In the present note we are concerned with the study of curvature-based comparison theorems on graphs related to main stochastic properties, such as the Feller property and stochastic completeness. We show that, under our main hypothesis, whilst previous results concerning stochastic properties are improved, it is not possible to obtain comparison theorems concerning volume growth. Finally, we prove an analogue of the Bishop-Gromov's relative volume comparison theorem and present a series of examples related to various possible notions of curvature.

中文翻译：

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关于图的比较定理的注记

在本说明中，我们关注与主要随机属性（例如Feller属性和随机完整性）相关的图上基于曲率的比较定理的研究。我们表明，在我们的主要假设下，虽然先前关于随机性质的结果得到了改善，但不可能获得有关体积增长的比较定理。最后，我们证明了Bishop-Gromov的相对体积比较定理的类似物，并给出了一系列与各种可能的曲率概念有关的示例。