9.1: Vertex Degrees
- Page ID
- 48344
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The in-degree of a vertex in a digraph is the number of arrows coming into it, and similarly its out-degree is the number of arrows out of it. More precisely,
Definition \(\PageIndex{1}\)
If \(G\) is a digraph and \(v \in V(G)\), then
\[\nonumber \text{indeg}(v) ::= \mid \{ e \in E(G) \mid \text{head}(e) = v\} \mid\]
\[\nonumber \text{outdeg}(v) ::= \mid \{ e \in E(G) \mid \text{tail}(e) = v\} \mid\]
An immediate consequence of this definition is
Lemma 9.1.2.
\[\nonumber \sum_{v \in V(G)} \text{indeg}(v) = \sum_{v \in V(G)} \text{outdeg}(v)\]
Proof. Both sums are equal to \(\mid E(G) \mid\). \(\quad \blacksquare\)
