Weighted degree

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In graph theory, the weighted degree ^[1] of a vertex $v$ in an edge-weighted graph $G$ is the sum of the weights of the edges incident to $v$, generalizing the degree to edge-weighted graphs.

Definition

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Definition 1 (Weighted degree)

Let $G=(V,E)$ be an ??edge-weighted graph?? with edge weight function $w:E\to \mathbb{R}$, and let $v\in V$ be a vertex. The weighted degree of $v$ is

$${\mathrm{deg}}_w(v):=\sum _{e\in E:v\in e}w(e)$$

Degree is a specialization of weighted degree where every edge weight is restricted to exactly $1$.

Notation

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The weighted degree of $v$ is denoted ${\mathrm{deg}}_w(v)$.

Directed graphs

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

References

1. [1]

   “Glossary of graph theory”, Wikipedia, Available: https://en.wikipedia.org/wiki/Glossary_of_graph_theory, Accessed: 2026-07-31 ↩