定向图上的圈/上圈斜投影
Cycle/cocycle oblique projections on oriented graphs
摘要：
众所周知，有向图的边向量空间可以用圈和上圈（别名割或键）来分解，并且圈和上圈空间的基可以通过向任意选择的生成树添加和移除边来生成。在本文中，我们表明，边缘向量空间也可以分解的循环和生成边的上循环（称为cochords），或上循环和生成边的循环（称为和弦）。从这个观察遵循的斜互补投影算子的建设。我们采用这种代数结构证明了几个性质的非加权Kirchhoff-Symanzik矩阵，编码之间的相互叠加的循环和上循环。特别地，我们证明了平面图的对偶矩阵具有相同的谱（直到多重性）。我们简要地评论如何建设提供了一个完善的形式化基尔霍夫的网格分析的电路，最近已被应用到通用热力学网络。
英文摘要：
It is well known that the edge vector space of an oriented graph can be decomposed in terms of cycles and cocycles (alias cuts, or bonds), and that a basis for the cycle and the cocycle spaces can be generated by adding and removing edges to an arbitrarily chosen spanning tree. In this paper we show that the edge vector space can also be decomposed in terms of cycles and the generating edges of cocycles (called cochords), or of cocycles and the generating edges of cycles (called chords). From this observation follows a construction in terms of oblique complementary projection operators. We employ this algebraic construction to prove several properties of unweighted Kirchhoff-Symanzik matrices, encoding the mutual superposition between cycles and cocycles. In particular, we prove that dual matrices of planar graphs have the same spectrum (up to multiplicities). We briefly comment on how this construction provides a refined formalization of Kirchhoff's mesh analysis of electrical circuits, which has lately been applied to generic thermodynamic networks.
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