Diagnosability of multiprocessor systems is an important research topic. The system and an interconnection network have an underlying topology, which is usually presented by a graph. In 2012, Peng et al. proposed a metric for fault diagnosis of the graph, namely, the n-neighbor diagnosability that restrains every fault-free node to contain at least n fault-free neighbors. It is difficult to get the n-neighbor diagnosability of the graph from the definition of the n-neighbor diagnosability. Afterwards, some sufficient and necessary conditions are given. It is also difficult to find the n-neighbor diagnosability of the graph from those results. In this paper, we show some new sufficient conditions for the graph to be n-neighbor d-diagnosable under the MM^∗ model. It improves the corresponding result of [Theoretical Computer Science 773 (2019) 107-114]