Given a connected graph G whose vertices are perfectly reliable and whose edges each fail independently with probability q  ∈ [0, 1], the (all‐terminal ) reliability of G is the probability that the resulting subgraph of operational edges contains a spanning tree (this probability is always a polynomial in q ). The location of the roots of reliability polynomials has been well studied, with particular interest in finding those with the largest moduli. In this paper, we will discuss a related problem—among all reliability polynomials of graphs on n vertices, what can we say about the rational roots? We prove that (for n  ≥ 2), the rational roots are −1, − 1/2, − 1/3,…, − 1/(n  − 1), 1. Moreover, we show that for n  ≥ 3, the root of minimum modulus among all graphs of order n is rational, and determine all roots of smallest moduli and the corresponding graphs. Finally, we provide the first nontrivial mathematical property that distinguishes, via reliability, the class of simple graphs (i.e., those without loops and multiple edges) from that of graphs in general.

中文翻译：

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所有终端可靠性的合理根源

给定的连通图G ^其顶点是完全可靠的，并且其边缘各自与概率独立地失败q  ∈[0，1] ，第（所有末端）可靠性的ģ是所得操作边缘的子图包含生成树的概率（这个概率始终是q中的多项式。可靠性多项式的根的位置已得到很好的研究，尤其对寻找模量最大的那些感兴趣。在本文中，我们将讨论一个相关的问题-在n个顶点上图的所有可靠性多项式中，关于有理根，我们能说什么？我们证明了（对于ñ ≥2），有理根为-1，-1/2，-1/3，...，-1 /（n  -1），1。此外，我们显示，对于Ñ  ≥3 ，最小模的次序对所有的曲线图中的根Ñ是合理的，并且确定最小模量所有的根和相应的曲线图。最后，我们提供了第一个非平凡的数学属性，该属性通过可靠性将简单图的类别（即，没有环和多个边的图）与一般图的类别区分开。