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Which Exterior Powers are Balanced? | Devlin Mallory
; Abigail Raz
; Christino Tamon
; Thomas Zaslavsky
; | Date: |
6 Jan 2013 | 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 $igwedge^{k} Sigma$ whose adjacency matrix is given by [
A({$igwedge^{k} {Sigma}$}) = 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 $igwedge^{k} Sigma$ 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. | Source: | arXiv, 1301.0973 | Services: | Forum | Review | PDF | Favorites |
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