# Finite simple undirected graphs

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

This entry covers finite simple undirected graphs. An edge is an unordered set of two distinct vertices, and degree counts distinct neighbors.

mko-graph · definition · v0.10.0



## Mathematical statement

A finite simple undirected graph G=(V,E) consists of a finite vertex set V and an edge set E of two-element subsets of V. There are no loops or multiple edges. N(v)={u∈V:{u,v}∈E}, deg(v)=|N(v)|, and the sum of all vertex degrees is 2|E|.





## Representative formula (exact TeX)

G(V,E)





## Assumptions


- V is finite with distinct vertex labels, and every edge has two distinct endpoints in V.
- An undirected edge ignores endpoint order. Neighbor and degree questions specify v∈V; the empty graph is permitted.





## Symbol scope


- G — finite simple undirected graph
- V — vertex set
- E — unordered edge set





## Intuition, definition, and misconceptions

Treat {u,v} and {v,u} as the same undirected edge. Normalize duplicate entries in a set representation; endpoint ordering and repeated list items do not create a new graph.

A vertex degree is its number of distinct neighbors; an isolated vertex has degree zero. In this simple-graph setting, every edge contributes twice to the total degree sum, giving the handshaking identity.

A graph in graph theory is distinct from a coordinate graph of a function, and a drawing does not determine adjacency. Directed graphs, multigraphs, loop-counting conventions, connectivity, and tree classification are outside this pulse.

Common errors include counting both endpoint orders as two undirected edges, or reading a crossing of drawn lines as an additional vertex.





## Adopted derivation guide

[Finite simple undirected graphs: neighbours, degrees and the handshake formula](https://ocme.evemisslab.com/en/derivations/derivation-mko-graph-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Set](https://ocme.evemisslab.com/en/mko/mko-set/) — Vertices and undirected edges form sets; duplicate representations add no edge.
- [Binary relations and set projections](https://ocme.evemisslab.com/en/mko/mko-relation/) — Adjacency is a finite symmetric relation without self-loops.





## Verification boundary

The finite edge sets and degrees can be recomputed completely; general graph theorems and the handshaking identity have no attached formal Evidence.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Bind SimpleGraph, finite neighbor sets, degree, the empty graph, and the exact scope of the handshaking identity.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/graph.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-graph/index.md)