Weighted in-degree

In graph theory, the weighted in-degree ^[1] of a vertex $v$ in an edge-weighted directed graph $G$ with weight function $w:E\to \mathbb{R}$ is the sum of the weights of the directed edges with $v$ as head, generalizing the in-degree to edge-weighted graphs. Its counterpart is the weighted out-degree.

Definition

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Definition 1 (Weighted in-degree)

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

$${\mathrm{deg}}_w^{-}(v)=\sum _{u:(u,v)\in E}w(u,v)$$

Notation

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

References

1. [1]

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