Let $X$ be an infinite, locally finite, connected, quasi-transitive graph without loops or multiple edges. A graph height function on $X$ is a map adapted to the graph structure, assigning to every vertex an integer, called height. Bridges are self-avoiding walks such that heights of interior vertices are bounded by the heights of the start- and end-vertex. The number of self-avoiding walks and the number of bridges of length $n$ starting at a vertex $o$ of $X$ grow exponentially in $n$ and the bases of these growth rates are called connective constant and bridge constant, respectively. We show that for any graph height function $h$ the connective constant of the graph is equal to the maximum of the two bridge constants given by increasing and decreasing bridges with respect to $h$. As a concrete example, we apply this result to calculate the connective constant of the Grandparent graph.

中文翻译：

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自回避行走的一般桥接定理

令 $X$ 是一个无限的、局部有限的、连通的、准传递图，没有循环或多条边。$X$ 上的图高度函数是适应图结构的映射，为每个顶点分配一个整数，称为高度。桥梁是自我回避的步行，这样内部顶点的高度受到起始顶点和结束顶点的高度的限制。自回避行走的数量和长度为 $n$ 的桥梁的数量从 $X$ 的顶点 $o$ 开始以 $n$ 呈指数增长，并且这些增长率的基数分别称为连接常数和桥梁常数. 我们表明，对于任何图高度函数 $h$，图的连接常数等于通过相对于 $h$ 增加和减少桥接给出的两个桥接常数中的最大值。作为一个具体的例子，