TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-07-31

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Handshake lemma: the sum of the vertex degrees is twice the number of edges

Statement

For every finite simple graph $G = (V, E)$ ,

$\sum_{v \in V} \deg_{G} (v) = 2 ∣ E ∣$

as an identity of natural numbers. It remains valid for the null graph, when both sides are $0$ .

Facts & Assumptions

Given: A finite simple graph $G = (V, E)$ .

[L1]

For a relation $R \subseteq X \times Y$ between finite sets, the sum of the row-fibre cardinalities and the sum of the column-fibre cardinalities both equal $∣ R ∣$ (Double counting: $\sum_{x \in X} ∣ R_{x} ∣ = ∣ R ∣ = \sum_{y \in Y} ∣ R^{y} ∣$ for a relation between finite sets).

[F1]

The row and column fibres of a finite incidence relation are finite and are defined by fixing the first or second coordinate (A relation $R \subseteq X \times Y$ between finite sets, its row fibres $R_{x}$ and its column fibres $R^{y}$ ).

[F2]

$\deg_{G} (v)$ is the number of edges incident with $v$ (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

Proof

technique · direct

1.1

Define the incidence relation $R : = {(v, e) \in V \times E : v \in e}$ . Its row fibre at $v$ is the set of edges incident with $v$ , so it has cardinality $\deg_{G} (v)$ by [F2].

givenF1F2

1.2

Every edge $e \in E$ is a two-element subset of $V$ , so its column fibre has cardinality $2$ .

givenF1

2.1

Applying [L1] to $R$ and using steps 1.1 and 1.2 gives $\sum_{v \in V} \deg_{G} (v) = ∣ R ∣ = \sum_{e \in E} 2 = 2 ∣ E ∣$ .

step 1.1step 1.2L1algebra∎

Depends on

Used by

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Cornell Mathematics, Graph Theory notes (standard reference, not scraped)