"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).
