# Finite directed graph

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

Under the loop-free simple convention, a finite digraph has a finite vertex carrier and an ordered-pair edge set without loops or duplicate same-direction edges.

mko-finite-directed-graph · definition · v0.10.0



## Mathematical statement

Each edge is (u,v) with u≠v; (u,v) and (v,u) are different edges. Edges form a set, so same-direction parallel duplicates are forbidden. In-degrees and out-degrees are separate, each summing over the whole graph to |E|. Successive walk steps must follow orientation. Explicit loop boundaries fail this declared model, not every possible digraph convention.





## Representative formula (exact TeX)

D(V,E)





## Assumptions


- Vertices are0 through n−1 with n≥1.
- Edges are canonical lexicographically ordered integer pairs, unique and loop-free.





## Symbol scope


- D — finite directed graph
- E — ordered edge set





## Intuition, definition, and misconceptions

Orientation is part of an edge meaning.

In-degree and out-degree totals each equal the edge count.

Loops and parallel edges need a different declared model.

Do not identify opposite oriented edges or reuse undirected degree conventions.





## Adopted derivation guide

[Orientation, all degrees and a walk audit](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-directed-graph-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Binary relations and set projections](https://ocme.evemisslab.com/en/mko/mko-relation/) — An edge is an ordered pair.
- [Set](https://ocme.evemisslab.com/en/mko/mko-set/) — Vertices and edges are sets.
- [Cardinality comparison of sets](https://ocme.evemisslab.com/en/mko/mko-cardinality-comparison/) — Degree counts have finite sizes.





## Verification boundary

The main slice excludes loops, parallel edges and undirected connectivity conventions.

not_formalized

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





## Next obligation

Add topological ordering and flow networks.





## 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/finite-directed-graph.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-finite-directed-graph/index.md)