"Neighborhood" is a word with many different levels of meaning in mathematics.

One of the most general concepts of a neighborhood of a point x in R^n (also called an epsilon-neighborhood or infinitesimal open set) is the set of points inside an n-ball with center x and radius epsilon>0. A set containing an open neighborhood is also called a neighborhood.

The graph neighborhood of a vertex v in a graph is the set of all the vertices adjacent to v generally including v itself. More generally, the ith neighborhood of v is the set of all vertices that lie at the distance i from v. The subgraph induced by the neighborhood of a graph from vertex v (again, most commonly including v itself) is called the neighborhood graph (or sometimes "ego graph" in more recent literature).

See also

Ball, Distance k-Graph, Graph Neighborhood, Moore Neighborhood, Neighborhood Complex, Open Neighborhood, Open Set, von Neumann Neighborhood Explore this topic in the MathWorld classroom

Portions of this entry contributed by Margherita Barile

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Balakrishnan, R. and Ranganathan, K. "Vertex Cuts and Edge Cuts." §3.1 in A Textbook of Graph Theory. New York: Springer-Verlag, p. 3, 1999.Buckley, F. and Harary, F. Distance in Graphs. Redwood City, CA: Addison-Wesley, p. 167, 1990.

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Cite this as:

Barile, Margherita and Weisstein, Eric W. "Neighborhood." From MathWorld--A Wolfram Web Resource.

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