A note on comparison theorems for graphs

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. (c) 2021 Elsevier Inc. All rights reserved.