ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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$\operatorname{ind}_{K_2}(G)=2|E(G)|$ under the induced-embedding convention

Statement

For every finite simple graph $G$ ,

$\operatorname{ind}_{K_2}(G)=2|E(G)|.$

Facts & Assumptions

Given: A finite simple graph $G$ .

[F1]

$\operatorname{ind}_{K_2}(G)$ counts induced embeddings of the two labelled vertices of $K_2$ into $G$ (The induced-embedding count $\operatorname{ind}_H(G)$ , Induced embeddings and induced copies of a graph).

[L1]

The sum of vertex degrees is $2|E(G)|$ (Handshake lemma: the sum of the vertex degrees is twice the number of edges).

Verification

technique · direct

1.1

An induced embedding of $K_2$ is exactly an ordered adjacent pair $(u,v)$ of vertices of $G$ .

2.1

Counting ordered adjacent pairs by their first vertex gives $\sum_{u\in V(G)}\deg(u)$ .

step 1.1

3.1

By the handshake lemma this is $2|E(G)|$ , proving the formula.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 31 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Swastik Kopparty, Local Structure: Subgraph Counts I (standard reference, not scraped)

ind⁡K2(G)=2∣E(G)∣\operatorname{ind}_{K_2}(G)=2|E(G)|indK2​​(G)=2∣E(G)∣ under the induced-embedding convention