 TOPICS  # Neighborhood

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

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

The graph neighborhood of a vertex in a graph is the set of all the vertices adjacent to generally 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 (again, most commonly including itself) is called the neighborhood graph (or sometimes "ego graph" in more recent literature).

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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## References

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.

Neighborhood

## Cite this as:

Barile, Margherita and Weisstein, Eric W. "Neighborhood." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Neighborhood.html