An outerplanar graph is a graph that can be drawn on the plane without crossing edges so that all vertices are on the infinite face. Most organic compounds have outerplanar graph structures. The number of matchings in the skeleton graphs of organic compounds is known as the topological index Z, introduced in the early 70s to investigate correlation between molecular structures and physical properties. This paper provides a simple formula that expresses the number of matchings, and more generally the matching polynomial, of an outerplanar graph by the Pfaffian of a certain skew-symmetrc matrix.

中文翻译：

---

匹配外平面图的多项式的Pfaffian公式

外平面图是可以在平面上绘制而没有相交边的图，因此所有顶点都在无穷大的面上。大多数有机化合物具有外平面图结构。有机化合物的骨架图中的匹配数称为拓扑指数Z，它是70年代初引入的，用于研究分子结构与物理性质之间的相关性。本文提供了一个简单的公式，该公式通过某个偏对称矩阵的Pfaffian表示外平面图的匹配次数，更普遍的是匹配多项式。