计数单圈的霍普夫代数
An Hopf algebra for counting simple cycles
摘要：
简单圈，也称为自回避多边形，是图上不允许访问任何顶点超过一次的圈。本文给出了一个精确的公式，用于计算任意有向图上任意长度的简单圈，该有向图涉及其诱导子图的和。这个结果源于一个霍普夫代数，我们明确地构造，并提供了进一步的手段计数简单的循环。最后，我们得到了一个更一般的定理，断言任何李幂等元可以用来枚举简单的循环。
英文摘要：
Simple cycles, also known as self-avoiding polygons, are cycles on graphs which are not allowed to visit any vertex more than once. We present an exact formula for enumerating the simple cycles of any length on any directed graph involving a sum over its induced subgraphs. This result stems from an Hopf algebra, which we construct explicitly, and which provides further means of counting simple cycles. Finally, we obtain a more general theorem asserting that any Lie idempotent can be used to enumerate simple cycles.
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