Neighbourhood

Definition

Let $G$ be a graph with vertex set $V$, and let $v \in V$. The neighbourhood of $v$ is the set $$N(v) = \{ u \in V : u \sim v \}$$ of the vertices adjacent to $v$. The elements of $N(v)$ are the neighbours of $v$.

Remarks

This is the open neighbourhood. It does not contain $v$, because no vertex is adjacent to itself. The closed neighbourhood is $N[v] = N(v) \cup \{v\}$. The degree of $v$ is $|N(v)|$ when $N(v)$ is finite. In the cycle with vertices $1, 2, 3, 4, 5$ in this order, $N(1) = \{2, 5\}$.

Explanations

L1 human

The neighbours of a vertex are the vertices joined to it by an edge. In a friendship graph, the neighbourhood of a person is the set of that person's friends.