Degree

In graph theory, the degree ^[1] of a vertex of a graph is the number of edges that are incident to the vertex.

Definition

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Definition 1 (Degree)

Let $G=(V,E)$ be a graph and $v\in V$ a vertex. The degree of $v$ is the number of edges in $E$ incident to $v$.

Definition 2 (Maximum degree)

Let $G=(V,E)$ be a graph. The maximum degree of $G$ is

$$\Delta (G)=\underset{v\in V}{\max }\mathrm{deg}(v)$$

Definition 3 (Minimum degree)

The minimum degree of $G$ is

$$\delta (G)=\underset{v\in V}{\min }\mathrm{deg}(v)$$

Notation

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The degree of $v$ is denoted ${\mathrm{deg}}_G(v)$, or simply $\mathrm{deg}(v)$ when the graph is unambiguous. The maximum degree of $G$ is denoted $\Delta (G)$, and the minimum degree is denoted $\delta (G)$.

Properties

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The sum of the degrees of all vertices of $G$ equals twice the number of edges, known as the handshaking lemma.

Directed graphs

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In a directed graph, the degree of a vertex $v$ splits into two directed notions: the in-degree ${\mathrm{deg}}^{-}(v)$, the number of directed edges with $v$ as head, and the out-degree ${\mathrm{deg}}^{+}(v)$, the number of directed edges with $v$ as tail.

References

1. [1]

   “Degree (graph theory)”, Wikipedia, Available: https://en.wikipedia.org/wiki/Degree_(graph_theory), Accessed: 2026-07-31 ↩