsigned. We prove sufficient and necessary conditions so that ∧k Σ is balanced. For k = 1, . . . , n − 2, the condition is that either Σ is a signed path or Σ 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 Σ is positive and each odd cycle in Σ is negative.