哪些外部力量是平衡的？
Which Exterior Powers are Balanced?
摘要：
符号图是指边被赋予（-1，+1）权的图。在这样的图中，圈的符号是其边的符号的乘积。一个有号图称为平衡图，如果它的邻接矩阵通过对角（-1，+1）矩阵的共轭与一个无号图的邻接矩阵相似.对于n个顶点的符号图$Σ$，它的外k次幂，其中k=1，.，n-1，是图$\bigwedge^{k} Σ$，其邻接矩阵由\[ A({$\bigwedge^{k} Σ$}) = P^{\dagger} A(Σ^{\Box k}) P, \]给出，其中P是到k重张量积空间$(\mathbb{C}^{n})^{\otimes k}$的反对称子空间上的投影，而$Σ^{\Box k}$是$Σ$与其自身的k重笛卡儿积.外部幂从任何图创建一个有符号图，甚至是无符号图。我们证明充分必要条件，使$\bigwedge^{k} Σ$是平衡的。对于k=1，..，n-2，条件是$Σ$是有符号路径，或者$Σ$是有符号循环，其对于奇数k是平衡的，或者对于偶数k是不平衡的;对于k=n-1，条件是$Σ$中的每个偶数循环是正的，并且$Σ$中的每个奇数循环是负的。
英文摘要：
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 $Σ$ on n vertices, its exterior k-th power, where k=1,..,n-1, is a graph $\bigwedge^{k} Σ$ whose adjacency matrix is given by \[ A({$\bigwedge^{k} Σ$}) = P^{\dagger} A(Σ^{\Box k}) P, \] where P is the projector onto the anti-symmetric subspace of the k-fold tensor product space $(\mathbb{C}^{n})^{\otimes k}$ and $Σ^{\Box k}$ 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 $\bigwedge^{k} Σ$ 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.
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