Graph Neighborhood

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.

REFERENCES

Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.