We expand on the work of Hosoya to describe a generalization of continued fractions called “tree expressions.” Each rooted tree will be shown to correspond to a unique tree expression which can be evaluated as a rational number (not necessarily in lowest terms) whose numerator is equal to the Hosoya index of the entire tree and whose denominator is equal to the tree with the root deleted. In the development, we use Z(G) to define a natural candidate ζ(G, v) for a “vertex topological index” which is a value applied to each vertex of a graph, rather than a value assigned to the graph overall. Finally, we generalize the notion of tree expression to “labeled tree expressions” that correspond to labeled trees and show that such expressions can be evaluated as quotients of determinants of matrices that resemble adjacency matrices.

中文翻译：

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顶点拓扑索引和树表达式，连分数的推广

我们扩展了 Hosoya 的工作，以描述称为“树表达式”的连分数的推广。每个有根树将显示对应于一个唯一的树表达式，该表达式可以被评估为一个有理数（不一定是最低项），其分子等于整棵树的 Hosoya 索引，其分母等于具有根删除。在开发中，我们使用 Z(G) 为“顶点拓扑指数”定义一个自然候选 ζ(G, v)，该指数是应用于图的每个顶点的值，而不是分配给整个图的值。最后，我们将树表达式的概念推广到与标记树相对应的“标记树表达式”，并表明这些表达式可以被评估为类似于邻接矩阵的矩阵行列式的商。