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The graph neighborhood of a vertex in a graph is the set of all the vertices adjacent to including itself. More generally, the th neighborhood of is the set of all vertices that lie at the distance from .
The subgraph induced by the neighborhood of a graph from vertex is called the neighborhood graph.
Note that while "graph neighborhood" generally includes vertices adjacent to together with the vertex itself, the term "graph neighbors" generally means vertices adjacent to excluding itself (e.g., Brouwer et al. 1989), so some care is needed when encountering these terms.
Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.
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