---
tag: 0T0F6CCS9W
kind: definition
name: Degree of a vertex
msc: [05C07]
defines: 0KS0KRDV4Y
uses: [01T3TMGTTE]
sources:
  - url: https://en.wikipedia.org/wiki/Degree_(graph_theory)
    locator: lead section
---

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] and let $v$ be a vertex of $G$ with a finite [[neighbourhood|01T3TMGTTE]] $N(v)$.
The degree of $v$ is the number
$$\deg(v) = |N(v)|$$
of vertices adjacent to $v$. It is also the number of edges that contain $v$.

## Remarks

A vertex of degree $0$ is isolated, and a vertex of degree $1$ is a leaf.
A graph is locally finite when each vertex has finite degree. Each finite graph is locally finite.
In a [[multigraph|01M5T0KDG7]], the degree counts edges with multiplicity, and a loop counts twice.

## Explanations

### L1 human

The degree of a vertex is the number of edges at that vertex.
In a friendship graph it is the number of friends of a person.